COMMON PART
Definition D1. System is every multitude that can be
presented by two elements: first – energetic sours, second – energetic consumer
or working object that consumes the energy for producing some product.
The product is preliminary selected or one
way, or periodically by desire of a human person. The product can by a
substance or an energy field.
Definition D2. Invariant is every system that produce a
product completely replaying in every time of the selected.
Figure 1.
On the figure 1. is imagined the schema of
a invariant system. About the signatures:
-
O is the producing object;
-
ES – the energetic source;
-
ER – the energetic resources;
-
x0 – the selected (inlet) product;
-
E – the quantity energy wanted from the inlet product;
-
x – the produced (outlet) product.
In this case the couple constant elements
ES and O aided with the variables ER, x0 and x makes about D1 a
system. The system is invariant about D2, because the condition
(1) x = x0
is unique necessary for existence of
invariability in the system.
In the reality that sort of systems don’t
exist, but exist the trend to create systems working with a minimum error
ε
(2) ε = lim(x0
– x) = 0, t-->∞
(for example: radar systems, by that the
turn of the antenna is followed by electronic ray over controlling screen,
generators of an electric system with equal numbers of poles turned in mode of
synchronisme etc.)
CRITERIA FOR EXISTENCE OF INVARIABILITY
About the necessary condition (1), by that
is defined the existence of invariability in a system, is clear when a system
is invariant. But what must be the parameters of the system for keeping the
condition (1)?
The answer of the question is in the
sufficient conditions for existence of invariability. Which are these
conditions?
Definition D3. It is differentiate three aspects of
energy:
-
En = En(x0) – really necessary for the
object energy;
-
Ek = Ek(x) – really consumed from the object
energy;
-
Es – really handed from the sours for the object energy.
Definition D4. Direction of the energy is the direction
of hers movement from the sours to the object (consumer).
Axiom A1. The observer of the energy movement is
“set foot” over the portable line between the sours and the object (consumer).
Axiom A2. The point for observing is accepted for an
energy balanced (neutral). In this point is not generated or consumed energy.
Axiom A3. Positive is the energy from the observer,
negative – toward him.
Axiom A4. The algebra sign + is for positive
direction of the energy, the sign – is for the negative.
Axiom A5. Over the multitude of energetic values are
valid the operations adding and subtracting that commutates. All other algebra
operations over this multitude lead to result outside the energy multitude.
Axiom A6. The variables x and x0 are in
the common case composite functions of the time t, who is measured on the calendar.
Axiom A7. The energetic component En is
controlling and she has only mathematical, not physical significance.
Theorem T1. The
multitude from energetic values forms an additive group.
Proof: According
to the axiom A4 every element from the energetic multitude has its opposite,
and according axiom A2 in some multitude exists single element – the zero (the
balance state of the system). Axiom A5 allows only additives algebra operations
that commutates.
From everything follows that the axioms
A2, A4 and A5 are axioms of an additive group and the theorem is proved.
Theorem T2. Functions xo(t), õ(t), Ån(xo), Ås(xo)
and Ek(x)
are in state of isomorphisme.
Proof: According to the axiom A6 xo
and x are
composite functions of the time t. If it’s true, they are only single values,
because nothing can not be in two places in one single moment of the time. They
are also opposites, because the time t is measured on the calendar. Then are
true the diagrams:
(3) t-->xo-->Ån(x0), t-->xo-->Ås(xo)
è t-->x-->Åk(x),
and it means that and the functions Ån(xo), Ås(xo) and Ek(x)are composite functions of
the time t:
(4) Ån = Ån(t), Ås = Ås(t) è Åk = Åk(t)
It means that the energetic values can be
only single in one single moment of the time. They are valid the diagrams:
(5) Ån(xo)-->t, Ås(xo)-->t
è Ån(xo)-->t,
and the theorem is proved.
Theorem T3. The system on the figure 1. is invariant
then and only then, when
(6)
Ån(xo) = Ån(xo)
Proof: For existence of invariability it’s
necessary the condition (1). From a other side, according T2 the functions Ån(xo) and Ån(xo) are in state of isomorphisme. Then are
valid the commutating diagram:
(7)
t-->xo-->Ån(xo)-->Åk(x)-->õ-->t
If the diagram (7) is true, the condition
(1) leads to the condition (6) or to the diagram:
(8)
xo = x-->Ån(xo)
= Åk(x),
and the theorem is proved.
Theorem T4. The multitude of
energetic functions about the figure 1. form towards the time t symmetric
towards the abscissa axis t (in plane Cartesian coordinates).
Proof: According
the theorem T1 the multitude of energetic functions forms an
additive group. From it follows that every energetic value is element who can
added with a from the three energetic component of the system - Ån, Åk and Ås.
According the axiom A3 and A4 Ån and Ås have negative
values and Åk – positive. Including and the neutral
(observing) point from the axiom A2 in our think, we can make the schema of the
invariant system in the air:
Figure 2.
From the figure 2. with common abscissa
axis we can design the graphs of Ån, Åk and Ås. According axiom A8 Ån
has only
mathematical or controlling sense. She announce of the energetic source which
energy is necessary of the object to be in mode of invariability. It is clear
the is necessary the condition:
(9)
Ån(t) = Ås(t)
Because Ån(t) and Ås(t) have equal – negative – algebra sign
toward the neutral point, their graphs (about figure 3) will be under the
abscissa axis and will be coincide.
Figure 3.
From a other side the graph Åk(t) will be over the abscissa axis,
because Åk is positive.
The theorem is proved.
MECHANICAL MODEL OF THE CONCEPT ENERGY
The sufficiently for existence of
invariability condition (6) for the system on the figure 1. requires to precise
physical the concept energy, for can to define the dynamical properties of the
invariant
Figure 3.
systems. With other words, it is necessary
to know all permissibly possible variants in the time of the energetic
components En, Es and Ek.
On the concept energy we shell see like a
unified measure of the different forms of movement of the matter. If the matter
exists only in movement, the energy will be an universal measure of existence
of the matter. Then every air of movement or physical phenomenon will possess a
energetic measure and all airs of energy (mechanical, heating, electrical,
chemical, nuclear etc.) have his logics to be measured with one and same
measure. She is named Joule, is marked with J and is internationally perceived
in the measuring system SI. All standards in the world are obligatory into
consideration with this system and our theory also will be.
As soon as all physical phenomenon have
unified measure, that sort of measure will have every production process and,
therefore, every production system, also and the invariant. Now it is clear why
the definition D1 for a system and D2 – for an invariant system are defines for
energetic system.
The elementary irreducible for observing
movement, so shows the human experiment, is the mechanical. Exactly for it is
accepted it to be a common viewpoint for every air (mechanical, heating,
electrical, chemical, nuclear etc.) movement.
The elementary irreducible for observing
mechanical movement is the movement of a material point, because the point is
the elementary irreducible concept in the space for the human imagination.
Exactly therefore the irreducible elementary from mathematical viewpoint
energetic functions, as we shell see down, will be the functions, describing
the dynamical variations of the movement of a material point and all other airs
of possible for existence energetic functions (heating, electrical, chemical,
nuclear etc.) will be a consequence from them.
Let accept the produce process to be a
material point, who consumes energy for moving.
Figure 4.
Axiom E1. There exists at least a Euclidean space
E(3) single sign defined from the immovable coordinate system (O, x, y,
z).
Axiom E2. In the space E(3) exists at least two non
incident points A and B (see the figure 4.)
Axiom E3. In the space E(3) exists at least a
trajectory g uninterrupted between the points A and B.
Axiom E4. In the space E(3) exists at least a material point M
non identical with A and B with mass m measured in kilograms (kg).
Axiom E5. The point M moves over the trajectory g
and for the time t, measured in seconds (s) over the calendar, traverses the
road s, measured in meters (m).
Theorem T5. The velocity v = ds/dt of the point M
measured in meter pro second (m/s) exists and it is a uninterrupted function of
the time t by hers movement along the entire road between the points A and B.
Proof: According to axiom E5 there exists the
function s(t). According to axiom E5 she is uninterruptible over all length of
the arc between the points A and B. Like a function of the time s(t) is single
sign. Then hers derivation v = ds/dt exists and it is also single sign. It is
valid the diagram:
(10) t-->s(t)-->v = ds/dt
By reason of the calendar measure of the
time t the diagram (10) is also opposite. Therefore we can her continue to the
air:
(11) t-->s(t)-->v = ds/dt-->t
The air of the diagram (11) speak that
exists an isomorphisme between the functions v(t) and s(t). Therefore from the
properties single sign and uninterruptible follows that v(t) is single sign and
uninterruptible.
The theorem is proved.
Theorem T5. The acceleration a = dv/dt of the point
M, measured in meter pro second pro second (m/s2) is a
uninterruptible function of the time t by hers movement along the entire road
between the points A and B.
Proof: According the theorem T5 there exists the
function v(t). Again according T5 she is uninterruptible by all time t, for
that the point M traverse the length s of the arc between the points A and B.
And again, according the theorem T5, v(t) is single sign. It is valid the
diagram:
(12) t-->s(t)-->v(t)-->a = dv/dt
Because of the calendar measure of the
time t the diagram (12) is also opposite. Therefore we can her to continue by
the air:
(13) t-->s(t)-->v(t)-->a = dv/dt-->t
The air of the diagram (13) speak that
exists an isomorphisme between the functions a(t) and s(t). Therefore from the
properties single sign and uninterruptible follows that v(t) is single sign and
uninterruptible.
The theorem is proved.
Axiom E6. The velocity v(t)
of the point M reach to values very much smaller from the velocity of the light
(300 000 000 m/s).
Axiom E7. Along hers all road s the point M meets the resistance
force f measured in Newton (N).
Axiom E8. the resistance force f is a vector sum of
the components fs, fv and fa measured in
Newton (N), that are proportional and collinear respectively of the road s, the
velocity v and the acceleration a.
Theorem T7. The functions s(t), v(t) and a(t), as the
linear – proportional of them functions fs(t), fv(t)
and fa(t) are integrative in the sense of Riemann.
Proof: According the diagram
(13) the functions s(t),
v(t) and a(t) are in state of isomorphisme. Than and the functions:
(14) fs(t) = Às(t)+Â, fv(t) = Ñv(t)+D è fa(t) = Åa(t)+F,
where A, B, C, D and F are constants (to
see axiom E8), are also in state of isomorphisme, respectively in relation to
s(t), v(t) and a(t). From it follows that if in the moment t0 the
material point M coincides with the point A (see the figure 4) and in moment t
– with the point B, because of the uninterrupted state in this interval the six
over indicated functions are integrative in the sense of Riemann in the
interval (t0,t). In force are the integrals:
t
t
t
⌠ ⌠ ⌠
(15) ⌡s(t)dt, ⌡v(t)dt, ⌡a(t)dt,
t0 t0 t0
t t
t
⌠ ⌠ ⌠
⌡fs(t)dt, ⌡fv(t)dt, ⌡fa(t)dt,
t0 t0 t0
The theorem is proved.
Theorem T8. There exists the Riemann – Stiltes
integrals:
(16) t
t
t
⌠ ⌠ ⌠
⌡s(t)ds(t), ⌡v(t)dv(t), ⌡a(t)da(t),
t0 t0 t0
Proof: It follows directly from the diagram (13)
and the theorem T7. As soon as between the functions s(t), v(t) and it’s
derivatives exists the state isomorphisme and they self are integrative in the
sense of Rieman, is clear that the integrals (16) exists.
Theorem T9. There exists ih the sense of Riemann the
scalar multitudes
(17) t
t
t
⌠ ⌠ ⌠
⌡v(t)s(t)dt, ⌡v(t)v(t)dt, ⌡v(t)a(t)dt,
t0 t0 t0
to the vectors v(t) è s(t), v(t) è v(t), v(t) è a(t).
Proof: As soon as the theorem T8 exists the integrals (16), theirs solutions can
have and the unfinished (from some viewpoints) air (17).
The theorem is proved.
Theorem T10. The vectors
s(t), v(t) è a(t) belongs of the space L2 from the
functions with integrative square.
Proof: According the axiom E4 the point M is
defined single sign from the vector r (see figure 4) in the space E(3). Trough
it the length of the road s traversed from the point M is defined from the
formula:
(18)
s = ((x-xo)2+(y-yo)2+(z-zo)2)1/2,
where with x, y and z we mark the
coordinates of the point M and with xo, yo and zo the coordinates of the point
A. The size of the velocity of the point M is defined from the formula:
(19)
v = ((dx/dt)2 + (dy/dt)2 + (dz/dt)2)1/2.
The acceleration a of the point M is
defined from the formula:
(20)
a = ((dx2/dt)2 + (d2y/dt)2 +
(d2z/dt)2)1/2.
The formulas (18), (19) and (20) defines
magnitude (size) of vector in space E(3). From other side the functions s(t),
v(t) and a(t) are according the theorem T9 with integrative square in the sense
of Riemann. As soon as it is true with integrative square in the sense of
Riemann are integrative also theirs scalar components x, y and z, dx/dt, dy/dt
and dz/dt, d2x/dt, d2y/dt and d2z/dt. But
every integrative according Riemann function is also integrative according
Lebegue or
(21) R2 = L2,
where R2 and L2 are the spaces of the
vector functions with integrative about Riemann and Lebegue functions.
The theorem is proved.
Theorem T11. The square of the vector functions s(t),
v(t) and a(t) is integrative according Lebegue in infinite limits.
Proof: In the theorem T7 was accepted that the
point M moves only in the interval (to,t). If by this condition we widen this interval
to the interval (-∞,∞), the integrals:
(22) t
t
t
⌠
⌠ ⌠
⌡s(t)dt, ⌡v(t)dt, ⌡a(t)dt
t0 t0 t0
will be transformed in integrals:
(23) ∞ ∞ ∞
⌠ ⌠ ⌠
⌡s(t)dt, ⌡v(t)dt, ⌡a(t)dt.
-∞ -∞ -∞
They will be convergent in theirs
integrative intervals (-∞,∞).
The theorem is proved.
Theorem T12. The integrals (23) are convergent
absolutely.
Proof: As soon as the vector functions s(t), v(t) and a(t) are integrative in the sense
(23), integrative are also in this sense and theirs modules (theirs absolute
values).
The theorem is proved.
Theorem T13. All said in the theorems T8,…,T12 for the
functions s(t), v(t) and a(t) is valid also for the functions fs(t), fv(t) and fa(t).
Proof: It comes like a consequence from the
theorem T7. The functions s(t) and fs(t), v(t) and fv(t), a(t)
and fa(t) are
linear and proportional and also in state of isomorphisme between them and
therefore they have the necessary common properties.
The theorem is proved.
Theorem T14. The scalar produce of the functions v(t)
and f(t):
∞
⌠
(24) <v(t),f(t)> = ⌡ v(t)f(t)dt
-∞
exists.
Proof: According the theorem T10 the functions
v(t) and f(t) are with integrative in the sense of Lebegue square. Therefore
they belongs of the space L2. The existence of scalar produce is a property of
this space.
The theorem is proved.
Theorem T15. There exists:
-
the norm of the vector v(t):
∞
⌠
(25)
|v(t)| = (⌡v(t)2dt)1/2;
-∞
-
the single vector:
(26)
v0(t) = v(t)/|v(t)|;
-
the distance between the vectors v1(t) and v2(t):
(27)
d = |v1(t) - v2(t)|.
Proof: The equations
(25), (26) and (27) are properties of the space L2.
The theorem is proved.
Axiom E9: The scalar produce:
∞
⌠
(28)
<v(t),f(t)> = ⌡ v(t)f(t)dt = E
-∞
of the vectors v(t) and f(t) defines the
consumed from the point M energy E, necessary for the hers movement along the
trajectory g. The energy is measured in Joule (J).
Axiom E10. The derivation:
(29)
S = dE/dt
of the scalar produce (28) of the vectors
v(t) and f(t) defines the consumed from the point M power S necessary for the
hers movement along the trajectory g. The power is measured in Watt (W).
ENERGETIC ALGEBRA. FIRST PART.
With the exhibition of the axioms
E1,…,E10 was construct a mechanical model of the concept energy and with the proofs of the theorems T5,…,T15 was defined the
necessary mathematical apparatus for the construction of the model. This
apparatus is sufficient for the calculating of the energetic balance of the
moving point M. But it is not sufficient to define the energetic balance of all
kinds of systems. For achieving of this target it’s necessary to construct a
duality between the mechanical model and the remaining kinds of energetic (heating,
electrical, chemical, nuclear etc.) models. It is possible to make over an
algebraic base.
If we
construct the algebraic structure (the algebra) Ω such that the multitude
of operations Ω , transforming the multitude A in the energetic multitude
E,
Ω
(30)
À: (o)--->Å
is valid in conjunction with the
vectors fs, fv and fa
and by them and the equations (28) and (29), the constructed mechanical model
will be with a duality similar of all kinds energetic models. Then it will be
possible to construct the common criteria for valuation of all kinds energetic
systems, also for its invariability.
Before beginning define the
algebraic properties of the equations (28) and (29), let to forget the concept
of the full invariability and to lead in some criteria for its disturbance.
Definition
D5. Effective
values of the functions v(t) and f(t) are the values:
t
⌠
(31) V(t) = (d/dt(⌡v(t)2dt))1/2 and
o
t
⌠
F(t) = (d/dt(⌡f(t)2dt))1/2
.
o
The equations (31) presents in reality
the opposite transformation of the squares of the magnitudes (norms) of the
vector functions v(t) and f(t) in the space L2. If we compare the equations
(31) with (28) and (29) we shell see that the produce:
(32)
V(t)F(t) = S(t)
presents the consumed from the
point M power if v(t) and f(t) are constants.
With other words, the constant magnitudes of the vectors v(t) and f(t)
defines a constancy of theirs effective values and also a constancy of the
consumed from the source power. Conversely, every one variation of the magnitudes of the vectors v(t) and f(t) leads
to a variation of theirs effective values and also of a variation of the
consumed from the source power. By this condition the definition D2 for a
invariant system is equal to:
Definition
D6: Invariant
is every system, that produces a product with effective values of its
parameters completely and in every time
answering of the selected.
Definition
D7: Invariant
is every system, that by an invariant selection of parameters of hers outlet
product keeps constant the effective values of hers parameters.
Let set the question: “What is the
time relation between the constancy and variation of the effective values of
the vector functions v(t) and f(t) in a energetic process?”
In a made by human (industrial)
process is not economical to be the transitive (variable) periods longer from the constant (normal) periods.
Then there not exists any opposite case.
What is in the nature?
Nobody can not to be acquainted
with all
infinity from the atom to the Universe.
But the Universe is uniform and elements out from the Mendeleev table not exists. From unremembered times the
planets, also and from the system of the Sun, turns invariably over Kepler
orbit, that are plain curves. The Polar star invariably – only with one degree
error – indicate the true North.
That means there is for us – the
common people – eternal thinks.
But, from an other face, the
uranium blaze up slow and sure to lead. The Sun make nuclear explosions because
of that it is predict, that after some billion years it will becomes extinct. On distances measured in light years there
explodes unstable stars.
That means there is for us – the
common people – not eternal thinks.
Who is more – the eternal or not
eternal?
The true answer, out from a Middle
Ages dispute in front of the Inquisition, is the reasonable
Axiom
E11. Every energetic transition in the
Nature is directed from a constant balanced state to an other, also constant
balanced state, as the time for the transition is disproportionately smaller
from the time of the equilibrium.
Theorem T16. Every invariant system can come out from hers
state of invariability for short time moments out of proportion with the moments of the
invariability.
Proof: It follows directly from
the axiom E11. Every variation in the system generated from starting, stop or
set of a new choice to outlet parameters of the system will presents a short
time transition with greater or smaller breach of the invariability.
Theorem
T17. The
energetic balance of the material point M in a state of transition can be
expressed with Fourier integral.
Proof: The state of transition
according the theorem T16 is a short time and therefore non periodic
phenomenon. From other side, according the theorem T12 the vector functions
v(t) and f(t) are integrative in sense of Lebegue in infinite intervals. It
means that v(t) and f(t) can be presented in the air:
∞
⌠
(33) v(t) =
(1/√2π)⌡V(iΩ)exp(iΩt)dΩ and
-∞
∞
⌠
f(t) =
(1/√2π)⌡F(iΩ)exp(iΩt)dΩ,
-∞
when with i is marked the imaginary
part of complex numbers, with Ω – a real number with quality of frequency,
who is measured in radian pro second (rad/s) or s-1 . V(iΩ) and F(iΩ) are the Fourier pictures
of the functions v(t) and f(t),
respectively:
∞
⌠
(34) V(iΩ) = (1/√2π)⌡ v(t)
exp(-iΩt)dt and
-∞
∞
⌠
F(iΩ) = (1/√2π)⌡ f(t)
exp(-iΩt)dt .
-∞
Providently the equation (28) and
according the theorem of Parseval
(35)
<v(t),f(t)> = <
V(iΩ) , F(iΩ) > = E,
it is clear that the energetic
balance of the point M is fully defined
by the use of the Fourier integral.
The theorem is proved.
Theorem
T18. The energetic balance of the
material point M in a state of non transition (invariability) can be expressed
with Fourier sequence.
Proof: The invariant state of
the point M according axiom E11 is disproportionately longer from
the time of its transition. Over this base we can think that the point move
invariantly from unremembered times (in the time interval -∞,∞). It means that in the interval
-∞,∞ the vectors v(t) and f(t) have constant effective values. If
we present those functions in Fourier sequence:
∞
(36) v(t) = Σvk(ikΩ) and
-∞
∞
f(t) = Σfk(ikΩ)
-∞
when = -∞ ,...,-2, -1, 0, 1,
2,..., ∞, are the Fourier
coefficients:
∞
⌠
(37)
vk = ⌡v(t)exp(ikt)dt and
-∞
∞
⌠
fk = ⌡f(t)exp(ikt)dt,
-∞
what are independent from the time
t, will exists, because the vector functions v(t) and f(t) are absolutely
integrative in the sense of Lebegue in infinite intervals. In that sort of case
we shell can express the scalar produce of v(t) and f(t) in the air:
∞ ∞
(38) <v(t),f(t)> = <Σvkexp(ikΩ), Σfkexp(ikΩ)>
-∞ -∞
The theorem is proved.
Theorem
T19.
The all energy consumed in balanced (invariant) state of the movement of the
point M can be calculated according the formula:
∞
(39) EP = (ΣVkFkcosφk)t
-∞
when Vk and Fk
are the effective values of the Fourier coefficients (the harmonious
components, the harmonious) of the vector functions v(t) and f(t) and φk is the phase difference between
them.
Proof: If we make to end the
right part of the equation (38), basing of the concept scalar produce in the
space L2, we shell obtain the air:
t
⌠ ∞ ∞
(40) <v(t),f(t)> = ⌡(Σvkexp(ikΩt), Σfkexp(ikΩt))dt
o
-∞ -∞
If the condition (36) exists the
integration interval (0,t) will be transformed in the interval (0,2kπ).
With other words, the time t is accepted like divided of the interval (0,2kπ).
Then the vector functions v(t) and f(t) will makes an orthogonal base in
the space L2. It means that:
t
⌠ ∞ ∞
(41) <v(t),f(t)> = ⌡(Σvkexp(ikΩt), Σfk+1exp-i(k+1)Ωt))dt
= 0
o
-∞ -∞
And it also means that the scalar produce (40) will be a sum
from produces of vk and fk with identical index,
answering of the air of the theorem of Parseval for Fourier sequence:
t
⌠ ∞
(42) <v(t),f(t)> = ⌡(Σvkfk)dt
o
-∞
According thecondititon (37) vk
and fk are complex number. Let them present in its exponent air:
(43) vk = |vk|exp(iδk)
and fk = |fk|exp(iθk),
when δk and θk
are constants with the quality of time. They are beginning phases of the
periodical varied harmonious with number k of the functions v(t) and f(t). If we replace the values of vk and fk from the equation (43) in the equation
(42) we shell obtain:
t
⌠ ∞
(44) <v(t),f(t)> = ⌡(Σ|vk|exp(iδk)·|fk|exp(iθk))dt
=
o
-∞
t
⌠ ∞
=
⌡(Σ|vk|·|fk|exp(iφk))dt ,
o
-∞
when φk = θk
- δk is the phase difference of the harmonious vk
and fk of the v(t) and f(t). If now we present:
(45) exp(iφk) +
exp(-iφk) = 2cosφk
and replace it in the end result
from the equation (44), the scalar produce (44) will obtain the air of the
transformed integral sum:
t
⌠
(46) <v(t),f(t)>
= 2 ⌡(Σ|vk|·|fk|cosφk)dt
o
Let calculate by formulas (31) the
effective values of the number k harmonious Vk and Fk of
the vector functions v(t) and f(t). They will be:
t
⌠
(47) Vk = (d/dt(⌡(vkexp(ikΩt)+
vkexp(-ikΩt))2dt))1/2 and
o
t
⌠
Fk =
(d/dt(⌡(fkexp(ikΩt)+ fkexp(-ikΩt))2dt))1/2
,
o
Now, let transform the complex
subintegral functions in the
trigonometric air:
(48) Vk(t) = 2|vk|
cos(kΩt+δk) and
Fk(t)
= 2|fk| cos(kΩt+θk) .
After a substitution of equations
(48) under the integrals (47) and calculating same integrals, the values of the number k harmonious Vk
and Fk will be:
(49) Vk = 2|vk|/√2 and Fk
= 2|fk|/√2
or according the equations also
they will be:
(50) Vk = Vk(t)/
√2 and Fk = Fk(t)/ √2.
This is the very well known result from the the theory of the
electricity, but in lieu of velocity we
must understand non sinus (deformed) current and in lieu of force - non sinus
(deformed) voltage.
Let now send back the equation
(46). If we replace in him the modules
of the Fourier exponential components with the effectives values of the
harmonious from the equation (49) and calculate the integral, we shell achieve
exactly the equation (39). It is also a
very well known result from the
theory of the electricity.
The theorem is proved.
Definition
D8. The
calculated over the equations (39), (42), (44) and (46) energy EP is
named active energy, because it is a single sign measure for action of every
sinus energetic source.
Remark: The definition is
accepted like a standard term.
Theorem
T20. In transitive mode the resistive force fs(t)
of the point M, that is proportional of the traversed from her road s, will be late with π/2 radians about phase
towards hers velocity v(t). From other side, the proportional of hers
acceleration force fa(t) leave behind the velocity also with π/2 radians.
In invariant mode similar conduct exists between the k-harmonious of the velocity
and resistive forces. It is not valid by k = 0.
Proof: In transitive mode the velocity of the point M is defined
according the equation (33). If we derivate (33) over the time t, and the true:
(51) i = exp(π /2) è -i =
exp(-π /2),
we shell obtain:
∞
⌠
(52) d/dt(v(t)) =
(1/√2π)⌡ΩV(iΩ)exp(iΩt+ π /2)dΩ
-∞
In invariant mode the velocity of
the point M is defined according the equation (36). If we derivate (36) over
the time t, we shell obtain:
∞
(53) d/dt(v(t)) = ΣkΩvkexp(ikΩvkt+
π /2)
-∞
Here the harmonious by k = 0 is
absent, because its derivate over the time t is zero.
If now we integrate over the time t
the equations (33) and (36), we shell obtain:
t ∞
⌠ ⌠
|t
(54) ⌡v(t)dt = (1/√2π)⌡
V(iΩ)(exp(iΩt- π /2)/ Ω)dΩ| and
o -∞ |o
t
⌠ ∞ |t
⌡v(t)dt = Σvk(exp(i(kΩt-
π /2)))/k Ω | , k ≠ 0.
o -∞ |o
Here the harmonious by k = 0 is
suspended from the second equation, because its module vk/k Ω
will infinitely great by k = 0.
The equations (52) and (53) defines
according the theorem T6 and axiom E8 a force proportional of the acceleration
a of the point M and the equations (54) according theorem T8 and axiom T5 – a
force proportional of the traversed from the point road s. The expressions:
(55) exp(i Ω t+ π /2) and exp(ik Ω t+ π /2)
speaks single sign in the equations
(53) for a leaving behind with π /2
radians toward the function v(t) and the expressions:
(56) exp(i Ω t- π /2) and exp(ik Ω t-π /2)
for a delay with π /2 radians.
The theorem is proved.
Theorem
T21. The
vector functions fs, fv and fa
makes in transitive mode of movement of the point M the two dimensional linear space F(2).
Proof: According the axiom E8
and the theorem T20 the forces fs and fa are in a state of contra phase. Theirs
resulting force fr is equal of the algebraic sum:
(57) fr = fa - fs
(see the equations (52) and (53)
compared with the equations (54) ). From
an other side also according the theorem T20 the force fv delays by phase
toward the force fa with π /2 radians and leave behind with same time the
force fs. If we represent in a figure
the diagram:
Figure 5.
We shell see that the result force
fr make with the force fv an angle of 90o. To see it exactly.
To finish the derivate over the
time t under the sign of integral in the equation (52) and the integrate over the time t under the sign of
integral in the first of equations (54). We shell obtain:
∞
⌠
(58) d/dt(v(t)) =
(i/√2π)⌡ Ω V(i Ω )exp(i Ω t)d Ω and
-∞
∞
⌠ ⌠
(59) ⌡v(t)dt = -
(i/√2π)⌡ Ω V(i Ω )exp(i Ω t)d Ω
-∞
∞
⌠
+
(i/√2π)⌡ ( V(i Ω )/ Ω )d Ω
-∞
The first of equations define the
force fa(t) and the second – the force fs(t). The last member from the equation
(59) is independent from the time t and, consequently, presents a static or
constant component of the force fs and not participate in the transitive
process of the movement of the point M.
This is the value of the force fs by t = 0 (in the beginning of the transitive
process). Let it sign with fso and to
accept it of zero. The resulting dynamical force fr(t) of the
transitive process will be:
∞
⌠
(60) fr(t) = (1/√2π)⌡ (Ω -1/ Ω
)V(i Ω )exp(i Ω t+ π /2))d Ω
-∞
The complex vector functions (33)
and (60) have different modules and
arguments. But for them we can write the equations:
(61) |fr(t)| = (Ω -1/ Ω)|v(t)|) and
arg(fr(t)) = arg(v(t))+ π /2.
The equations (61) for second way
proves the reality of the diagram on the figure 5. It means that the square of
the vector sum of the forces fr and fv is:
(62) |f(t)|2 = |fr(t)|2+
|fv(t)|2
And if the bylinear form of the
equation (62) is zero, the vector functions fs,
fv è fa can make in transitive
mode of movement of the point M the two dimensional linear space F(2). It will be
subordinate of the axioms of an ordinary Euclidean two dimensional vector
space.
The theorem is proved.
Theorem
T22 (space E(2)). The energy E consumed
from the point M in transitive mode of her movement presents a vector in the
two dimensional vector space of the energies E(2).
Proof: According the theorems
T17 and T20 the scalar produce between the velocity v(t) of the point M and the
resulting force fr(t):
t
⌠
(63) <v(t),fr(t)> =
⌡(Ω-1/Ω)V(iΩ)V(-i Ω)dt =
o
t
⌠
=
⌡(Ω-1/Ω)|V(iΩ)|2exp(i π /2)dt
o
has a quality of energy. This is
the energy Er(t) proportional of the force fr(t). From other side the scalar produce:
t
⌠
(64) <v(t),fr(t)> =
⌡V(iΩ)V(-i Ω)dt =
o
t
⌠
=
⌡|V(iΩ)|2dt
o
also has a quality of energy that
is proportional of the force v(t). Let mark it with Åv(t). Both energetic
components have the property:
(65) År(t)| = (Ω-1/Ω)|Åv(t)| and
arg(År(t))) =
arg(Åv(t))+π /2
By this last condition the square
form:
(66) |Å(t)|2==|År(t)|2+
|Åv(t)|2
will possess bylinear form equal to
zero. But according the theorem T1 the multitude of energetic value makes an
additive group. Therefore the functions Åv(t)
and År(t) makes in transitve mode
of movement of the point M a orthogonal base in the two dimensional and linear
space E(2). It will be subordinate of the axioms of an ordinary Euclidean two
dimensional vector space.
The theorem is proved.
Theorem
T23 (space S(2)). The power S consumed from the point M in transitive mode of its
movement present a vector in the two dimensional space of the powers S(2).
Proof: After derivate over the
time t of the equation (63) we shell obtain the following complex function:
(67) Q(t) =
(Ω-1/Ω)|V(iΩ)|2exp(i π /2),
and after derivate of the equation
(64):
(68) P(t) = |V(iΩ)|2
The equations (67) and (68) are in
a state of isomorphisme toward the equations (63) and (64) because the action
derivate is isomorphe toward the action integrate. The same equations have the
quality of power and, excepting it, because of the state of isomorphisme with
the equations (63) and (64) they makes an additive group. Also because of the
state of isomorphisme the square form:
(69) |S(t)|2 = |P(t)|2+
|Q(t)|2
will possess bylinear form equal to
zero and the functions P(t) and Q(t) will makes in transitive mode of the
movement of the point M an orthogonal base in the two dimensional and linear
space S(2). This space will be in state of isomorphisme with the space E(2) and
it will be subordinate of the axioms of an ordinary Euclidean two dimensional
vector space.
The theorem is proved.
ENERGETIC ALGEBRA.
PART TWO
Theorem
T24. The phase corner between the effective values of the k-harmonious Vk and Fk
of the velocity v(t) and the force f(t) (see the equation (39)) is
subordinate of the equation:
(70) -π /2 ≤ φk ≤
π /2
Proof:
If we suppose in contrast to the equation (70), it will be consumed from the point M according the
equation (39) active energy can be negative. But this contradict of the proved
in the theorem T4 necessary for the state of invariability condition, that the
consumed from the object (in the case, the point M) active energy is always
contra of the active energy generated from the source. The cases, when the
object return the translated toward him energy back to the source (mode of
recuperation, dynamical stop and contra including) are characteristic for
transitive mode and they can be discuss separately.
The theorem is proved.
Theorem
T25. The energy
proportional of the road s(t) and of the acceleration a(t) of the point M
(reactive energy) in invariant mode is numeric equal to zero.
Proof:
The proportional of the of the acceleration a(t) force fa(t) is
calculated after a derivate the equation (36) over the time t, accepting the
derivation constant CD (Ns2/m) equal to one:
∞
(71) fa(t) = CDd/dt(Σvkexp(ikΩt))
=
-∞
∞
= ΣikΩvkexp(ikΩt) =
-∞
From other
side, the proportional of the road s
resistant force fs(t) is calculated (after an integration over the time t of
the equation (36), accepting the integrating constant CI (N/m) equal
to one:
t
⌠ ∞
(72) fs(t) = CI⌡(Σ vkexp(ikΩt))dt
=
o-∞
∞
= Σ(1/ikΩ) vkexp(ikΩt)
-∞
The common – let
name reactive – resistive force fr(t) of the point M will be the vector sum:
(73)
fr(t) = fa(t) + fs(t) =
∞
= iΣ(kΩ-
(1/kΩ)) vkexp(ikΩt) =
-∞
∞
=
2iΣ(kΩ-(1/kΩ)) vkexp(ikΩt)
1
The case k=0 is
excluded from the sum, because the derivate in the right part of the equation
(71) will be zero and the integral in the right part of the equation (72) will
be indefinite. From it follows that the zero harmonious don’t participates in
the reactive resistant force fr(t) of the point M.
The reactive
energy EQ will present the scalar produce:
(74)
ÅQ = <v(t),fr(t)>.
Foreseeing the equations (72) and (73) also and the fact that by
existence of the condition (36) the integrative interval (0, t) is
translated in the interval (0, 2kπ) and the functions exp(ikΩt) forms
an orthogonal base in the space L2, the scalar produce (74) will obtain the
air:
2kπ
∞⌠
(75) ÅQ =2i(Σ⌡(kΩ-(1/kΩ))|
vk |2dt)
1 o
The sum of
integrals in the right part of the equation (75) is zero, because every k-th
integral has the solution:
2kπ
⌠
(76) 2i (kΩ-(1/kΩ))| vk
|2⌡dt =
o
2kπ
|
=
2i (kΩ-(1/kΩ))| vk |2it| = 0 –
0 = 0
o
In the complex
plane (0, Re, Im) the produce i2kπ is equal to zero.
The theorem is
proved.
Definition
D9. The energy EQ proportional of the road s(t) and the
acceleration a(t) of the point M is named reactive energy. It is, as we shell
see down below, single sign measure for non effectiveness (reaction) of every
sinus energetic source (see the theorem T29).
Theorem
T26. Every harmonious of the reactive energy EQ consumed
from the point M oscillate with a frequency 2kΩ,
who is twice greater from the frequency of the corresponding harmonious vk
of the velocity v(t) or fk of the resistant force f(t).
Proof:
The equation (75) proves that the reactive energy EQ
consumed from the point M in invariant mode presents a sum from harmonious. If
in the equation (76) we do the integration in the current interval (0,
2kπ), it will signifies that we search the value of the k-th harmonious ÅQk
of the same reactive energy in every point of the interval (0, 2kπ). This
value will be defined from the behavior of the function:
(76)
ÅQk(t) = 2i(kΩ-(1/kΩ))| vk |2
kΩt
The function (76)
presents a vector in the complex plane (0, Re, Im), who has the variable in the
time t module 2(kΩ-(1/kΩ))| vk |2· kΩt and also the variable in the time t argument
i. The behavior of the of the function ÅQk(t) can be follows by the
next table:
kΩt i
ikΩt
---------------
0 0
0
π/2 i
i
π 0
0
3π/2 -i -i
2π 0
0
In other words, if the vector:
(77) 2(kΩ-(1/kΩ))| vk
|2exp(i2kΩt)
describes a circumference with radius 2(kΩ-(1/kΩ))| vk
|2 in the complex plane (0, Re, Im), the vector:
(78) i2(kΩ-(1/kΩ))| vk
|22kΩt
will be its
projection over the imaginary axe Im. It changes its value in the interval (-i2(kΩ-(1/kΩ))|vk|2, i2(kΩ-(1/kΩ))|vk|2)
with velocity (frequency) 2kΩ. This frequency is twice greater than the
frequency kΩ, with that in complex plane (0, Re, Im) turns the vectors vk or fk
(see the equation (36)) of the k-th harmonious of the velocity v(t) or the
resistance force f(t) of the point M.
The theorem is
proved.
Theorem
T27. Consumed from the point M in invariant mode reactive energy EQk(t)
has minimum by kΩt = - π/2 and maximum by kΩt = π/2.
Proof:
For the k-th harmonious of the reactive energy EQk(t)
this affirmation is an evident true from the table in the
theorem T26. According it the vector (78) will be minimal by kΩt
= - π/2 and maximal by kΩt = π/2. This vector presents the
subintegral function in the equation (75), over that is calculated in invariant
mode the consumed from the point M reactive energy EQk(t). As soon
as the integrals in the right part of this equation have a maximum, their sum
will be also maximum and in contrast – if the integrals have minimum, their sum
will be also minimum.
The theorem is
proved.
Theorem
T28. Calculated over the equation (75) reactive energy is fictional (it is not preserved in the system).
Proof: According the theorem T25 the equation (75) is
identically equal to zero and according the theorem T26 the reactive energy ÅQ(t)
presents a sum from harmonious. Therefore these energetic harmonious oscilates
between the point M and the energetic source without making any useful work.
With other words, they not provokes any changes of the velocity of the point M
and coerces the source to canges the generated
from him power to consume the point M the necessary for its velocity v(t)
energy. That signifies the reactive energy EQ not participate in the
energetic balance of the end product of the system (in our case, the velocity
of the point M) and therefore it is not preserved
in the system.
The theorem is proved.
Theorem
T29. The reactive energy consumed in invariant mode from the point M
and calculated over the equation:
2kπ 2kπ
∞⌠ ∞⌠
(79) EQ = (Σ⌡vk f-kdt)
= (Σ⌡ vk f+kdt) =
-∞o -∞o
∞
= (ΣVkFksinφk)t,
1
when with f+k
and f-k are
respectively substituted:
(80) f-k = | fk| exp(ikΩt-π/2) and
(81) f+k = | fk| exp(ikΩt+π/2).
Proof:
According the theorem T19 all consumed from the point M energy by
its movement in invariant mode is identical with the active energy EP
calculated over the equation (39). It is according the definition D8 a measure
for effectivity of the system.
From other side
the oscillations of the reactive energy according the theorem T28 decreases the
effectivity of the system and the connecting energetic line “source – consumer”
decreases its conductivity. Therefore is necessary a measure for this non
effectivity that the measuring instruments for active energy can’t display. In
contrast it is impossible to specify devices
for its removing.
As soon as the equation (39) is a measure for effectivity of the system,
the equation:
(82) ÅQ = (ΣVkFksinφk)t
will be a measure
for its non effectivity. This is true, because according the equation (70) in
the theorem T24 the corner φk belongs of the interval (-π/2,
π/2) and therefore over ther the function cos φk increases
- sin φk decreases. And by condition that the effective values
Vk and Fk of the harmonious of the velocity v(t) and of
the resistive force f(t) are constant, the equation (80) is an alternative of
the equation (39). It can’t be other exepting an element from the common
measure for non effectivity of the system.
Let now calculate
the integral part of the equation (79), accepting that vk leaves behind over phase fk. For
single sign of our reasons let accept that the phase corner φk is
measured in direction from vk to fk. Then it will be
positive, when vk leaves behind over
phase fk and negative – in contrast. We shell obtain:
2kπ 2kπ
∞⌠ ∞⌠
(83) EQ = (Σ⌡ vk
f-kdt) = (Σ⌡|vk
||fk|exp(i(δk- θk+ π/2)dt,
-∞o -∞o
when with δk
and θk we marked the accepted by the equation (43) beginning
phases of vk and fk.
Having provision also and the accepted in the equation (44) that the phase
corner is equal to:
(84) δk- θk
= φk,
2kπ
∞⌠
(85) EQ = (Σ⌡|vk
||fk|exp(i(-φk + π/2)dt =
-∞o
2kπ
∞⌠
= (Σ⌡|vk
||fk|(exp(i(-φk + π/2) + exp(-i(-φk
+ π/2))dt =
1 o
2kπ
∞⌠
=2(Σ⌡|vk
||fk|cos(-φk + π/2)dt =
1 o
2kπ
∞⌠
=2(Σ⌡|vk
||fk|sinφk dt.
1 o
The harmonious
with number zero in the trigonometric part of the equations (85) are ignored
according the proved in the theorem T25 necessity.
Having provision
also and the equation (49) for the effective values Vk and Fk
of the harmonious of the functions v(t) and f(t) expressed in trigonometric air
in the equation (85), we shell obtain exactly the end result from the equation
(79).
The theorem is
proved.
Theorem
T30. The produces from square forms in the air:
∞ ∞
(86) (Σak2) (Σbk2)
1 1
are subordinate of
the equation:
∞ ∞ ∞ ∞
(87) (Σak2) (Σbk2)
= (Σakbk)2 + (Σapbq
- apbq)2
1 1 1 1 p
≠ q
Proof:
Let make the multiplication:
(88) (a12 + a22)(b12
+ b22) = a12b12
+ a22b22 + a12b22
+ a22b12
If we add and subtract in the equation (88) the expression 2a12b12a22b22
and form a new group from the members, the equation will
obtain the air:
(88.1) (a12 + a22)(b12
+ b22) =
= (a1b1 + a2b2)2
+ (a1b2 - a2b1)2
And continuing
over the method of the full mathematical induction, by comparing the members
with identical index, we can compose the produce:
(89) (a12
+ a22 +...+ai2 +...)(b12
+ b22 +...+bi2 +...) =
∞ ∞ ∞ ∞
= (Σak2) (Σbk2)
= (Σakbk)2 + (Σapbq
- apbq)2
1 1 1 1 p
≠ q
The theorem is
proved.
Theorem
T31. Consumed from the
point M active energy EP can be presented from the expression:
t=2kπ
∞⌠
(90) EP = Σ⌡|vk||fvk|dt
=
-∞o
t=2kπ
⌠ ∞ ∞
= ⌡(( Σ|vk|2)
( Σ|fk|2cos2φk )
-
o -∞ -∞
∞
- ( Σ|vp||fq|cosφq
- |vq||fp|cosφp)2)1/2dt,
-∞ p
≠ q
when according
with yhe equation (46) from the theorem T19 we substituted:
(91) |fk|cosφk
= | fvk|,
defining over this
manner the modul of the k-th harmonious fvk(t) of the prorportional
of the velocity of the point M resistant force fv(t).
Proof:
If in the end result from the equation (89) we move the end right
member with an invers sign in leftwing and if we change the upper limit of the
sum with -∞ and, also, in place of ak we write |vk|
and in place of bk we write |fvk|, we shell obtain the
equation (90).
The theorem is
proved.
Theorem
T32. Consumed from the
point M reactive energy Eq can be presented from the expression:
t=2kπ
∞⌠
(92) Eq = Σ⌡|vk||frk|dt
=
-∞o
t=2kπ
⌠ ∞ ∞
= ⌡(( Σ|vk|2)
( Σ|fk|2sin2φk )
-
o -∞ -∞
∞
- ( Σ|vp||fq|sinφq
- |vq||fp|sinφp)2)1/2dt,
-∞ p
≠ q
when according
with yhe equation (85) from the theorem T29 we substituted:
(93) |fk|sinφk = | frk|,
defining over this
manner the modul of the k-th harmonious frk(t) of the reactive
resistant force fr(t).
Remark: The
harmonious with number zero in the equation (92) is excluded.
Proof:
If in the end result of the equation (89) we move the end right
member with an invers sign in leftwing and if we change the upper limit of the
sum with -∞ and, also, in place of ak we write |vk|
and in place of bk we write |frk|, we shell obtain the
equation (92).
The theorem is
proved.
Axiom
E12. The moving medium of the
point M is homogeneous and have constant parameter.
Axiom E13. The moving
medium of the point M generates friction.
Definition D10. The value μ measured in Ns/m, who transforms
linear the velocity v(t) of the point M in active resistant force fv(t) over
the equation:
(94) fv(t) = μv(t),
we shell name
active resistance or linear dynamical viscosity. It is a real number and characterize the medium in who
move the point M and, exepting, it define the active energetic wastes of the
source of the system. By homogeneous and
having constant parameter medium μ is constant. We can accept the in the
text we have preliminary according:
(95)
μ = 1 Ns/m
Theorem
T33. The right member from the end result in the equation (90) is equal
of number zero.
Proof:
Let accept that the condition (95) is not observed and, therefore,
μ = 1. Than the condititon (94) has its
common air. We can in the right member from the end result in the equation (90)
substitute:
(96) |fvq| = μ|vq|
and |fvp| = μ |vp|
Substituting (96)
in the right member from the end result in the equation (90), we shell see that
this member is equal to zero.
The theorem is
proved.
Theorem
T34. Maximum consumed in invariant mode of moving of the point M energy
ES is defined by the equation:
t=2kπ
⌠ ∞ ∞
(97) ES = ⌡((Σ|vk|2)
(Σ|fk|2))1/2dt
o -∞
-∞
Proof: As
soon as theorem T33 right member from the end result in the equation (90) is
equal of number zero, this equation will be:
t=2kπ
∞⌠
(98) EP = Σ⌡||vk|||fk|cosφkdt
=
-∞o
t=2kπ
⌠ ∞ ∞
= ⌡((Σ|vk|2)
(Σ|fk|2|cosφk2)))1/2dt
o -∞
-∞
The function EP will obtain a maximum when its trigonometric
part is equal to one. With other words, when cosφk = 1 or
φk = 0. It is confirmed also and from the equation (39) in the theorem T19. As soon as is true, we
can put in equation (98) φk = 0. The end result will be same
with the right part of the equation (97).
The theorem is
proved.
Theorem
T35. The equation (97) defines also and the minimum of the consumed
reactive energy in invariant mode of movement of the point M.
Proof:
The direct valuation of the equation (82) speaks that by φk
= 0 the consumed reactive energy in invariant mode of movement of the point M
will equal to zero. This is its real physical minimum. The algebraic minimum –
the maximal negative value of EQ – is not a object of this theorem.
The theorem is
proved.
Theorem
T36. (space E(3)). The energy E consumed from the point M
(the energetic balance of the system “source – point”) in invariant mode of its
movement presents a vector in the three dimensional linear energetic space
E(3).
Proof: Let
derivate over the time t the equation (90), having provident the true that
according theorem T33, the last member from the end result in this equation is
zero, also and the true (91). We shell obtain:
(99)
P = dÅP/dt = (SP2)1/2,
when we substituted:
∞ ∞ ∞ ∞
(100) SP2 = (Σ|vk|2) (Σ|fvk|2) = (Σ|vk|2) (Σ|fk|2cos2φk)
-∞ -∞ -∞ -∞
If we derivate
over the time t the equation (92), having provident the true (93), we shell
obtain:
(101) Q = dÅQ/dt = (SQ2-
RQ2)1/2 ,
when we
substituted:
∞ ∞ ∞ ∞
(102) SQ2 = (Σ|vk|2)
(Σ|frk|2) = (Σ|vk|2) (Σ|fk|2sin2φk)
-∞ -∞ -∞ -∞
∞
(103)
RQ2 = (Σ|vp||frq| - |vq||frp|)2 =
-∞ p ≠ q
∞
= (Σ|vp||fq|sinφq - |vq||fp| sinφp)2.
-∞ p ≠ q
If now we compose
the sum:
(104) P2 + Q2 = SP2
+ SQ2 - RQ2 = S2 –
R2,
it will folow
that:
∞ ∞
(105) S2 = SP2
+ SQ2 – RQ2 = (Σ|vk|2) (Σ|fk|2)
and
-∞ -∞
∞
(106)
R2 = RQ2 = (Σ|vp||fq|sinφq
- |vq||fp| sinφp)2.
-∞ p
≠ q
Or in the end will
be the true:
(107) P2 + Q2 + R2
= S2 or
(108) S = (P2 + Q2 +
R2)1/2.
If the equations
(90) and (92) have the physical quality of energy and they are transformed by derivate
over the time t to the equations (99)
and (101), in other words:
d/dt d/dt
(109) EP(90) →
P(99), EQ →
Q(101),
than the
equations:
t t
⌠ ⌠
(110) ⌡Pdt = EP ⌡Qdt = EQ,
o o
who according the
diagram (109) presents theirs invers isomorphe transformation, evidently will
have also the quality of energy. In such case and the sum:
t t
⌠ ⌠
(111) ⌡Sdt = ⌡(P2
+ Q2 + R2)1/2dt
o o
will have the
quality of energy, because it will have a result identical with the result in
the equation (97), who also has the quality of energy. And if over this cause
we unite the equations (97) and (111), it will be clear that the entire energy
ES, who the source transfers of the point M is defined by the
equation (111). It has exepting both
defining by the equation (110) active - EP and reactive - EQ
components, also and the restful (deformation) component ER. It is
defined by the equation:
t
⌠
(112) ⌡Rdt = ER.
o
Forseeng that in
invariant mode of movement of the point M the powers P, Q and R are independent
from the time t, the integral equations (110) and (111) will have the
solutions:
(113)
Pt = ÅP, Qt = ÅQ, Rt = ÅR
As soon as follows
that the integral equation (111) will have the solution
(114) St = ÅS
The left part of
the equation (114) can be substituted in the equation (111). Then will be
trivial clear that
(115) ÅS2 = S2t2
= P2t2 + Q2t2 + R2t2
Three member
square form (115) has a bylinear form equal to zero. Exepting this, according
theorem T1 its three member makes an additive group. And from the last facts
follows that ES is a three dimensional vector of the components EP,
EQ and ER and, therefore belongs of a linear three
dimensional space in that EP, EQ and ER makes
an otthogonal base. We can mark this space with E(3). The spase E(3) will be
subordinate of the axioms of an ordinary Euclidean space.
The theorem is
proved.
Theorem
T37. Calculated by the equation (112) restful energy ER is
ficticious (it is not preserved in the system).
Proof:
According the equation (92) the restful energy ER
presents its right end member. Its growth or decrease depends from the phase
corner φk or, exactler, from the function sinφk.
When every phase differences φk between the harmonious vk(t)
of the velocity v(t) of the point M and
the harmonious fk(t) of its resistant force f(t) are equal to zero,
it follows that and ER will be also zero. In contrast, when φk
is equal to π/2 radians, ER will have a maximum. The sign of ER
(positive or negative) depends also from φk.
The same behavior
towards φk has also and the reactive energy EQ.
Therefore, the phase properties of ER are equal to the the phase
properties of EQ. Every k-th harmonious of EQ oscilates
with same frequency - 2kΩ with then oscilates also and k-th harmonious of
ER. As soon as every unabridged by 2π period the harmonious of
ER are equal to zero, the energy ER is ficticious.
The theorem is
proved.
Definition
D11. The value:
(116) σm = mkΩ,
who is a modul of
the imaginative number imkΩ, transforming linearly k-th harmonious vk(t)
of the velocity v(t) of thepoint M in the k-th harmonious fak(t) of
the proportional of the acceleration a(t) resistant force fa(t) over
the equation:
(117) fak(t) = imkΩvk(t)
= iσmvk(t),
has a quality of
resistance. Let name it reactive inertial resistance. Here m is the mass of
point M, who is measured in kg or nm/s2. According axiom E6 follows
that we can accept m like a constant. We can also accept beforehand that in the
text the mass of the point M is:
(118)
m = 1 Ns/m2 = 1 kg
Like of the active
resistance μ, the reactive inertial (mass) resistance is measured in Ns/m.
Axiom
E14. The middle in that move the
point M possess an elasticity.
Definition
D12. The value:
(119) σl = 1/ÑlkΩ,
who is a modul of
the imaginative number 1/ikΩCl, transforming linearly k-th
harmonious vk(t) of the velocity v(t) of the point M in the k-th
harmonious fsk(t) of the proportional of the road s(t) resistant
force fs(t) over the equation:
(120) fsk(t) = vk(t)/ikΩÑl
= -iσlvk(t)
has a quality of
resistance. Let name it reactive elastic resistance. Here:
(121) εl = 1/Ñl
is a linear modul
of Young for the elasticity of the middle in that move the point M. It is
measured in N/m and its reciprocal value – the coefficient of the elasticity of
the middle Cl – in m/N.
Let down below we
present the standard modul of Young G defining the elastic flexibility of the
middle Δl/l over the low of Houck:
(122) Δl/l = Fl/GS,
when with Δl
we mark the rectilinear elastic flexibility of the
middle if the point M passed the right section l, with l – the length of this
section measured in m, with F – the internal (proper) elastic force of the
middle measured in N and with S – the cross section (measured in m2)
of the point M if for an instant we accept that the point M is transformed in
flat body moving perpendicular of l. By
this condition the modul of Young εl will be:
(123) εl = dG/dl
By a middle with constant parameter εl
is a constant. We can accept that in the text we preliminary are consonantal
with the true:
(124) εl = 1 N/m and
(125) Cl = 1
m/N.
Like of the active
resistance μ, the reactive elastic resistance σl is
measured in Ns/m.
Theorem T38. Calculated over the equation (97) energy ES
is the maximal consumed by the moving in invariant mode point M energy.
Proof: As soon as according the theorem T37 ES
is a vector in the space E(3), the equation (97) is necessary to define its
modul.
From other side:
-
the theorem T34
proves that the equation (97) defines the maximum of the energetic effectivity
EP;
-
the theorem T35
proves that the equation (97) defines the minimum of the energetic non
effectivity EQ;
-
the theorem T37
proves that the phase behavior of ER coincides with the phase
behavior of EQ.
So the equation (97) also defines also the energetic non effectivity ER.
Let compose the series of:
(126) φ:
-π/2<φk1 < φk2 <…< φkn = 0 < φkn+1 <…< π/2
(127) ÅP: 0 <
ÅP1 < ÅP2 <...< ÅPn = max > ÅPn+1
>... > 0
(128) ÅQ: max > ÅQ1 > ÅQ2 >...>
ÅQn = 0 < ÅQn+1 <... < max
(129) ÅR: max > ÅR1 > ÅR2 >...>
ÅRn = 0 < ÅRn+1 <... < max
Here the energies
are present with its absolute values. Looking attentiver in the middle of the
four series or, exactly to say, estimating simultaneously the values of ÅPn, ÅQn and ÅRn
by φkn = 0, for the series (127), (128) and (129) there make
the conclusion, that:
(130) sup(ÅP) = inf(ÅQ)
= inf(ÅR) = ÅS
The theorem is
proved.
Theorem
T39. (space S(3)). The power S consumed from the point M (the
power balance of the system “source – point”) in invariant mode of its movement
presents a vector in the space S(3).
Proof:
After a square rooting, derivating by the time t and once again
squaring of (115) we shell obtain the square form:
(131) S2 = (dÅS/dt)2
= P2 + Q2 + R2
The equation (131)
is in state of isomorphisme toward the equation (115), because the energetic
components ÅP, ÅQ and ÅR are functions of the time t
and according the theorems T5 and T6 the action derivation is isomorphic toward
the action integration. The same derivated components have according axiom E10
the quality of power and, exepting it, because of their isomorphisme with the
components in the equation (115) they makes an additive group. Over again
because of the isomorphisme the square form from powers (131) will pssess
bylinear form numerc equal to zero, and
the functions P, Q and R will makes in invariant mode of movement of the point
M an orthogonal base in the three dimensional space S(3) in that the power S is
a vector. This space will be isomorphic toward the space E(3) and it will be
subordinate to the axioms of an ordinary Euclide three dimensional vector
space.
The theorem is
proved.
Theorem
T40. Calculated over the equation
(97) power S is the maximum possible consumed by the point M in
invariant mode of its movement.
Proof:
As soon as the three dimensional space S(3), in that the power S
is a vector, according the theorem T39 is isomorphic of the energetic space
E(3), in that according the theorem T36 the energy ES is a vector,
the proved in the theorem T38 for ES is in force also and for the
power S.
The theorem is
proved.
Theorem
T40. Calculated over the equation
(97) power S is the maximum possible consumed by the point M in
invariant mode of its movement.
Proof:
As soon as the three dimensional space S(3), in that the power S
is a vector, according the theorem T39 is isomorphic of the energetic space
E(3), in that according the theorem T36 the energy ES is a vector,
the proved in the theorem T38 for ES is in force also and for the
power S.
The theorem is
proved.
Theorem T41 (space
Å(2)). The private case by that is possible in the equations (90) and (92) to
replace:
(132)
p = q,
(133)
k = 1,
with other words,
when the point M is in a sinuous invariant mode of movement, the consumed
energy by the point M (the energetic balance of the system “source – point)
presents a vector in the two dimensional linear energetic space E(2).
Proof:
The respect of the condition (132) demands the consumed by the
point M restful (deformation) energy ER from the equation (106) to
be equal to zero. Than the equation (111) can be writed in the air:
t t
⌠ ⌠
(134) ES = ⌡ Sdt = ⌡(P2 + Q2)1/2dt
o o
and the equation
(115) - in the air:
(135) ÅS2 = S2t2
= P2t2 + Q2t2.
The equations
(134) and (135) don’t eliminates the
criteria for existence of an energetic space, that the theorem T36 demands,
because its members possess the quality of energy, subordinated already of
these demands. These members are two, that signifies, that the energetic space
E(3) from the theorem T36 is transformed from three dimensional to two
dimensional. We can mark with E(2).
From other side,
the respect also and the condititon (133) demands the transformation of the
equation for active energy (39) in the equation:
(136)
ÅP = VFcosφt,
and the end result
from the equation for reactive energy (79) – in the equation:
(137)
ÅQ = VFsinφt,
when V and F are
the effecrive values of the velocity v(t) and the resistant force f(t) of the
point M, calculated over the equations:
t
⌠
(138) V =
(d/dt(⌡(2|v|cos(Ωt+δ))2dt)))1/2 and
o
t
⌠
F =
(d/dt(⌡(2|f|cos(Ωt+θ))2dt)))1/2
o
Foreseeing that the exponent moduls |v| and |f| of the
harmonious of the velocity v(t) and the resistant force f(t) of the point M
according the equations (49) are double way littler about its trigonometric
(real measured) moduls (magnitudes):
(139)
2|v| = vmax = V√2 è 2|f| = fmax = F√2,
it follows
according the equations (138) that the single harmonious of the velocity
defines its behavior over the time t over the equation:
(140)
v(t) = vmaxcos(Ωt+δ),
and the single
harmonious of the resictant force defines its behavior over the time t over the
equation:
(141) f(t) = fmaxcos(Ωt+
θ).
Here, like in the
equations (48), δ and θ are the beginning phases of the velocity v(t)
and the resictant force f(t) and their difference:
(142) δ - θ = φ.
The equations
(140) and (141) are well known trues from the electrical engineering. In
practice is accepted the coordinate systems for design of these equations to
have its beginning with π/2 radians forward or backward. So by beginning
phases equal to zero in the beginning of the coordinate system the oscillations
to have value zero. By this condition the equations (140) and (141) obtains the
best knowing air:
(143) v(t) = vmaxsin(Ωt+δ)
è f(t) = fmaxsin(Ωt+θ),
from that comes
the term “sinous invariant mode”.
Theorem is proved.
Theorem
T42 (space E(1)). The private case by that is possible in
the equations (90) and (92) to replace:
(144)
p = q,
(145)
k = 0,
with other words,
when the point M is in a continous invariant mode of movement, the consumed
energy by the point M (the energetic balance of the system “source – point)
presents a vector in the one dimensional linear energetic space E(1).
Proof:
The respect of the condition (144) demands the consumed by the
point M restful (deformation) energy ER from the equation (106) to
be equal to zero and the respect of the condition (145) – to be equal to zero
the reactive resistance σm and σl (see the
equations (116) and (119) and the condititon for the 0-th harmonious in the
theorem T25). With other words, the mass of the point M is inertial ignored and
the middle of its movement is non elastic. From that follows that the point M
don’t consumes also and reactive energy EQ (see the equation (92) in
the theorem T25) and – like consequence – according equation (111) the three
dimensional linear energetic space E(3) is transformed in one dimensional linear
energetic space E(1), because the last two members will be zero. Exepting all
it, the respect of the condition (145) signifies physically that the velocity
v(t) of the point M according the equation (36) is a non periodic value,
because it is independent from the frequency Ω. It will be very good
visible if in the equation (36) formal we replace the respected condition
(145). That will be:
(146) v(t) = 2vo = V = const.
f(t) = 2fo = F = const.
With other words,
the point M move with a constant in the time t velocity, that generate a
constant in the time resistant force. As soon as it is so, the concept phase
loses its sense. Something more, the condition (145) formal makes equal to zero
the imaginary part in the equations (36), that brings to avalanche
consequences: the beginning phases in the equations (43) to be equal to zero
and from them – the phase difference in equation (138) also be equal to zero.
In sich case, the consumed from the point M energy will be defined by the
equation:
(147)
Å = VFt,
when V and F are
the effective values of the velocity v(t) and of the resistant force f(t) of
the point M calculated in the equations (138) by:
(148) Ω = 0, δ = 0 and θ = 0.
The theorem is
proved.
Theorem
T43 (space E(1). The consumed from the point M energy (the
energetic balance of the system “source – point”) presents a vector in one
dimensional space E(1) also and by sinous invariant mode of its movement, by the
condition:
(149) σm = σl,
(see the equations
(116) and (119)).
Proof:
By respect of the condition (149) the complex imaginary number:
(150)
iσ = iσm + 1/iσl = i(σm
- 1/σl) = 0
It according the
equations (117) and (120) signifies that the single harmonious of the reactive
resistant force fr(t) of the point M (the conditions (132) and (133) remains)
will be defined by next equation:
(151) fr(t) = fa(t) - fs(t) = iσv(t) = i(σm - 1/σl)v(t)
= 0
(see the equation
(73)). Than, according the equation (75), all consumed from the point reactive
energy will be equal to zero. It comes the phenomenon resonance between the
inertial (kinetic) and elastic
(potential) energy of the point M. The point consumes only active energy for
overcoming of the friction. It is very good known real harmonic oscillator,
finding applications in the clock pendulum of Hygens, the oscilating cycle of
Marconi, setting the beginning of the radio diffusion etc.
By resonance the phase difference in the equation (142)
will be:
(152)
δ - θ = φ =
0,
because the single
in action – proportional of the velocity resistant force fv(t) of the point M
is in phase with the velocity v(t) (see the equations (36) and (94)). It
permits us to substitute (152) in the equations (135) and (136). In result all
consumed from the point M energy according the equation (134) will be defined
from the equation:
(153) Å = ÅS = St = Pt = VFt,
that is identical
with the equation (147) in the theorem T42. Over this manner the two
dimensional space E(2) is transformed in one dimensional space E(1).
The theorem is
proved.
Theorem
T44 (spaces S(2) and S(1)). By transforming of the energetic space
E(3) in the spaces E(2) and (E1) in conjunction
with them the power space S(3) is transformed in the spaces S(2) and S(1).
Proof: The spaces E(2) and E(1) are defined single sign
from its bases:
(154)
Å(2):(ÅP, ÅQ), Å(1):(ÅP).
Afrer a derivation
of the energetic bases (154) according the axiom E10 we shell obtain theirs
isomorphic bases from powers:
(155)
S(2):(P, Q), S(1):(P).
The theorem is
proved.
Definition
D13. By respect the condition:
(156)
f(t) = 0
it comes an
without energy kinematical balance. The point M has not mass or its mass is not
inertial and it moves in an ideal middle without resistance. Over it actuates
the velocity of the source vg(t) that is accepted without wastes and is
transformed in v(t). The system “source – point” is kinematic invariant, if the
effective value V of the velocity of the point M is constant. With other words:
(157)
vg(t) = v(t) = V = const.
Definition
D14. By respect of the condition:
(158)
v(t) = 0
it comes an without energy static balance. The point M is in without motion. Over it actuates the external force of the source fg(t), that is balanced
by the internal tensions of the matter concentrated in the point. The
resultante of these tensions is the force f(t). The system “source – point” is staticaly invariant, if the effective value F of the resistant force f(t) of the point M is a constant. With other
words:
(159)
fg(t) = f(t) = F = const.
Definition
D15. By coincidently respect of the conditions (156) and (158) it comes
a full energetic or non energetic insulation of the point M from the source.
The system “source – point” is destructed. The point M is in an conservative
invariability if its internal energetic exchange don’t leads to any structural
changes (crystalization, polarization, chemical reaction, nuclear destruction
etc.) of the concentrated internal matter.
Definition
D16. By respect of the conditions:
(160)
μ = const, σm = const è σl = const,
with other words,
by a constant mass of the point M, that moves in a homogeneous and isotropical
middle, the system “source – point” is linear. By non respect of the conditions
(160) the system is non linear.
Theorem
T45. Invariant mode exists than and only than, when the system is
linear.
Proof:
Let it makes the complex number:
(161)
Γ = μ + i(σm - σl).
Its module:
(162) |Γ| = (μ2 + (σm
- σl)2)1/2
will be constant
when is respectful the condition (160). By this condition it will be constant
also and its argument:
(163)
δ = arc tg((σm - σl)/μ)
Over again by this
condition will be constant the effective value of every k-th harmonious fk(t)
of the resistant force f(t), because according the equations (94), (117) and
(120) follows that:
(164)
fk(t) = fvk(t) + i(fak(t) - fsk(t))
= Γvk(t)
The value Γ
has the quality of resistance according the equations (162) and (164). It is
measured in Ns/m. Let like in the electrical engineering it be named impedance
and the difference σm - σl that also has the
quality of resistance – reactance.
If according the
equations (36), (47) and (49) we calculate for invariant mode of the point M
the effective value F of its resistant force f(t), it will be:
∞ ∞
(165)
F = (d/dt<f(t),f(t)>)1/2 = (Σ|fk|2)1/2 = (Σ|Fk|2)1/2
-∞
o
With other words,
the constant impedance can bring to constant effectives values of the
harmonious and over there – and of the entire resistant force f(t). It, from
its side, according the equation (105) makes possible to be constant the full
energy consumed from the point M. Or, with other words, the energetic balance
of the system “source – point” can be constant, if (see the equation (164)) the
effective value V of the velocity v(t) of the point M, defined by the equation:
∞ ∞
(166) V = (d/dt<v(t),v(t)>)1/2 = (Σ|vk|2)1/2 = (Σ|Vk|2)1/2
-∞ o
is constant. Contrariwise, according the theorem T3 the constant energetic flow Åg(i) of the source leads to constant energetic
consummation Ec(i) of the point M, and from there to constant effective value V
of its velocity v(t), when the system “source – point” is invariant.
The theorem is proved.
DUALITY OF THE ENERGETIC ALGEBRA. UNIVERSALITY
OF THE THEORY OF INVARIANT SYSTEMS
Definition
D17. The linear vector
space:
(167)
L(i): (l1, l2,...,li)
is dual of the
space:
(168)
M(j): (m1, m2,...,mj),
when the vectors l1,
l2,...,li and m1, m2,...,mj
makes orthogonal bases in the spaces L(i) and M(j), if every vector L from the space L(i) can be
transformed linear in the vector M in the space M(j). With other words, there
exists the constant β ≠ 0, such that:
(169)
L = βÌ
When β >
0, the spaces are covariant dual and when β < 0 – contra variant dual.
Theorem
T46. The space L(i) is dual of the space M(j), if its dimensions:
(170)
dim(L(i)) = i = j = dim(Ì(j)).
Proof:
The respectful of the condition (169) wants a linearity between
the modules and the arguments of the vectors L and M. With other words, it is
necessary the modules to be subordinated of the condition:
(171) |L|
= (l12+ l22+...+li2)1/2
=
= β (m12+
m22+...+mi2)1/2= β |Ì|,
and defining the
arguments single vectors ei and ej to be subordinated of
the conditions:
(172)
ei = li/|li| = βmj/β|mj|
= ej or
(173)
eiej = 1
The respectful of
the conditions (171), (172) or (173) is possible only by respectful of the
condition (170).
The theorem is
proved.
Theorem
T47. The system “source – point M” is invariant only then, when the
energetic spaces Eg(i) of the source and Ec(j) are contra variant dual.
Proof:
According the theorem T3 it is necessary every energetic vector Ec
from the space Ec(j) to be equal and opposite to every energetic vector Eg from
the space Eg(i). With other words it is necessary:
(174)
Åc + Åg = 0
And it is
possible, when the spaces Åc(i) and
Åg(i) are according the equation (178) contra variant dual. By this condition
for respectful of the equation (174) is necessary the coefficient of duality:
(175)
β = -1
so that:
(176)
Åg = βÅc = -Åc
The theorem is
proved.
Theorem
T48. The system “source – point M” is invariant only then, when the
energetic spaces Eg(i) of the source and Ec(j) are with equal dimensions in the
transitive and in the invariant mode of movement of the point.
Proof:
According the theorem T47 for existence of invariability it is
necessary a duality between the energetic spaces Eg(i) and Ec(j) and according
the theorem T46 the duality between the energetic spaces Eg(i) and Ec(j) exists
when is respected the condition (170). With other words, when:
(177)
dim(Åg(i)) = i = j = dim(Åc(j))
The theorem is
proved.
Theorem
T49. The system “source – point M” is invariant only then, when the
power spaces Sg(i) of the source and Sc(j)
of the consumed power point M are with equal dimensions in the
transitive and in the invariant mode of movement of the point.
Proof:
According the theorem T39 the spaces Sg(i) and Sc(j) are
isomorphical with the spaces Åg(i) and
Åc(j). From it follows that if we derivate over the time t the energetic spaces
in the equation (177), its dimensions will be not changed. Then will be true
the equation:
(178)
dim(Sg(i)) = i = j = dim(Sc(i))
The theorem is
proved.
Theorem
T50. The impedance in the equation (170) presents a vector in the two
dimensional linear space Γ(2).
Proof:
According the equation (171) the square form:
(179)
|Γ|2 =μ2 + (σm - σl)2
has a bylinear
form equal to zero.
According the
equations (94), (117) and (120) the values μ, σm and σl
transforms linear the k-th harmonious vk(t) of the velocity v(t) of
the point M in the harmonious fvk(t), fak(t) and fsk(t),
that have the quality of force. But the multitude of forces:
(180)
{fvk(t), fak(t) - fsk(t)}
according the
axiom E8 are adding member of a vector sum and, therefore, they makes an additive
group. Then also and the linear of
(180) multitude:
(181)
{μ, σm - σl}
will makes an
additive group. As soon as is so, the values μ and σm - σl
will makes an orthogonal base in a two dimensional vector space. Let name it
Γ(2).
The theorem is
proved.
Theorem
T51. In transitive mode of movement of the point M the two dimensional
linear space F(2) of its resistant forces is covariant dual of the impedance
space Γ(2).
Proof:
According the definitions D10, D11 and D12 the impedance multitude
(181) characterize completely the object consuming energy and according the theorem
T45 this object can have an invariant behavior when this multitude don’t
depends from the time t and the velocity v(t) of the point M. In an invariant
mode, when according the equation (36) the velocity of the point M has a
constant effective value, because it depends from the constant frequency
kΩ, it is so. But in a transitive mode, when according the equation (33)
the velocity of the point M depend from the changing in infinite limits
frequency Ω, the variable effective value of this velocity leads according
the equations (116), (119) and (170) to the variable in the time t impedance:
(182)
Γ = μ + i(σm - σl) = μ + i(mΩ - 1/ÑlΩ),
because the
frequency Ω changes simultaneously
with the time t. Independently from it, the square form:
(183)
|Γ|2 = μ2 + (σm - σl)2
will have a
bylinear form equal to zero. According the equations (59) and (60) it signifies
that the impedance space Γ(2) by the equations:
(184)
F(iΩ) = ΓV(iΩ) = μV(iΩ) + i(mΩ-1/ÑlΩ)V(iΩ)
is transformed in
the force space F(2). According the theorem T46 and the equal algebraic signs
of μ, σm and σl
in the equations (182) and (184) these spaces are covariant dual.
The theorem is
proved.
Definition
D18. Energetic algebra is the multitude of operations:
(185)
Ω: (∙Γ, <v,f>),
when ∙Γ
presents a multiplication over an impedance and <v,f> - the scalar
produce between the velocity v(t) of the point M and its resistant force f(t),
that transforms yhe multitude of velocities and impedances:
(186)
À: (v, Γ)
in the energetic
multitude:
(187)
ÅS: (ÅP, ÅQ) èëè ÅS: (ÅP, ÅQ,
ÅR)
Or, with other
words, energetic algebra Ω is the composition from operations:
(188)
Ω = ∙Γ ○ <v,f>
in the diagram:
Ω
(189)
À: (v, μ, σm, σl)--->ÅS
Theorem
T52. Independently from the physical quality of the energy (mechanical,
warmth, electrical, magnetic, electro – magnetic, chemical, nuclear etc.) all
energetic equations, describing consummation or generation of energy, that are
based on an axiomatic dual of the axiomatic E1, E2,…,E14, are dual covariant of
the possible energetic equations, describing the consummation or generation of
energy in the system “energetic source – material point”.
Proof:
According the axiomatic E1, E2,…,E14 was an energetic algebra
Ω, with its aid are defined the equations, calculating the energetic
balance of the system “source – material point M”. The behaviors of these
equations was defined from the theorems T5, T6,…,T49. With their aid was
explored in the time all possible energetic states of the mentioned system.
There was made a
difference between two characteristic modes of behavior of the system –
transitive and invariant, that are described by the behavior of the function
v(t) of the velocity of the point M. This behavior depend from the parameters
(in this case, the mass m) of the point and of the middle in that it moves (in
this case, its viscosity μ and its elasticity Cl).
The entire
energetic theory of the system “source -
material point” is described shortly from the diagram (189). According this
diagram the performance of the consuming energy object – the multitude A
defines isomorphicaly by the algebra Ω its energetic necessity ES,
with other words, the minimal energy that must possess the source, for the
possibility of existence of the system. And the algebra Ω is isomorphical,
because the elements of the multitudes A and ES are constant or
single sign functions of the time t.
Here is the
energetic theory of the possible most elementary mechanical system, with other
words, of the most elementary energetic process – the process of energetic
consummation of material point. Every other energetic process will be described
from mathematically sophisticated theory illustrated from the diagram:
α
(190)
Â:(w, Õ)--->ÅS
Here B is the
characteristic multitude of the consuming energy object, that is composed by
the function:
(191)
w = w(t),
that defines its
behavior and the function:
(192) Õ = {õ1, õ2,...,õi},
that defines the
internal (proper) and external (of the ambient middle) parameters. With α
we mark the algebra that from the characteristic multitude B defines the
multitude ES of the energetic necessity of the object.
The algebra α
will be isomorphical, because the new axiomatic – let name it G1, G2,...,Gj – that will subordinates the
multitude B, will demands its elements to be single sign functions of the time
t. Other air functions of the time in the nature don’t exists.
The elements of
the multitude ES will be same like in the diagram (189), because
every energy is measured in Joules (J) independently from the nature
(mechanical, warmth, electrical, magnetic, electro – magnetic, chemical,
nuclear etc.) of the energetic process.
As soon as the
energetic measure remains the same, it follows that exists the composition:
(193)
Ω = α ○ β-1
such that:
β Ω
(194)
Â--->À--->ÅS
But the multitudes
ES are covariant dual, if we interpret the definition D17 only from
the consuming side of the possible energetic spaces. Then also and the diagram:
β Ω
(195)
Â--->À--->ÅS
α
Â-------->ÅS
will be dual
(covariant or contra variant) of all possible energetic diagrams making a functor
with them. Dual will be also and the axiomatic E1, E2,…,E14 of all possible
energetic axiomatic, as and the theory T5, T6,…,T49 of all possible energetic
theories.
The theorem is
proved.
Theorem
T53. The axiomatic of the invariant systems A1, A2,…,A7 as well as the
theory of invariant systems T1, T2,…,T4 are dual of every other axiomatic or
theory of the invariability.
Proof:
The axioms A1, A2,…,A7 stands the base of any production process,
whose product x according the theorems T1, T2,…,T4 depend only from the
energetic flow Ek that realy is consumed by the object.
According the
theorem T52 the energetic flow Ek is calculated over equations dual
between them. From other side the same flow Ek according the theorem
T3 is defined isomorphicaly by the behavior of the product x, with other words,
by the function:
(196)
õ = õ(t)
(see the diagram
(3)), as for existence of invariability according the theory T5, T6,…,T49 it is
necessary also and a constancy of the parameters of the object.
Then the theorem
T52 permits us to compose the isomorphical diagram:
α
(197)
Ñ:{õ(t), Õ}--->Åê(t)
when with α
we mark the energetic algebra transforming the performance of the object in its
minimal energetic necessity, that the source must always ensure. Here X is the
multitude:
(198)
Õ = {õ1, õ2,...,õi}
from parametes of
the object that already we accepted they are constant.
The diagram (197)
will be dual (covariant or contra variant) of all possible system diagrams,
making a functor with them, because the isomorphical algebra leads to the dual
of all possible energetic multitudes - the multitude Ek. Dual will
be also and the axiomatic A1, A2,…,A7 of all possible axiomatic as and the
theory T1, T2,…,T4 of all possible theory.
The theorem is
proved.
With the
proof of the theorem T53 it bring to an end the exposition of the theorem of the
invariant systems or, briefly said, the theory of the invariability. It is a
classical (non relative) theory and it is valid everywhere, when we can speak
for a movement dual from algebraic viewpoint of the movement in the mechanics
of
SOME INVARIANT
PROPERTIES OF THE MATTER
This part of the
scientific work includes in it self applications of the theory of the invariant
systems, that presents it like a viewpoint to some properties of the matter,
independently from the manner of its scientific observation – in laboratory or
in the nature. In both cases it will be make an interpretation of properties of
the matter that are not a result from a technologic process and, in this sense,
they exists objectively and independently from the human. The technologic
process helps only for its discovery and exploration.
The applications
presents well known but look trough the prism of the theory of the
invariability examples from the physics. Over this manner there aim to outline
the useful area for applications of the theory like a scientific approach by
exploration of physical phenomena.
Before to expose
the examples for applications it is necessary to lead in the concept invariant transitive mode. Its existence
is prompted by the theorem T4, in that are mentioned systems by which the
process moves over a previous setting variable line in the time. But in the
theorem T4 a transition is not mentioned. Let make the next:
Definition
D19. Invariant transitive process (mode) is this that pass over a line
changing in the time t from the beginning (t = 0) to the end (t →
∞) of the transition always over an identical manner.
Theorem
T54. Invariant transitive mode is possible only then when the impedance
Γ of the system (see the equation (161)) depends only from the frequency
Ω (see the equations (116) and (119)), with other words, the system must
be linear.
Proof:
The mentioned equations (116) and (119) relates to a constant
(invariant) mode, by that the frequency Ω is accepted constant.
If in the
equations (116) and (119) we substitute k = 1 and accept that the frequency
Ω º (-∞, ∞), the impedance Γ from the equation (161) will
be related for a transitive mode. And if the mass m from the equation (116) and
the module of the elasticity 1/Cl from the equation (121) are
constant, the impedance depend only from the frequency Ω.
The system
according the definition D16 is linear and the velocity v(t) will be derived
and integrated linearly over the time t (see the equations (52) and (54)) and
it according the theorem T21 signifies that by an identical spectrum V(iΩ)
of the velocity will exists an identical spectrum F(iΩ) of the force, with
other words, an invariant transitive velocity v(t) will generates an invariant
transitive resistive force f(t).
The theorem is
proved.
Example
Ex 1 (gravitation). The system “Earth – gravitated body” is invariant about the values velocity of the
body in relation to its mass, weight – in relation to the time of gravitation
and percussion force by the contact of the body with the Earth – in relation
the time or the altitude of gravitation (incidence).
Proof:
On the figure 6 is represented the system “Earth – gravitated body”
in the state, when the corps is over the Earth on altitude h = ho and its
incidence is forthcoming and, when is on altitude h = 0 and already is
incident.
Figure 6.
In both cases it
is in a static balance so that on the force G of its weight is opposed the
support reaction FG and it remains immovable. And because the body
is immovable, its consumed from the Earth (kinetic) energy is
(199)
Åc = <v,G> = <v,FG> = 0.
It is equal to
zero, because the velocity v in both cases is zero. The static balance of the body
is invariant, because all value in the equation (199) are constant. This case
is not interesting, because it make a model of non producing process. Let see
the case when the corps is free from the upper support and it is on altitude h,
according the inequality:
(200)
ho ≥ h ≥ 0.
Let accept that
the distance from the altitude ho to the Earth when h = 0 is traversed for the time
to, in other words, according the inequality:
(201) 0 ≤ t ≤ to.
We shell introduce
the next axioms that are true by definite conditions proved from the physics.
Axiom
G1: There exists at least one body with constant mass m (kg).
Axiom
G2: The Earth attracts the corps from the altitude ho to the altitude
0 with a constant acceleration g (m/s2).
Axiom
G3: By the attraction the corps will move to the Earth with an ignored
resistance of the air and without any wind.
Axiom
G4: The corps will be accelerated (opposed of the Universe inertia) by
the force of the contra inertia G = mg (N).
Axiom
G5: The acceleration of the Earth g never and nowhere when there is
an action of the Earth gravitation can’t be isolated or behind a screen placed.
According the
axioms G1,…,G5 we will proof the next help auxiliary theorems:
Theorem
LG1. The Fourier picture Fa(iΩ) of the exercised over
the body in the interval (0, to) acceleration g is:
(202)
Fa(iΩ) = g(exp(-iΩt) -
1)/(-iΩ∙√2π).
Proof:
From the definition for the Fourier picture (see the equations
(34)) the Fourier picture of the function:
(203)
a(t) = g
will be:
∞
⌠
(204) Fa(iΩ) = (1/√2π)∙⌡g(exp(-iΩt)dt
-∞
The solution of
the integral (204) will result to the equation (202).
The theorem is
proved.
Theorem
LG2. Beginning its flight to the Earth in the moment to the body will
acquires the velocity v(t) with the effective value according the equation:
(205)
v(t) = V = gto,
with other words,
the body has make a transitive process (it moved in a transitive mode).
Proof:
According the theorem T6 the velocity v(t) of the body will be:
to
⌠
(206)
v(t) = ⌡gdt
o
The solution of
the integral (206) will result to the equation (205). From other side, if over
the rule (31) we calculate the effective value V of the velocity v(t) of the body
it also will resulte to the equation (205). That signifies the effective value
V is a variable in the time t. The body moved in a transitive mode.
The theorem is
proved.
Theorem
LG3. The Fourier image Fv(iΩ) of the velocity
v(t) of the gravitated in the interval (0, to) body is:
(207)
Fv(iΩ) = g(exp(-iΩt) - 1)/(-iΩ∙√2π∙iΩ)
Proof:
As soon as the velocity v(t) of the body presents the integral of
its acceleration a(t), depending on the equations (58) and (59) from the
theorem T21, that proves two specialy
feature behaviors of the Fourier image, we arrive to the conclusion
that:
(208)
Fv(iΩ) = Fa(iΩ)/(iΩ)
Substituting the
expression for Fa(iΩ) from the equation (202) in the equation
(208) we shell result exactly the equation (207).
The theorem is
proved.
Theorem
LG4. The movement of the body in the time interval (0, to) or equally
in the altitude interval (ho, 0) is made in an anti inertial middle,
characteristic with the impedance:
(209)
Γ = σm =imΩ
Proof:
According the axiom G4 the body will be accelerated by the Earth
with the force of its weight G = mg. According the Fourier image Fa(iΩ)
of the acceleration g over the equation (202), the Fourier image FG(iΩ)
of the this force will be:
(210)
FG(iΩ) = mFa(iΩ) = mg(exp(-(iΩ)t) -
1)/(- iΩ∙√2 π)
The expression
(210), however, presents the produce of the Fourier image Fv(iΩ)
of the velocity v(t) of the body from the equation (209) and the impedance from
the equation (209). That signifies that the middle of the gravitational field
in that move the body is characterized single sign and invariant (see the axiom
G1) by the impedance from the equation (209). It will be valid the substituting
electrical scheme on the figure 7.
Figure 7.
The theorem is
proved.
Theorem
LG5. In the moment to or in altitude h = 0 the body will strike with
energy:
(211)
ÅS = mg2to2/2
and power:
(212)
S = d/dt(ÅS) = d/dt(mg2t2/2) = mgto
t = to
Proof:
According the equations (207) and (209) the energetic system “Earth –
gravitated body” will be characterized by the function:
Ώ
(213)
À: (Fv(iΩ), Γ
= σm = imΩ )--->ÅS,
when A is the
characteristic multitude of the system, composed by the Fourier image Fv(iΩ)
of the velocity v(t) of the body and the impedance Γ, that it meets in its
transitive movement. The letter Ώ over the arrow is different over sense
from the frequency Ω in the bracket of the characteristic multitude A.
Over the arrow Ώ has the sense of the defined by the equation (188)
energetic algebra and presents the composition:
(214)
Ώ = ∙Γ ◦ <Fv(iΩ),FG(iΩ)>
With other words,
the invariants of the system m and g completely describes its characteristic
multitude (213).
Let now we
calculate the energetic balance of the gravitated body. It is:
∞
⌠
(215) <Fv(iΩ),FG(iΩ)>
= ⌡[mg(exp(-iΩto)-1/(-iΩ∙√2π)∙
-∞
_____________________
∙g(exp(-iΩto)-1)/( Ω2-√2π)]dΩ
=
∞
⌠
= [(mig2)/π]∙⌡[(1-cosΩto)/Ω3]dΩ
=
-∞
∞
⌠
=2[(mig2)/π]∙⌡[(1-expΩto)/Ω3]dΩ
=
-∞
= mg2to2/2 = ÅS
The end result
from the equation (215) coincide with the assertion (211). If we it derivate
over the time t we shell achieve the assertion (212).
The theorem is
proved.
Theorem
LG6. The moment to the strike of the body with the Earth is defined by
the altitude ho over the equation:
(216)
to = √2gho
Proof:
If following the logics of the diagrams (10) and (11) – see the
theorem T5 – we integrate over the time t the equation (205), we shell
obtain:
to
⌠
(217) s(t) = ⌡gt∙dt = gto2/2
= ho
o
Transforming the
equation (217) like a evident function of the time t we shell obtain the
equation (216).
The theorem is
proved.
Proof
of the example (the systematic invariabilities): Let present the equations (206) and
(205) by a characteristic functional diagram similar of the diagram (213). It
will have the air:
Ώ
(216)
Â: (t,g)--->v,
when the
Ώ-algebra presents the scalar produce:
t
⌠
(217)
Ώ = <1,g> = ⌡gdt
o
The characteristic
multitude B of the system according the diagram (216) is independent from the
mass m of the body and according the equation (217) – also and its
Ώ-algebra. Therefore, the multitudes (216) and (217) are compesed by
invariants toward the mass of the corps and its gravitation will be
according the equation (205) with a
constant acceleration. The velocity v(t) of the gravitation is dependent only
from the prolongation of the downfall
to.
According the
axiom G4 the weight G of the body is defined over the equation:
(218)
G = mg
or its
characteristic representation will be:
Ώ
(219)
Ñ: (m,g)--->G
Here the
Ώ-algebra is composed only from the operation multiplication over the
acceleration g. With other words,
(220)
Ώ = ∙g
The characteristic
multitude C of the system according th expression (219) is independent from the
time t and according the equation (217) – also from the time t - is independent and its Ώ-algebra.
Therefore the multitudes (216) and (217) are composed by invariants toward the
time of the downfall to. From that follows that and the strike force over the
Earth also will be invariant toward him. And as soon as it is invariant toward
the time to, it will be invariant also toward the altitude ho of the downfall
of the body.
The transition of
the body from the altitude ho to the Earth surface (the altitude h = 0)
according the theorem T54 is invariant over all indexes of the primary
assertion.
The example is
proved.
It is very well-known the fact
that the proof of the example Ex 1. has its start from the experiments of Galilei
before more of 400 years. That, however, entirely not be obstructed of
The equation (211) compared with the equation (217) speaks that the
strike energy of the body over the Earth is proportional of the height of its
downfall and the equation (212) compared with the equation (205) – that the
power of this strike depends of the velocity of this downfall. And if we repeat
the experiments of Galilei from before 400 years let to fall stones from a
constant altitude (for example, the altitude of the Tower in
The energy of the gravitation Earth field, so we shell
see in the next part, can be utilized for protective needs, because according
the axiom G5 the gravitation field cannot be isolated or behind
a screen placed. The gravitation operative energy exists always and everywhere.
Example
Ex 2 (electrical
conductivity). The system “source – electrical consumer” is
invariant in non transitive mode about the values: currant, voltage, power and
energy by a stable source (its voltage is changed only by desire of person) and
constant thermal exchange between the electrical circuit and the environment.
Proof: Let we lean
on the next axioms:
Axiom AC1. There exists metals and metal alloy sheet with a constant vertical
section.
Axiom AC2. The valeni electrons of the metals and the metal alloy are free and
moves chaotically and every from them traverses by 20o C measured in
meters road:
(225) L = 1/T,
when T is the
absolute temperature to then is heated the metal. For different metals this
road is different.
Axiom AC3. The time τ of the road L of the free valeni electron in the metal
(the free road) we name period of the electron, that is defined by the
equation:
(226) τ = L/vT,
when vT is the medial velocity with that
the electron traverses its free road. For different metals this velocity
is different.
Axiom AC4. The chaotic movement of the free electrons don’t makes generation or
consumption of energy.
Axiom AC5. Every electron, free or not, possess a constant material mechanical mass
m, measured in kg and a constant field electrical mass e, measured in Coulombs
(Q), presenting the common name of Ampere ∙seconds (As).
Axiom AC6. There exists energetic source that is capable to transform the chaotic
movement of the free electrons in one direction movement over equidistant
trajectories.
Axiom AC7. The trajectories over that can moves the electrons presents lines of a
space distribution of the energy of the source, but they are named not
energetic, but – lines of force.
Axiom AC8. The energy of the
source has a field form of its existence and the performances of its lines of force defines with synonymous
and fully its energetic field.
Axiom AC9. The source possess
two poles: positive (anode), that attracts the free electrons and negative
(cathode), that they repulses.
Axiom AC10. If we connect the
two poles – the anode and the cathode – of the source with the sheet metal or
metal alloy it will exercise over every free electron the force:
(227) Få = eE,
when with E we mark the intensity of the
emitted from the source energetic field, measured in W/Am.
Axiom AC11. The intensity E of
the field of the source is constant and invariant about the behaviors of the
metal or metal alloy.
Axiom AC12. Against the
generated from the source force Fe the free electron opposes the motive
inertial force:
(228)
Fà = ma,
when with a we
mark the constant acceleration that will obtain the electron under the action
of the force Fe.
And now let we prove the next help
theorems:
Theorem TC1:
The acceleration that will obtain the electron under the action of the
force Fe will be:
(230)
à = åÅ/m
Proof: If we accept that
under of the action of the field of the source the electron moves over someone
line of force, it will be in a dynamical balance. It signifies that the
exercised over him according the equation (227) force of the source Fe and the
inertial motive force Fa (see the equation (228)) are equal. Then if we make
equal the two sides of the equations (227) and (228) we shell achieve the
equation (230).
The theorem is proved.
Theorem TC2. In the interval of
the moment 0 of the switching of the source to the moment t the electron will
achieve the velocity:
t
⌠
(231)
vÅ = ⌡(åÅ/m)dt = (åÅ/m)t
o
Proof:
After a direct integration over the time t of the acceleration
from the equation (230) we shell achieve the equation (231).
The theorem is
proved.
Theorem TC3. In the interval of the moment 0 of the
switching of the source to the moment t the electron will achieve the road:
t
⌠
(232)
h = ⌡vÅdt = eÅt2/2m
o
Proof:
After a direct integration over the time t of the velocity from
the equation (231) we shell achieve the equation (232).
The theorem is proved.
Theorem TC4. The time through that the electron possess
the full probability to be in movement is the time of its free road τ (see
the equation (226) of the axiom AC3).
Proof:
The time of the free road τ of the electron in the equation
(226) is invariant toward the intensity E of the field of the source.
Therefore, the free electron can’t move more for long from its period τ.
The theorem is
proved.
Theorem TC5. For the time τ of its free road the
accelerated from the field of the source electron will moves with a middle
velocity:
(233)
v = h/τ = eÅτ/2m
Proof:
Substituting in the right part of the equation (232) the time t
with the period τ of the free electron and executing the intermediate
action in the equation (233) we shell achieve the end result of this equation.
The theorem is
proved.
Theorem TC6. In the interval of the moment 0 of the
switching of the source to the moment t by middle velocity from the equation
(233) the electron will acquire (transfer) the electrical energy:
t t
⌠ ⌠
(234)
Wå = ⌡v∙Fådt =
⌡(eÅτ∙eÅ/2m)dt.
o o
Proof:
The scalar multiplication of the velocity of the electron that
according the equation (233) it acquired from the field of the source over the
force exercised over him according the equation (227) will bring exactly to the
equation (234).
The theorem is
proved.
Theorem TC7. In the interval of the moment 0 of the
switching of the source to the moment t the volumetric density dW/dV of the
transferred in the metal energy will be:
t t
⌠
⌠
(235)
dW/dV = nWå =⌡(ne2Å2τ/2m)dt =
⌡Γ∙Å2d t,
o o
when with n we
mark the number of electrons in an unity volume from the metal and the value
(236)
Γ = ne2τ/2m,
we name specific
electrical conductivity of the metal, because from it depends the value of the
transferred through him energy. The exact physical sense of Γ will be
shown down below in the text. The metal will be named electrical conductor or,
shortly, conductor.
Proof:
Multiplying with n the both parts of the equation (234) we shell
achieve the both intermediate results from the equation (236) and substituting ne2τ/2m
equal to Γ, we shell achieve the end result from the same equation. From
other side, the produce nWe has the quality of energy in unity volume (W/m3) and if with
the character V we mark the volume of entire conductor, it is true that in the
left part of the equation (235) we have the right to write the dW/dV.
The theorem is
proved.
Theorem
TC8. The electrical mass (charge) of the entire conductor is:
(237)
q =
when
with l we mark the entire length of the conductor, measured in meters (m) and
with s – the area of the vertical section, measured in m2.
Proof: The produce ne has
a quality of specific volumetric electrical mass of the conductor. Therefore,
the produce
The
theorem is proved.
Theorem
TC9. The velocity i of the transfered through the conductor electrical
mass we name electrical current or shortly – current, that is measured in
Amperes (A) and it is equal to:
(238)
i = dq/dt = nes∙dl/dt = nesv.
Proof:
The bilateral derivation over the time t of the equation (237)
will bring to the result (238)
The theorem is
proved.
Theorem
TC10. The density δ of the current lines – the lines of force of
the field of the source, that will be named in future electrical source – over
that in the conductor moves the entire electronic flow is:
(239)
δ = i/s =
Proof:
The quotient i/s has the quality of a density that is measured in
A/m2. Therefore, if we divide on s the right parts of the equations
(238) and (236) we shell reach to the equation (239).
The theorem is
proved.
Theorem
TC11. The density δ of the current lines in the conductor is
defined from the value:
(240)
Γ = ne2τ/2m,
that we have name
in the theorem TC7 specific electrical conductivity or shortly – conductivity,
over the equation:
(241)
δ = ΓÅ
Proof:
Substituting with Γ in the equation (239) the expression ne2τ/2m,
we shell achieve the equation (241), that presents the low of Ohm in derivative
air.
The theorem is
proved.
Theorem
TC12. The conductivity Γ of the conductor is invariant toward the
field (the intensity E of the field) of the source.
Proof:
In the definition for the concept conductivity in the equation
(240) the intensity E of the field don’t participate. From other side,
according the axiom AC11 the intensity E of the source is invariant from the
behaviors of the metal – named also conductor. And it sign that the field of
the source don’t changes the conductivity of the conductor.
The theorem is
proved.
Theorem
TC 13. The value σ, that is reciprocal of the specific conductivity
Γ according the equation:
(242)
σ = 1/Γ
and that we name
specific electrical resistance of the conductor or shortly - specific
resistance is invariant toward field (the intensity E of the field) of the
source.
Proof:
As soon as in the right part of the equation (242) the value
Γ is invariant toward field (the intensity E of the field) of the source,
that signifies that also and its reciprocal value σ from the equation
(242) is invariant toward the same value E.
The theorem is
proved.
Theorem
TC14. The entire transferred by the conductor energy W is defined by the
equation:
t t
⌠ ⌠
(243) W = ⌡δÅsl∙dt = ⌡iudt,
o o
when with u we
mark the value:
(244)
u = Ål,
that is
named fall of voltage made by the conductor or voltage over the conductor. The
measuring unit for the voltage are Volts (V) like for the potential U of the
electrical source (see the axiom AC13).
Proof: If the end result
in the equation (235) is marked with the character A, it will be that:
t
⌠
(245)
À = ⌡ΓÅ2dt
o
From other side,
according same equation, the energy transferred trough the entire volume of the
conductor (the entire conductor) will be:
V l
⌠ ⌠
(246)
W = ⌡ÀdV = ⌡Àsdl = Àsl
o o
Substituting in
the end result from the equation (246) A from the equation (245) and, expecting
the conductivity δ of the conductor from the equation (241) we shell
achieve the intermediate result from the equation (243). And if in the same
intermediate result from the same equation we expect also the new value u from
the equation (244), we shell achieve the end result from the equation (243).
The value u from
the equation (244), according the axiom AC10, is measured in Watts/Ampere (W/A)
or shortly – in Volts (V).
The theorem is
proved.
Theorem
TC15. The entire transferred trough the conductor energy W is defined over
the equation:
t t
⌠ ⌠
(247)
W = ⌡i2Rdt =
⌡(u2/R)dt,
o o
when the value R,
defined over4 the equation:
(248)
R = u/i = σl/s,
presenting the low
of Ohm, we name resistance (impedance) of the conductor. It is measured in
Volts/Ampere or shortly – in Ohms (Ω).
Proof:
If according the trues (239), (241) and (242) the intermediate
result of the equation (243) can be equivalently transformed: t t t
⌠ ⌠ ⌠
(249)
W = ⌡δÅsldt = ⌡(δ2sl/Γ)dt =
⌡δ2s2(ål/s)dt =
o o o
t
⌠
=⌡i2Rdt =
<i,iR>
o
If according the
trues (241), (242) and (244), the same equation can be transformed in such a
way:
t t t
⌠ ⌠ ⌠
(250)
W = ⌡δÅsldt = õ⌡(Å2sl/Γ)dt =⌡Å2l2(s/σl)dt
=
o o o
t
⌠
= ⌡(u2/R)dt
= <u,u/R>
o
The theorem is
proved.
Here we shell
mark, that as soon as the value R – the resistance of the conductor – is measured
in Ohms (Ω), the value σ – the specific resistance of the conductor –
according the equation (248) must be measured in Ohms∙meter (Ωm).
From other side, the reciprocal value of the resistance, that we name
conductivity of the conductor:
(251) G = 1/R
is measured with a
measure named Siemens (S). After a transforming of the equation (250),
according the conditions (242) and (251), it is true that the entire
transferred trough the conductor energy W is equally defined also over the
equation:
t t
⌠ ⌠
(252)
W = ⌡(u2G)dt = <u,uG> =⌡(u2(s/σl)dt,
o o
from that it
follows that the conductivity G of the conductor is equal to:
(253)
G = s/σl = sΓ/l
According the
equation (253), the specific conductivity Γ of the conductor must be
measured in Siemens/meter (S/m).
Ώ-
and α-algebra. Existence and applicatios .
Theorem
TC16. There exists δ-algebra describing the transformation of the
thermal energy toward the conductor (the absolute temperature of the conductor)
in mechanical energy of the chaotic movement of its free electrons possessing
the velocity vT and the period τ.
Proof:
Surveing the equations (225) and (226), we arrive to the
conclusion that is real the diagram from consecutive transformations:
∙T-1 ∙1/vT
(254)
T------>L------->τ
That sign that the
composition:
(255)
δ = ∙T-1 ◦
∙1/vT
makes the
transformation of the characteristic multitude X in the multitude τ over
the diagram:
δ
(256)
Õ:{T, vT}--->τ
The elements of
the composition δ makes a multiplicative algebraic group. Let name it
δ-algebra.
The theorem is
proved.
Theorem
TC17. There exists the invariants:
- Γ-algebra
describing the transformation of the energy of the source in electrical force
Fe exercised over a free electron in the conductor over the diagram:
Γ
(257)
G:{e,E}---->Fe, Γ = ∙Å
(see the equation
(227));
- π-algebra
describing the opposition of the mechanical mass m of the electron against its
electrical mass e by the force Fa over the diagram:
π
(258)
P:{m,a}---->Fa, π = ∙a
(see
the equation (228));
-
Σ algebra describing the dynamic balance of the electron over the diagram:
Σ
(259)
F:{Fe,Fa}---->{Fe=Fa,Fa=Fe}, Σ = “=”
- μ-algebra
describing the acceleration of the electron over the diagram:
μ
(260)
F:{Fe,Fa}---->{a,eE/m, (a=eE/m)}, μ = ∙1/m
(see
the equation (230));
- the
composite θ2-algebra describing the traversed over the action
of the field of the source free road h of the electron over the diagram:
t
θ2 ⌠
(261)
À:a---->h, θ = ⌡∙dt
o
(see the equations
(231) and (232));
- φ-algebra
describing the middle velocity v of the electron under the action of the field
of the source over the diagram:
φ
(262)
Í:{h,ç}--->v, φ = ∙1/τ
(see
the equation (233));
-
Φ-algebra describing the transferred through the conductor by one electron
energy We over the diagram:
Φ
(263)
U:{v,Få}--->Wå, Φ = < , >
(see
the equation (234));
-
ε-algebra describing the transferred trough the conductor by they all free
electrons energy W over the diagram:
ε
(264)
V:{Wå}--->W, ε = ∙nsl,
(see
the equations (237), (238),…,(243) or in the equation (243) to substitute
δ from (241) and after that – Γ from (240));
Proof: Following the
logics of the theorem TC16 the diagrams (257), (258),…,(264) makes algebraic
structures and they deserves theirs names –algebra. They are invariant because
theis characteristic multitudes X, G, P, F, A, H, U, and V are composed from
independent one towards other variables.
The
theorem is proved.
Theorem TC18. The multitude:
(267)
Ώ:{δ, Γ, π, Σ, μ, θ2, φ, Φ, ε}
makes Ώ-algebra over the sense of the definition D18. Since the elements
of Ώ don’t makes a consecutive composite sequence, let we name that air of
multitude interrupted algebra.
Proof: After a
survey of the definition D18 there arrives to the trivial conclusion that the
theorem TC18 is true.
The theorem is proved.
Theorem TC19. There exists β-algebra over the sense of the theorem T52 describing
the transformation of the velocity v of a free electron in the conductor in the
currant i flowing through him and the electrical force Fe exercised over the
electron by the field of the source in the voltage u on its poles over the
diagram:
β-1
(268)
{v, Fe}---->{i, u},
over that sort of
manner that the equation (243) remains in force.
Proof:
According the theorem TC18 the Ώ-algebra transforms the
characteristic multitudes X, G, P,
F, À, H, U and V in the energetic
multitude W over the diagram:
Ώ
(269)
Y:{X, G, P, F, À, H, U, V}--->W,
as the multitude W is presented in the air:
t
⌠
(270)
W = nsl∙<v,Få> = nsl∙⌡(eEτ/2m)∙eEdt
o
If in the equation (270) we make the multiplication under the symbol
integral and after a regrouping we substitute as in the equation (238):
(271)
vnse = (eEτ/2m)∙nse = i
and as in the
equation (244) also substitute:
(272)
Få∙(l/e) = El = u,
it will come out that the assertion (268) is true. However it follows to
accept that:
(273)
β-1 = {∙nse, ∙l/e}
The theorem is
proved.
Theorem
TC20. There exists α-algebra over the sense
of the theorem T52 presenting the composition (see the equation (193)):
(274)
α = β ◦ Ώ,
through that the
multitude from electrical values {i, u}
is transformed in the energetic multitude W over the diagram:
α
(275)
{i, u}--->W
Proof:
According the theorem TC19 the multitude from values Y with a
mechanical measure is transformed in the energetic multitude W over the diagram
(269). The same multitude trough the β-1-algebra is transformed
over the same theorem in the multitude {i, u} in an electrical measure. With
other words, we can compose the diagram:
β Ώ
(276)
{i, u}<----Y---->W.
That sign that
there exists the composition:
α
(278)
{i, u}--->W
The theorem is
proved.
Invariants
and invariabilities.
Theorem
TC21. The α-algebra is invariant by a thermal balance of the
conductor.
Proof:
According the theorem TC16 the δ-algebra is non invariant,
because the velocity vT of the free electron in the conductor depend
from its absolute temperature T. According the theorem TC17 all remaining
elements (algebras) of the multitude Ώ (see (267) from the theorem TC18)
are invariant. With other words, the Ώ-algebra will be invariant only if
the absolute temperature T of the conductor is constant (the conductor is in a
thermal balance). By the same condition according the theorem TC19 will be
invariant also and the β-algebra and according the theorem TC20 – and the
α-algebra.
The theorem is
proved.
Theorem
TC22. The scalar produces:
(279)
<u,i> = <i,iR> = <u,uG> = W
are invariant by
thermal balance of the conductor and they belongs to the energetic space E(1).
Proof:
The energetic equation (279) is proved in the theorem TC15 and
therefore is true. According the equation (251) from the same theorem the
values conductivity G and resistance R of the conductor are reciprocal and,
therefore, if one is invariant, invariant is also and other.
According the
equation (248) the resistance R of the conductor depends from its longitude l,
its vertical section s and the specific resistance σ of conductor
material. And according the theorem TC19 these three values are elements from
the multitude Y. They are invariant by thermal balance of the conductor. By the
same condition are invariant also and the Ώ- and the α-algebra, that
make real the circuit from equations (279).
According the
equations (279) we can write:
(280)
(<u,i>)2 = (<i,iR>)2 = (<u,uG>)2
= W2,
that according the
theorem T42 speaks that the energetic balance of the electrical system is
calculated in the space E(1).
The theorem is
proved.
Proof
of the example (the systematic invariabilities): According the
conditions of the assertion of the example the system “source – electrical
consumer” is invariant.
Proof:
Let represent trough the figure 8. the electrical system “source –
consumer”.
Figure 8.
By an stable
energetic source (its voltage is changed only over a desire of person) and
constant thermal exchange between the electrical circuit and the environment
that sort of that the temperature of the resistance Rc of the
electrical consumer and Rl, making the common resistance R = Rc
+ Rl of the system, remains constant, the system will be invariant
over its currant and voltage. And after a derivation of the equations (250) and
(252) over the time t, obtaining:
(281)
P = ui = i2R = u2G,
we shell reach to
the conclusion that the system is invariant also and over its consumed power.
The example is
proved.
Example
Ex 3 (quantum invariability). The atom system “proton – electron”
of the hydrogen possess invariant states of its full energy (the Hamiltonian of
Schroedinger) about the velocity of the proton and the electron.
Proof:
Let set the next axioms:
Definition
D20. Atom is the smallest particle of a chemical element possessing its
chemical properties.
Axiom
AQ1: The atom is composed from nucleus with positive electrical mass
(charge) and moving around the nucleus negative charged electrons.
Axiom
AQ2: The atomic nucleus is composed by elementary particles named
nucleons.
Axiom
AQ3: The nucleons bringing the positive charge of the nucleus are named
protons and electrical neutral nucleons – neutrons.
Axiom
AQ4: The absolute value of the electrical mass of the proton and the
electron are equal.
Axiom
AQ5: The number of the protons in an invariant (non ionized) atom is
equal of the number of the circles round of its nucleus electrons.
Axiom
AQ6: The number of the neutrons in the atomic nucleus is equal or
higher from the number of its the protons. An exception makes the atom of the
hydrogen, the helium and other recently discovered atoms with deficit of
neutrons.
Axiom
AQ7 (first postulate of Bor): There exists non transitive states
of the atom in that it not emits electromagnetic energy, independently that the
electrons around him possess accelerations.
Axiom
AQ8 (second postulate of Bor): In non transitive state of the
atom every its electron moves over circular orbits around its nucleus, as its
kinetic momentum has discreet (quantum) values, according the condition:
(282)
Mq = mevnrn = n/2πh,
when me
is the mass of the electron, vn – its velocity over its orbit, rn
– the radius of its orbit, n – the number of its orbit that is a integer,
different from zero number, h – the constant of Plank.
Axiom
AQ9 (third postulate of Bor): There exists transitive states of the
atom by that its electron traverses from orbit m to orbit n and back. By m <
n the electron absorbs, and by m > n – emit a quantum electromagnetic energy
Q in the range of the light frequencies over the equation:
(283)
Q = hΩmn = Qm - Qn
as the quantum
frequency Ωmn is:
(284)
Ωmn = (Qm - Qn)/h
Axiom
AQ10. The postulates of Bor not have any dual similarity with the
exposed to this example theory of the invariability, because they are
explicable on the basis of the quantum mechanics.
Axiom
AQ11. The quantum mechanics is not a consequence from the classical
mechanics on the basis over that is constructed the theory of the
invariability.
Axiom
AQ12. Orbit of the electron is not any plain curved line like the conic
sections over that moves the celestial bodies but it is electrical Coulomb
field in air of charged “cloud” in that probably can be the electron. The
maximum density of the “cloud” (the density of the probability) is on distance
from the nucleus equal of the radius over the equation (282).
Axiom
AQ13. Non transitive according the first postulate of Bor states of the
atom presents stationary states of the movement of its electrons in a Coulomb
field over probable trajectories that are described from the solution of the
Schroedinger equation for every electron. (Here the equation is not indicated).
Axiom
AQ14. The orbital energetic components Qm and Qn in
the equation (283) we name kinetic energies of the states of the electron. They
are accepted with negative signs.
Axiom
AQ14. The maximum positive, calculated over the equation (283) energy
over that the electron quits the field of the nucleus of the atom, transforming
it in a positive ion, we name energy of the ionization of the atom. The maximum
negative energy over the same equation that attracts the electron toward the
field of the atom we name energy of the connection of the electron with the
nucleus. Both energy have equal absolute values.
Axiom
AQ16. The emanated from the electron energies by an ionized atom possess
a continuous (analogous) frequency spectrum.
Axiom
AQ17. The atom of the hydrogen contains a proton in its nucleus and an
electron around him.
Definition
D21. The value characterizing the capability of the electrical field to
traverses trough electrical non conductive matter (dielectric) we name
dielectric permeability. It is measured in Farads/meter (F/m). Its physical
interpretation is very complex and here is not made.
Axiom
AQ18. The proton and the electron creates around they self a Coulomb field with intensity:
(285)
Å = å/4πεr2,
when with e we
mark the electrical mass of the proton or the electron, with ε – the
dielectric permeability of the electric field in vacuum and with r – the
distance between them over the equation (282).
Axiom
AQ19. The movement of the proton and the electron in the space is made
by variable mass m and constant impulse mv.
Axiom
AQ20. The probable space trajectories of the proton and the electron are
defined over the mentioned in the axiom AQ13 equations of Schroedinger, in that its impulses pp
and pe are the negative complex numbers:
(286)
(-ih/2π)Dr1 = pp and
(287)
(-ih/2π)Dr2 = pe,
when with D we
mark the operator:
(288)
d/dx + d/dy + d/dz = D
with the property:
(289)
D2 = d/dx2 + d/dy2 + d/dz2,
and with r1
and r2 – the radius – vectors of the proton and the electron:
(290)
r1 = r1(x, y, z) è r2 = r2(x,
y, z)
Definition
D22. The energy that is defined from the exactly traversed road
(trajectory) of the proton and the electron in the space is named kinetic and
the energy, depending from the distance between them – potential.
Theorem
TQ1. The proton and the electron in the atom of the hydrogen creates
electrical Coulomb fields with potential energy:
(291)
P = - e2/4πεr
Proof:
According the axiom AC9 the positive (see the axiom AQ1) the
electrical mass of the proton attracts the negative (see the axiom AQ1) of the
electron with the force (see the equation (227) in the axiom AC10):
(292)
F+ = eE+,
when with E+
we mark the intensity of the field of the proton and with e – the electrical
mass of the electron. Conversely, the negative electrical mass of the electron
will attracts the proton with the force:
(293)
F- = eE-,
when with E- we mark the intensity of the field of
the electron and with e – the electrical mass of the proton.
As soon as the
electrical mass generates an electrical field, the equal according the axiom
AQ4 electrical mass will generates equal over intensity electrical fields.
Therefore,
(294)
E+ + E- = 0 and F+ + F- = 0,
with other words,
the system “proton – electron” is in a kinematic (see the definition D13)
balance. The nucleus and the electron probably moves under the influence of
theirs proper electrical fields, but theirs resulting force remains zero almost
always, with an exception of a discrete number moments. And for almost always
to calculate the energetic reserves of the system “proton – electron” let we
allow a potential possibility to generate of a movement changing the distance r
between them. That signs that the potential energy P of the system “proton –
electron” will be proportional of the maximum probable distance that the
electron may be traversed approaching to the nucleus. This energy will present
the scalar produce:
(295)
P = <vE, F->,
when with vE
we mark the potential velocity of the electron toward the nucleus. It will be
equal:
(296)
vE = dr/dt
From an other side
according the equations (293) and (294) the potential force F- will
be equal to:
(297)
F- = å2/4πεr2
If according the
equation (295), foreseeing the trues (296) and (297) and making the scalar
produce:
t t
⌠ ⌠
(298) <vE, F-> = ⌡(dr/dt)∙(å2/4πεr2)dt
= ⌡( å2/4πεr2)dr
o o
we shell achieve
exactly the result (291).
The theorem is
proved.
Theorem
TQ2. The kinetic energy Qp of the proton and Qe –
of the electron are calculated over the equations:
(299)
Qp = (-h2/8π2mp)∙D2r1
and
(300)
Qå = (-h2/8π2må)∙D2r2,
when we mark with
mp and me – respectively, the mass of the proton and the
electron in the conditions of repose.
Proof:
According the definition D21 the kinetic energy is a measure for
probable movement and according the axiom AQ19 the system “proton – electron is
characterized with constant in the time impulse. And because according the
axiom AQ19 in movement the mass m of the proton and the electron changes but
the impulses remains constant, there follows to eliminate the concept inertial
mass and to change him with the concept inertial impulse. For this purpose we
make a chain from equivalent energetic transformations:
(301) Q = <v,mdv/dt> = <mv,dv/dt> =
<p,(1/m)dp/dt> =
t p
⌠ ⌠
= ⌡(p/m)(dp/dt)dt =
⌡(p/m)dp = p2/2m
o o
And now if in the end result from the equation (301)
instead the general signs for impulse p and mass m we carry in the sign pp
of the impulse of the proton from the equation (286) or pe – of the
electron from the equation (287) and the mass mp by repose of the
proton or me – of the electron, we shell exactly achieve the kinetic
and energetic equations (299) and (300).
The theorem is
proved.
Axiom
AQ21. The kinetic Q and the potential P energy describes completely the
energetic balance of the non transitive state of the system “proton - electron” of the hydrogen atom.
Proof
of the example (the systematic invariabilities): The full energy
(the Hamiltonian of Schroedinger):
(302) H = QP + QE+ P
of the system
“proton – electron” of the hydrogen atom is invariant and belongs to the
energetic space E(1).
Proof:
According of kinetic and energetic equation (301) we can compose
the chain functional diagram:
∙v ∙1/2 ∙1/m
(303)
m--->p----->p/2------>p/2m = Q,
from that we can
compose the Ώ-algebra:
(304)
Ώ1 = {∙v, ∙1/2, ∙1/m}
such that trough
him the characteristic multitude À:{v,
m} can be transformed in the kinetic and energetic equation Q over the
diagram:
Ώ1
(305)
À:{v, m}---->Q
If in the equation
(304) and the diagram (305) instead the general signs for mass m and velocity v
we carry in the sign mp of the mass by motion of the proton or me
– of the electron and the velocity vP of the proton or vE –
of the electron, we shell achieve in the both cases – for the proton and for
the electron – an invariant Ώ-algebra and an invariant characteristic
multitude A, because in the both cases the mass by repose don’t depend from the
velocity. Therefore the kinetic energy Q is invariant in relation of the
velocity of the proton and the electron.
According the
potential and energetic equation (295) we can compose the diagram:
Ώ2
(306)
Â:{vE, F-}---->P, Ώ2 = < , >
from that we see
that the characteristic multitude Â:{vE, F-} through the Ώ-algebra Ώ2
is transformed in the potential and energetic multitude P. And here the
characteristic multitude  and the Ώ-algebra Ώ2 are
invariant, because according the equations (285) and (293) the resistant force
F- of the electron non depend from the supposed velocity vE
with that it may be moved overcoming the distance r between him and the
nucleus.
Therefore the
potential energy P is invariant about every velocity of the proton and the
electron.
According the
axiom AQ21 in its non transitive state the system “proton – electron” of the
hydrogen atom don’t possess other energetic components excepting the kinetic
and the potential. And it signs that the equation (302) describes the full
energetic balance of the system “proton – electron” of the hydrogen atom, the
energy H really belongs of the space E(1) and it deserves to be named full. It
is also invariant toward the velocity of the proton and the electron according
the diagrams (303) and (306).
The example is
proved.
With the proofs of
the three examples was proved the existence of invariant properties of the
matter from the scales of the cosmos to the scales to the atom by the both
modes of its existence – material and field, over classical and quantum
energetic measure. And because the energy is an uniform measure for all modes
of the movement of the matter and the matter exists only in movement and also
matter is infinite, it follows to accept that it is impossible the existence of
an invariant process without energy balanced systems in the material world. The
reasonable engineer must only study from the infinite number invariant natural
systems the conditions for existence of every from them. Only over this way the
engineer can synthesize every invariant process – the ideal of every creator.
It will be fully
clear in the next part.
INVARIANT SYSTEMS
IN THE TECHNOLOGY
In this part of
the scientific work are indicated some applications of the natural invariants
systems in the technology. As be mentioned and proved the common mortal man –
also and the engineer – has not other choice excepting to be subordinated to
the all-powerful nature and to study the conditions by that can be exists the
invariants of every one from the its infinite number systems to be capable to
create something useful for the society from the air of the invariant
production process. It is clearly that an ideal invariability cannot be
achieved, but the aspiration for perfection, as shows the millennial human history, is naturally built in and the
society generously rewards it.
“If you can make a
mouse-trap better from that of the neighbor, the world will make a path to your
house”. This winged wisdom of the
philosopher Emerson may be follows to be true everywhere and for every
man. Let we attempt at least to indicate who neighbors what sort of mouse-trap
are make better. And to make it better from the neighbor.
Example
Ex 4 (gravitational protecting relay). The gravitational
protection of a nuclear reactor against explosion is invariant over
fast-action, motive (stressed) force, power and energy, if is invariant its
shooting spring and its bearing has an ignored friction.
Figure 9.
Proof:
On the figure 9 is imagined a nuclear reactor from the type of E.
Fermi with anti-explosive protection that on the picture a) is in excluded
state and on the picture b) – in included. The positioning number of the figure
corresponds on:
1. Reactor
2. Source
of neutrons
3. Protecting
staff
4. Active
part of the staff
5. Shooting
the staff springy device
6. Latch
for retain the springy device in state of readiness for shoot
In the state a)
when the active part of the protecting staff
don’t cover the source of neutrons the reactor works in its nominal
mode. The neutrons bombards the nuclear “fuel”, the nucleus of the atoms in the
“fuel” in air of avalanche divides they self, there dissociates thermal energy that is accepted
from a thermal agent (in the case, water) and the water go in a steam generator
that generate steam for the necessities of consumers (turbines etc.).
The dividing
nuclear power of the reactor is regulated by regulating staffs that covers a
part of the radiating surface of the neutron source to stage selected from the
working with the reactor personnel. Here this staffs are not imagined.
In the state b),
when the active part of the protecting staff covers entirely the source of
neutrons, the reactor is emergency stopped. To this state it is reached when
the regulating staffs begins to lose their capability to support the nominal
dividing power. Than the protecting dosimeters around the reactor transmits an
emergency signal for pulling the latch “aside”, the shooting springy device
make shoot. The protecting staff flies off downhill traversing the road from
elevation zero to elevation ho accelerated from the spring and the
Earth gravitation and falls freely in its nest traversing the road from the
elevation ho to the elevation H accelerated only from the Earth.
The road of the
protecting staff from the elevation zero to the elevation H is subordinate of
the next characteristic time-table:
(307) VELOCITY AND FORCE OF THE PROTECTING
STAFF
|
Elevation,
interval, h |
Time,
t, s |
Road,
h,m |
Velocity,
m/s |
Force,
f, N |
|
0 |
0 |
0 |
0 |
G+Fo |
|
H≤h<ho |
t≤t<to |
H≤h<ho |
(Ca+1)gt |
G+F |
|
ho |
to |
ho |
(Ca+1)gto |
G |
|
ho≤h<H |
to≤t<t |
ho≤h<H |
(Cato+t)g |
G |
|
H |
tH |
H |
(Cato+tH)g |
G |
Let following the
example Ex 1 to explain the characteristic values velocity and force for the
different intervals from the time t and the road h of the staff.
a) h = 0 and t =
0.
In this moment the
staff is still immovable and its velocity v is equal to zero and over him acts
simultaneously the springy force Fo and the force of its weight G.
If we accept that the springy force Fo changes linearly according
the length of its strain (stretch out), in this moment it will have its maximum
value cho, when with c we mark the springy constant measured in
Newtons/meter (N/m). The resulting force over the staff will be:
(308)
f = Fo + G = cho
+ mg,
when with m we
mark the mass of the staff and with g – the Earth acceleration.
b) Í ≤ h
< ho and 0 ≤ t < to.
In this interval
the staff already moves and under the action of both forces – of the spring and
the weight – gathers the velocity:
(309)
v = vg + va = gt + cd∙vg
= (1 + ca)gt,
when with vg
we mark the gravitating component of the velocity and with ca – a
set by us normative coefficient for acceleration the movement of the staff with
a goal it to reach to elevation H with the minimum permit delay tH.
With other words, if we normalize the delay tH of the staff there
must the maximum middle velocity vH of the staff to be subordinated
of the criterion:
(310)
vH ≥ H/tH.
Then according the
equation (309), substituting v with vH, there must:
(311)
ca ≤ (H/gtH2) – 1.
Over the staff
continues to act simultaneously the force of its weight G and the springy force
to, as we accept, that G remain constant trough entire road H of the
staff and Fo decreases proportional of the stretch out h,
transforming it self in the time function:
(312)
f = G + cho - (cho/to)t
(see the figure
(10)).
Figure 10.
c) h = ho and t = to.
In this moment the
spring is fully stretch and its force is equal to zero. That signs that over
the staff acts only the force G of its weight. The velocity of the staff has
reach the value:
(313)
v = vo = (1 + ca)gto
that is named
initial velocity of the staff.
d) ho ≤
h < Í and to ≤ t < tÍ
In this interval
over the staff acts only the force G of its weight and its velocity is a sum
from the components va and vg as the accelerating
component va retains its value for the moment to and the
gravitational component vg grows linearly toward the time t, so
that:
(314)
v = va + vg = g(cato+t)
e) h = Í, t = tÍ.
In this moment
over the staff acts only the force G of its weight and the gravitational
component vg of its velocity has reach its maximum value gtH,
so that the velocity of the staff has reach the value:
(315)
v = vH = g(cato+tH),
that we shell name
end velocity of the staff.
Let from the
characteristic over the time t time-table (307) of the staff we compose its
characteristic over the parameter iΩ (spectral) time-table for the
intervals:
à) 0 ≤ t
< t --> to.
In this interval
according the properties of the Fourier transformation the velocity will have
the spectrum:
(316)
V(iΩ) = ((ca + 1)g∙exp(-iΩt)-1)/Ω2∙√2π
toward that the
staff will exercises accelerating force with the spectrum:
(317)
F(iΩ) = (((G + Fo)∙exp(-iΩt)-1))/-iΩ∙√2π)
-
- (((Fo/to)∙exp(-iΩt)-1))Ω2∙√2π)
According the
equation (35) from the theorem T17 in this interval the staff will gather the
initial motive energy:
(318) Åo(t) = <V(iΩ),F(iΩ)>
= (ca + 1)g[(t2/2)∙(G
+ Fo) - Fot3/3to].
b) to ≤
t < t --> tÍ.
This interval is
with a non zero beginning and calculated in that spectrums will have very
complex air. For a simplification of our work let accept the moment to
like a new beginning, from that we begin to observe the movement of the staff.
If we make the translation, substituting in the equation (314):
(319)
t = τ = T - to,
that signs that
the translating function:
(320)
Φ: T = τ + to = t + to
the interval b)
can be transformed over the diagram:
Φ-1
(321) to ≤ t < t -->tH---->
0 ≤ T - to < T - to-->Φ-1(tH)
in the interval:
c) 0 ≤ T - to
< T - to --> Φ-1(tH).
In this interval
the velocity will have the spectrum:
(316-1)
V(iΩ) = ((cagto∙exp(-iΩτ)-1)/-iΩ∙√2π)
+
+ ((g∙exp(-iΩτ)-1)/Ω2∙√2π),
toward that the
staff will render an accelerating force with the spectrum:
(317-1)
F(iΩ) = (G∙exp(-iΩτ)-1)/-iΩ∙√2π
For the interval
c) the staff will acquire the energy:
(318-1)
Å(τ) = G(cagtoτ
+ gτ2/2) or
(319-1) Å(T-to) = G(cagto(T-to)
+ g(T-to)2/2),
that in the
contrariwise transformed interval b) will be:
(320-1)
Å(t) = Φ-1(Å(τ)) = Gg[cato(t-to)
+ (t2-to2)/2]
Arriving to the
elevation H, the staff will strike with its lower end in its nest, taking in
the foundation of the reactor the deforming (motive) energy:
(321-1) Å = Å(to) + Å(tH) = (ca
+ 1)g[(to2/2)∙(G + cho) -
-choto2/3]
+ Gg[cato(tH-to) + (tH2-to2)/2]
If we differentiate
over the time t the equations (318-1) and (320-1) and after that we change in
them t, respectively, with to and tH we shell have also
and the end deforming (motive) power of the staff over the foundation of the
reactor:
(322)
S = S(to) + S(tH) = Gg[(ca+1)to
+ (cato + tH)]
The elasticity and
the strike robustness of the staff must by projecting be controlled over the
energy (321-1) and the power (322), independently that the reactor is
calculated to be undamaged even by direct bombing attack. The both criteria are
independent one toward other.
Proof
of the invariabilities: Let we present the strike energy from the
equation (321-1) like a variable in the time t energy:
(323) Å(t) = (318-1) + (320-1) =
= (ca + 1)g[(t2/2)∙(G
+ cho) - chot3/3to] +
+ Gg[cato(t-to)
+ (t2-to2)/2]
By condition that
the springy constant c of the shooting device is really invariant in the time
t, all letters in the right part of the equation (323) excepting the letter t,
as indicates the example Ex 1, signs constants. The system for protecting the
reactor against explosion according the definition D16 is linear and,
consequently, according the theorem T54 is invariant. The protecting staff of the
reactor makes an invariant transition from elevation zero to elevation H by
every disappear of the operating voltage to the latch (see the figure 9) of the
shooting device.
The
example is proved.
With the proof of
the example Ex 4 there is proved that the gravitational principle for
anti-explosion protection of nuclear reactor remains invariantly the surest,
because the gravitational field of the Earth can’t be never and not anywhere to
be isolated or behind screen placed (see the axiom G5) and the springs from
special for the goal steels haves an invariant deforming force for practically
unlimited time.
And by condition
that the reactor has minimum three protecting staffs, practically will be
necessary especially trained and especially organized sabotage group to provoke
a nuclear explosion, even of the oldest type reactor – this of E. Fermi.
Not so stands the
question with the reactors from the type “academician Kourchatov” by that the
protecting staffs are moved by electrical motors. Here the unhappiness a la
Chernobil wants only an unlimited dictators over the top of the state
authority. The sabotage can made from some electrical beginners. In the
interval of the nuclear unhappiness the effect of
Example
Ex 5 (invariant protection against short circuit). The protection of
an electrical transmission line against short circuit, tuned in its Joule heat
in the time of the circuit is with an invariant selectivity.
Proof:
On the figure 11 is imagined a three phases electrical
transmission line between a city substation 110/10 KV and a substation of the
metropolitan subway in
Figure 11
On the figure is
marked with:
CB1 and CB2 –
outlet protecting circuit-breakers in the city substation and inlet protecting
circuit-breaker in the subway substation that automatically turns off the
energetic currant by short circuit in the points A and B;
lA and
lB – distances from the side 10 KV of the city substation to the
points A and B – probable places of short circuits;
A, B and C -
probable places of short circuits over the electrical transmission line.
Definition
D23. Short circuit over the electrical transmission line is every
lasting galvanic connection between its conductors that stops after it self the
transference of electrical energy.
The electrical
transmission line and attached to him systems are subordinated of the follows:
Axiom
AS1: The consumed from the trains freight current io changes
its effective value Io in the time t according the equation:
(324)
Io(t) = U/√3∙(Zo + ZG(t)),
when U is the
voltage of the electrical transmission line (in the case – 10 KV), ZG
– the impedance of the traction electrical circuit on the side 10 KV in the
subway substation, Zo – the impedance of the electrical transmission
line (see the low of Ohm – the equation (248)).
Axiom
AS2: The current over the electrical transmission line ilB
after determine of the short circuit in the point B changes its effective value
IlB in the time t over the equation:
(325)
I1B(t) = U/√3∙Z,
when Z is the
impedance:
(326)
lim (Zo + ZG(t))-->Zo,
t-->∞
with other words,
the impedance of the line after the short circuit that is provoked a by-pass of
the traction impedance.
Axiom
AS3: Trough the points A and C from the schema (ahead and after the
circuit breaker CB2) flows identical, described in the axioms AS1 and AS2,
currents.
Axiom
AS4: Transformed over the time of the short circuit in the point B
electrical in heat energy (see the equation(247)) in the theorem TC15):
tk tk
⌠ ⌠
(327)
Wk = ⌡I1B(t)2Rîdt = ⌡[(U/√3)2/Rî]dt,
o o
when with tk
we mark the moment toward the beginning of the short circuit (t = 0), in that
the circuit-breaker CB2 turn off the line, change disparagingly its active
resistance Ro.
Axiom
AS5: The inductivity Lo of the electrical transmission line
and its resistance Ro are linear distributed over its entire length.
Axiom
AS6: Selective protection of the electrical transmission line signs
preserving of its conductors from damages under the influence of the heat of
Joule (327) so that by short circuit in the point B the circuit-breaker CB2 to
turn off the energetic flow from the city substation before the circuit-breaker
CB1 to make also too.
Axiom
AS7. The circuit-breakers CB1 and CB2 not refuses simultaneously by
same short circuit.
Axiom
AS8: The apparatuses from that is composed the protecting system of the
electrical transmission line are invariant with a guarantee.
Theorem
TS1: In the interval:
(328) 0 < t ≤ tk
has its force the regularity:
(329)
dIo(t)/dt < dI1B(t)/dt, t-->0.
Proof:
Over definition the derivative in the equation (329) is presented
by the limits:
(330) lim{[Io(tk) - Io(0)]/(tk
- 0)} = dIo(t)/dt
tk-->0
(331) lim{[I1B(tk) - I1B(0)]/(tk
- 0)} = dI1B(t)/dt
tk-->0
Comparing the
currents from the equations (324) and (325) we arrive to the natural conclusion
that the current of the short circuit I1B is greater from the
freight current Io because the impedance Z after the short circuit
in the equation (326) is smaller from the freight impedance Zo + ZG(t)
in the equation (324). From other side in the beginning of the short circuit
when t = 0 between the freight current Io and the current of the
short circuit I1B because of the single sign and the continuity of
the current like a function of the time t (see the theorem T2) there exists the
equation:
(332) I1B(0) = Io(0).
Therefore
numerator in the fraction (331) is greater than the numerator in the fraction
(330) and foreseeing the equal denominators of same fractions, the assertion
(329) is true.
The theorem is
proved.
Theorem
TS2: In the time interval (0, tk) the thermal energy WkA
by short circuit in the point A is greater than the thermal energy by short
circuit WkB in the point B.
Proof:
According the definition D23 the short circuit stops the transfer
of electrical energy after the point in that is made. Therefore the entire
generated from the substation energy after the short circuit begins to be
consumed from the part of the electrical transmission line whose length is
limited between the substation and the point of the short circuit. In that sort
of case for the full energy EkA by short circuit in the point A and
EkB by short circuit in the point B are in force the equations:
tk
⌠
(333)
ÅêÀ = ⌡(U/√3)∙I1A∙dt and
î
tk
⌠
(334) ÅêB = ⌡(U/√3)∙I1B∙dt
and
î
when with IkA
we mark the current of short circuit in the point A.
From other side,
if we allow that over the time of the short circuit in the point B begins short
circuit in the point A, over the logics of the equation (326) for the impedance
ZA of the electrical transmission line to the point A is in force
the regularity:
(335)
lim (ZÀ + ZÀÂ(t))-->ZÀ,
t-->∞
when with ZAB
we mark the impedance between the points A and B. From the equation (326) and
the axiom A35 is trivial clear that:
(336)
(ZÀ + ZÀÂ) > ZÀ
and over the low
of Ohm between the currents of short circuit I1A and I1B
there exists the dependence:
(337)
I1A = U/√3∙ZÀ > U/√3∙(ZÀ
+ ZÀÂ) = I1Â
If we substitute
the values of I1A and I1B from the inequality (337)
respectively in the equation (333) and (334) they will achieve the air:
tk
⌠
(338)
ÅêÀ =⌡(U/√3)2/ZA∙dt and
o
tk
⌠
(338)
ÅêB =⌡(U/√3)2/(ZA+ZAB)∙dt
o
The sub-integral
functions of the equations (338) and (339) are fractions with equal nominators
and the denominator in the equation (339) according the transition (335) is
greater than the denominator in the equation (338). That signs that:
(340)
ÅêÀ > ÅêB.
And if we
determine the active (see the theorem T19) energy of the short circuits that is
transformed in heat of Joule, there will be:
(341)
WêÀ = ÅêÀ∙cosφ > ÅêB∙cosφ
= WêB,
when:
(342)
cosφ = RA/ZR = (RA+RAB)/(ZR+ZAB)
= Ro/Zo
is the ratio of
the resistance of the electrical transmission line in the section “substation –
point A” and “point A – point B” to the impedance in same sections. Like a
consequence from the axiom AS5 this ratio is constant. If it be substituted
respectively in the equations (338) and (339) they will achieves the similarity
of the equation (327) to the axiom AS4.
The theorem is
proved.
Proof
of the invariability: Figure 12 presents a scheme of the
protecting system of the electrical transmission line against short circuit in
the point B.
Figure 12.
The protecting
actions of the system forms the consequence:
-
the current iB trough the point
B is measured by the kilo-ammeter -1 that defines its effective value IB;
-
the measured effective value IB
is transferred simultaneously of the derivative instrument – 2 and of the relay
– 5;
-
the derivative instrument – 2 derivates IB
over the time t, the derivative dIB/dt is transferred to the
algebraic adding instrument – 3. There it is compared with the preliminarily
selected from the setting instrument – 4 derivative (dIB/dt)o;
-
according the theorem TS1 the value (dIB/dt)o
is selected greater than the value of the derivative dIo/dt of the
effective value Io if the freight current io If
(343) dIB/dt > (dIB/dt)o,
there is a short circuit
and the relay – 5 will switch on the information line to transmit
the value IB for a next manipulation;
-
IB is raised to second power by
the square instrument – 6 and the value IB2 is multiplied
over the coefficient RB by the multiplying instrument – 7. The
coefficient RB is proportional of the active resistance RA
+ RAB of the electrical transmission line from the substation to the
point B. The outlet signal IB2RB after the multiplying
instrument – 7 by this condition will be proportional of the thermal power
heating the electrical transmission line by short circuit in the point B (see
the equation (327));
-
the signal IB2RB
is integrated over the time t from the integrator – 8 on its outlet according
the equation (327) is transfer a signal proportional of the energy WkB
heating the electrical transmission line by short circuit in the point B;
-
the signal WkB is compared in
the adding instrument – 9 with the preliminarily selected by the setting instrument
– 10 value WkBo that is proportional smaller than from
the permit energy heating the electrical transmission line so that is really
respected the condition of the axiom AS4. When
(344)
WêÂ
> WêÂî;
the
outgoing from the adding instrument – 9 signal WkB commands of the circuit breaker CB2 to turn
off the energetic flow trough the point B to the subway substation. If the
command is not executed the electrical transmission line will be turn off by
the city substation and its burning will be avoided;
- if the circuit
breaker CB2 refuses to turn off the energetic flow the transferred (will be
best by fiber optical path) and to the adding instrument – 11 signal WkB
will be compared with the preliminary selected by the by the setting instrument
– 11 signal kcWkBo. Here independently that is
selected:
(345) kc > 1,
the signal kcWkBo
also like WkBo is proportional smaller from the permit energy heating the electrical
transmission line so that really is respected the condition of the axiom AS4.
When
(346) WêÂ
> êñWêÂî
the outgoing from
the adding instrument – 11 signal WkB command of the circuit breaker to turn off the
energetic flow trough the point B to the subway substation so far away before
the energy WkB to be grow to the energy WkA (see the
theorem TS2). According the axiom AS7 the circuit breaker executes the command.
The electrical transmission line is turn off from the energetic system and
protected against blaze, explosion and other non provident even for the system
consequences.
Theorem
TS3. The protecting system is selective.
Proof:
According the theorem TS2 the energy WkA by
short circuit
in
point A heats
maximum the electrical transmission line for the time of the circuit tk.
That signs that the electrical transmission line must be constructed with
conductors enduring surely sufficiently greater thermal overloads. We accept
that it is true. By this condition the circuit breaker CB1 is set for turn off
by selected energy:
(347) WêÀî > WêBî.
By this setup if a
short circuit comes in the point B, the energy WkB will provokes a
turn off the breaker CB2, when WêB > WkBo.
The electrical transmission line will be heated engulfing the energy WkBo
over the regularity:
t t
⌠ ⌠
(348) WêBî = ⌡I1B(t)2(RA+RAB)dt
> ⌡I1B(t)2RAdt < WêÀî
o o
when with tkB
we mark the moment of the turn off the breaker CB2. With other words, for the
time tkB the electrical transmission line will not engulf an energy
that will heat the part to the point A like by short circuit in this point.
If the circuit
breaker refuses the turn off and the information line to the adding instrument
– 11 don’t exists, the heating of the electrical transmission line will
continue as long as it engulfs the energy WkAo over the
regularity:
t t
⌠
⌠
(349)
WêÀî = ⌡I1B(t)2RAdt
> ⌡I1B(t)2RAdt
o o
when with tkA
we mark the moment of the turn off of the breaker CB1. The same effect will
come if a short circuit appears in the point C, out from the subway substation
(see the axiom AS3).
The sub-integral
functions in the inequality (349) are equal. Therefore:
(350)
tkA > tkB
or with other
words the breaker CB1 will delay toward CB2. According the axiom AS6 that signs
that the system will be protected selectively.
The theorem is
proved.
Theorem
TS4 (proof of the invariability): The selectivity of the protecting
system is invariant
Proof:
According the theorem TS3 the selectivity of the protecting system
is achieved by setting of every circuit breaker to turn off the energetic flow
to the subway substation over a signal proportional of the Joule heat, engulfed
from the electrical transmission line by a short circuit made directly after
him. If in the common case we mark this place like the point X, following the similarity
of the inequality (348) we must set the breaker before the point X to turn off
the energetic flow to the subway substation proportionally over the energy:
tkX
⌠
(351)
WêÕî = ⌡I1Õ(t)2rlXdt,
o
when with IlX
we mark the effective value of the current i1X by short circuit in
the point X, with tkX – the moment of the turn off of the breaker,
with r – the linear resistance of the electrical transmission line measured in
Ohms/meter (Ω/m), with lX – the length of the electrical
transmission line from the city substation to the point X.
Following the
similarity of the diagram (275) from the theorem TC20 we arrive to the
conclusion that there exists an α-algebra trough that is made the
transformation of the characteristic multitude w in the energetic multitude WêÕî
over the diagram:
α
(352)
w:{I1Õ, rlX}--->WêÕî, α
= < , >
According the
axiom AS4 the resistance rlx is independent from the current IlX.
Therefore the characteristic multitude w is invariant. Invariant is also and
the α-algebra. And by guarantee for invariability of the apparatuses of
the protecting system (see the axiom AS8) the selectivity of the protecting
actions is also invariant.
The theorem and
the example are proved.
The system on the
figure 12 to day don’t exists and the selectivity of the protection of the
electrical transmission lines “middle voltage” remains a great world problem.
But the author of this theory is an optimist and he hope in the future of the
automation science like the writer-fantast Jules Verne in the submarines and
the travels to the Moon.
Example
Ex 6 (invariability of laser ray): The laser ray is
invariant over force and frequency by a constant inversion of its “density of
population” and constant loss of the amplification in its quantum system.
Proof:
Let we rest against the next definitions and axioms:
Definition
D22: Quantum particle is every material point that moves over the lows
of the quantum mechanics also and over the axioms AQ7,…,AQ14 likening the
movement of the electrons in the field of the atom.
Axiom
AQ22: The state by that the quantum particle possess its minimum kinetic
energy Qo (see the equation (283)) is primary (fundamental)
invariant state and all remaining states by that:
(353)
Qo < Q1 < Q2 < ,...,< Qn
are secondary
(meta-) invariant states.
Axiom
AQ23: The meta-invariant states are achieved trough an exterior
energetic flow because of that they are named excited.
Definition
D25: The quantity Nn of quantum particles in an volume
everyone from that possess kinetic energy (stands on the energetic level) Qn
we name density of population of the level Nn.
Definition
D26: If in someone volume the quantum particles makes the densities of
population Nm and Nn
and:
(354) Nn > Nm,
we say that the
middle of this volume is active and the difference:
(355) Nn - Nm > 0,
we name inversion
of the densities of population.
Definition
D27. System from quantum particle (quantum system) we name every
multitude from density of population.
Axiom
AQ24 (postulate of Einstein): For every quantum system with
densities of population Nn and Nm exists the
probabilities A – for free (spontaneous)
and B(qY) - for stimulated by an exterior source transition
from the state Qn to the state Qm by that an energy in
air of light is emit.
Here with the qY
we mark the transitive energetic density (J/m3) of the system.
Axiom
AQ25 (low of Boltzmann): By a thermodynamic balance the densities
of population Nm and Nn in a quantum system are in the
correlation:
(356) Nm/Nn = exp((Qn
- Qm)/kBT) = exp(hΩnm/2πkBT),
when with kB
(J/oK) we mark the constant of Boltzmann, with T (oK) –
the absolute temperature of the populated with quantum particle middle and with
Ωmn – the quantum frequency (see the equation (284)). By that:
(357)
Nn < Nm
Axiom
AQ26: There exists at least a point with interior energy that is totally
transformed in a sine (monochromatic) light.
Axiom
AQ27: The interior power S (W) of the point makes a light flow Φ
that is measured in lumens (lm) over the equation:
(358) Ô = cfS,
when with cf
(lm/W) we mark a transforming coefficient characterizing the luminous source
(in the case, the point).
Axiom
AQ28: The flow Φ illuminates an unit of area from the interior
surface of an circumscribed around the point sphere with a radius r (m) with
force I (lm/m2).
Axiom
AQ29 (differential low of Buger – Lambert): The luminous force
Io that is available on a distance ro from the point
decreases with the difference Io - I > 0 on a distance r toward
the point over the equation:
(359)
- (I - Io) = αI(r - ro), r > ro
when with α
we mark the coefficient of luminous engulf (lm/m) of the middle around the
point.
Theorem
TQ2-1 (Integral low of Buger – Lambert): The luminous force
I on the distance r from the point is defined over the equation:
(360)
I = Io∙exp[-α(r - ro)]
Proof:
If in the equation (359) we make after a transformation the limit
transition:
(361)
lim[-(I - Io)/I] = α∙lim(r - ro),
I->Io r->ro
after another
transformation will be real the linear and homogeneous differential equation:
(362)
dI/dr + αI = 0.
If we integrate
the equation (362), accepting the beginning condition from the axiom AQ29 that
(363) I(ro) = Io,
we shell achieve
the result (360).
The theorem is
proved.
Theorem
TQ3: The kinetic energetic density q (lm∙s/m3) by a
transition (luminous emission or engulf) of a quantum system with a density of
population N is defined over the equation:
(364)
q = NhΩ/2π
when Ω is the
frequency of the emit or engulf light.
Proof:
It is obvious result from the concept quantum system (see the
definition D27) and the third postulate of Bor (see the equation (283)).
The measuring of
the energetic density in lm∙s/m3, from an other side, is a
result from the axiom AQ27.
The theorem is
proved.
Theorem
TQ4. Quantum system with a predominated possibility (probability) for
stimulated luminous emission and ignored possibility for spontaneous luminous
emission can be controlled in invariant mode over the frequency Ω of the
emit light. In the opposite case - by system with predominated possibility
(probability) for spontaneous luminous emission and ignored possibility for
stimulated luminous emission – it is impossible.
Proof:
According the postulate of Einstein (see the axiom AQ24) the stimulated
emission depends from the energetic density of the system. That signs that the
equation (364) is valid for him.
The equation (364)
is discreet linear if we allow that the density of population N of the system
is constant. That signs that if N are the particles losing a part of the
energetic density q, this density depend from the frequency Ω of the emit
light.
With other words
the equations from the air (364) belongs of some discreet linear energetic
space. It is single measured and we can mark him with ε(1). This discreet
space in actuality presents a point in the space E(1) from the classic theory
of the invariability. With other words, a system controlled over the functional
dependence (364) will be invariant (see the theorem T45) for some discreet
values of the frequency Ω. From other side, again according the axiom
AQ24, the probability for spontaneous luminous emission not depends from the
energetic density of the system and an invariant control of the spontaneous
emission is impossible.
The theorem is
proved.
Axiom
AQ30: Quantum systems with spontaneous emission not presents any
technological interest because of the proved in the theorem TQ4 impossibility
for an invariant control over the frequency Ω of the emit light. Over this
case in the next part of the text under an emission there must understand only
stimulated emission.
Theorem
TQ5: The density of the luminous power dq/dt (lm/m3) by
transition (emission or engulf) of a quantum system is defined over the
equation:
(365)
dq/dt = (dN/dt)∙hΩ/2π.
Proof:
It is a obvious result from the differentiate over the time t of
the equation (364).
Theorem
TQ6: The density dq/dt of the luminous power by transition (emission or
engulf) of a quantum system is numerically equal of the gradient dI/dr of the
luminous force I of the system over direction of the radius – vector r in the
space and it is defined over the equation:
(366)
dq/dt = dI/dr = (dN/dt)∙hΩ/2π.
Proof:
According the axiom AQ28 the luminous force I is measured in lm/m2,
that signs that the gradient dI/dr from the equation (366) will be measured in
lm/m3 and it will have the quality of density of luminous power.
The theorem is
proved.
Theorem
TQ7: The density dq/dt of the luminous power by transition (emission or
engulf) of a quantum system is defined also and by the equation:
(367)
dq/dt = σ(Ω)NI,
when with σ(Ω)
we mark the effective transitional section (m2) that is transverse
for every luminous ray and it characterizes the quantum system
Proof:
The right part of the equation (367) presents a physical value
measured in lm/m3 and therefore it has the quality of luminous
power.
The theorem is
proved.
Theorem
TQ8: The balance of the density of the luminous power Pa by
engulf toward the density of luminous power Pe by emission in the
quantum system with densities of population Nm and Nn is
defined over the equation:
(368)
σmn(Ωmn)NmI + σnm(Ωnm)NnI
= 0,
when with σmn
we mark the effective transitional section of the system by engulf and with
σnm – by emission.
Proof:
The both addend in the equation (368) according the theorem TQ7
possess the quality of densities of luminous power, by that the first – by
transition of the density of population Nm to Nn
(accompanied with engulf) and the second – back to front, from Nn to
Nm (accompanied with emission). Therefore we can mark they with Pa
and Pe. Foreseeing that the frequency Ωmn of the
engulf according the third postulate of Bor is equal over module and opposite
over sign of the frequency Ωnm
of the emission, there follows that:
(369)
Pa + På = 0.
The theorem is
proved.
Definition
D28: Quantum system by that:
(370)
σmn = σnm = σ
we name non
degenerated and back to front, by:
(371)
σmn ≠ σnm
the system is
degenerated.
Axiom
AQ31: There is accepted that examined in the next text system is non
degenerated over the sense of the definition D28.
Theorem
TQ9: Stimulated emission of luminous energy is possible only then when
the quantum system is active (see the definition D26).
Proof:
Foreseeing the axiom AQ31 the equation (368), defining the density
of the power balance “emission – engulf” in a quantum system will obtain the
air:
(372)
σ(Ω)[Nm - Nn]∙I = 0
If in the equation
(372) we differentiate the luminous force I over the radius – vector r, with
other words, if we define the gradient dI/dr of the luminous force I in
direction r from the space it will come out that:
(373)
σ(Ω)[Nm - Nn]∙dI/dr = 0
The integration of
the differential linear equation (373), accepting the beginning condition:
(374) I(0) = Io
it will achieve
the result:
(375)
I = Io∙exp(-σ(Ω)(Nm - Nn)r)
Comparing the
equations (375) and (360) come out that according the integral low of Buger –
Lambert the value:
(376)
α = σ(Ω)(Nm - Nn)
presents a
coefficient of exponential amplification of the luminous ray Io over
the direction r when:
(377) Nn > Nm,
with other words,
when the system is active and a coefficient of exponential damping of the same
ray when:
(378) Nn < Nm,
with other words,
when the system is not active.
With other words,
when the system is active, directed to him luminous ray coerces the system
again to emit this ray, amplifying its force I exponentially to direction r and
back to front – when the system is not active, the same ray is engulfed
exponentially over opposite direction.
The theorem is
proved.
Definition
D29: Device containing in it self a quantum system, that amplifies a
luminous ray over the integral low of Buger – Lambert by observing the
conditions (376) and (377), is named Light Amplifier by Stimulated Emission of Radiation (LASER).
Axiom
AQ32: The coefficient of amplification α of the quantum system of a
laser decreases additively with the value β because of loss of luminous force generated from
relaxation (natural return of the thermal balance over the equation (356)),
saturation etc. Over this cause the equation (375) acquires the real air:
(379)
I = Io∙exp((α - β)r).
Axiom
AQ33: There accept that the coefficient β of the loss in the
amplification of the laser is constant.
Proof
of the invariability: Figure 13 presents the closed volume of
the quantum system of a laser.
Figure 13.
The volume has a
constant section over all its length r and it is fill with substance composed
from particles with the energetic levels Qm and Qn making
the density of population Nm and Nn. The inversion Nm
< Nn in the volume is supported constant by the stimulating
energetic flow Qo to him, by that the inlet luminous ray Io
that has the frequency Ω quit the volume reaching the value I over the
equation (379) with the same frequency. By these conditions the quantum system
is invariant.
Proof:
If in the equations (364) and (365) formally we substitute the density
of population N with the difference Nm - Nn in them will
not comes any qualitative change but will come its exact air:
(380)
q = (Nm - Nn)∙hΩ/2π and
(381)
dq/dt = (hΩ/2π)∙d(Nm - Nn)/dt =
dI/dr
Therefore, we can
arrive to the conclusion that the equation (379) is a result from the diagram:
d/dt ≡ int(r)
(382)
(380)--->(381)--->(366)------>(379),
when with int(r)
we mark the integrate of the differential equation (381) foreseeing its
identity with the equation (366).
The parameters of
the equations making the diagram (382) by respecting the beginning conditions
of the example and according its axioms and theorems are values invariant
toward the force I and the frequency Ω of the laser ray. Invariant trough
them is also the algebra:
(383)
d/dt ◦ ≡ ◦ int(r)
The example is
proved.
With the proof of
the examples Ex 4, Ex 5 and Ex 6 is proved the existence of possibilities to
create equipments working in an invariant mode over the lows of the classical
and the quantum mechanics. For every air of equipment are indicated limitation
conditions by whose respect is possible the technological performances of the
equipment to be invariant.
And as soon as there is limitation conditions for existence of the invariability, that signs that exists exact limits towards that the invariability don’t exists. Over what kind of criteria this limits to be defined? It will be indicate in the next part of the theory.
LIMITS OF THE
INVARIABILITY
This part of the
scientific work presents the finished state of the theory of the invariability.
In the part are indicated the criteria according that the invariability reaches
the limits of its existence. Here are also indicated the methods for achieving
of impermissibility for passing of these limits.
Already (see the
beginning text in the part “Energetic Algebra. Part One”) are indicated that
over scientific prognosis even the Sun after billions years will become
extinct. It unconditionally directs us to the melancholy thought that the
eternal for us pronounced and execrated solar shining has a limited eternity in
the space and the time of the universal infinity. And it signs that the
invariants of the universal infinity have same limits. And as soon as they –
the natural invariants have limits, there follows and theirs applications in
the technology to be considered with them.
Let we step across
still invisible for us limits of the invariability and let we find outside
that.
Theorem
T55: System with impedance coefficients μ, σm and
σl changes in the time t according the equations:
(384)
μ = μ(t), σm = σm(t), σl
= σl(t),
is non invariant.
Proof:
If we allow the opposite we shell enter in a contradiction with
the truth of the theorems T45 and T54 that demands for the existence of an
invariant and invariantly transitive mode a respect of the equation (160). With
other words, the impedance coefficients μ, σm and σl
must be constant.
The theorem is
proved.
Theorem
T56: Non invariant system can’t be controlled.
Proof:
To here under control tacitly was understand a concept over the
sense of the definition D2 for an invariant system. With other words, control
is a realized human action defining the energy (see the axiom A7) of the system
so that will be available a product completely and over every time
corresponding of the set. Therefore, controlled is only a system that is
invariant.
The theorem is
proved.
With the proof of
the theorem T56 seemingly was come to the conclusion that practically nothing
can’t be controlled because in the common sense the matter is non linear.
Control, however,
is possible even of systems subordinated of the equations (384) – moving by
variable friction, mass and elasticity of the middle in that they moves. For
example, all vehicles with proper energetic source – automobiles, ships,
plains, missiles etc. They are not only controlled, but its control constantly
is perfected. And it is the general sense of the progress in the technology.
The conclusion:
There is conditions for control also and of non invariant systems. Which are
they and haw they appears?
That will be
visible in the next rows of the text
Definition
D30: System for that the equation (384) is in force by repose (v = 0)
is statically non invariant (non working aging). Back to front – by motive (v
≠ 0) the system is dynamically non invariant (working aging).
Definition
D31: Dynamically non invariant system for that in its working period T
ª (0, to) the impedance coefficients μ, σm and
σl possess the property:
(385)
μ(0)= μ(to), σm(0)=σm(to),
σl(0)=σl(to),
with other words,
they changes only over the working time, is elastic. Back to front, if the
equations (385) are not true and μ, σm and σl
changes in the working time, the system is plastic.
Definition
D32: The impedance coefficients μN, σmN
and σlN of a dynamically non invariant system that it possess
in its preliminary period (to beginning of its work) T ª (-∞, 0) and they
remains sufficiently long time constant we name nominal impedance coefficients.
Definition
D33: The differences DN:
(386)
DN:{μN - μ(t), σmN - σm(t)
and σlN - σl(t)}
we name additive
degrees of non invariability.
Definition
D34: The ratios RN:
(387)
RN:{μN / μ(t), σmN / σm(t)
and σlN / σl(t)}
we name
multiplicative degrees of non invariability.
Axiom
E15: The equations (384) presents continuous functions with limited
variation in the time t in the working period of the system T ª (0, to)
Axiom
E16: The degrees of non invariability are permit if in the working
period T ª (0, to) of the system are in force the limits:
(388)
DN--->const., RN--->const.,
by t--->to.
Definition
D35: Under characterizing of a system there understand the definition
of its impedance.
Theorem
T57: System with permit degrees of invariability can be characterized
with its nominal impedance:
(389)
ΓN(Ω) = μN + i(σmN
- σlN) ª Γ(2)
Proof: If the
degrees of non invariability of the system are permit over the sense of the
limitation conditions (388), they are ignored and in the working period T the
nominal impedance coefficients μN, σmN and σlN
of the systems can accept constant. The equation (389) will can in force and
the system will can be characterized like linear with the defined over him
nominal impedance ΓN(Ω).
The theorem is
proved.
Definition
D36: The nominal impedance ΓN(Ω) of the system is
named General Technological Characteristic (GTC).
Theorem
T58: System whose working period TΣ presents the union:
(390)
ÒΣ: (0, to) U (0, t1) U,..., U (0,
ti)U,...,
and its degrees of
invariability are permit for every period
Ti ª (ti-1, ti)
has for GTC the row of impedances:
(391)
ΓΣN = ΓNo, ΓN1,...,
ΓNi,...,
whose every member
ΓN characterizes the respective period Ti.
Proof:
As soon as the working period TΣ of the system
presents an union of continuous intervals from the time t and in every from
these intervals the degrees of non invariability of the system are permit, it
can work continuously in entire its working period TΣ, passing
from the linear characteristic ΓNi-1 in the linear
characteristic ΓNi. Therefore, the linear characteristics
ΓNi con be united in the row (391).
The theorem is
proved.
Definition
D37: System permit a dynamic state (v ≠ 0) is efficient. Back to
front – by inadmissibility of that sort of state (v = 0) – the system is not
efficient.
Axiom
E17: The energetic source of the system transfers its energy of the
consumer acting over him with the force fG(t) that depends from the
properties of the source.
Axiom
E18 (third low of
(392)
fG(t) + f(t) = 0
Theorem
T59: System whose GTC ΓΣN (see the equation (391))
continuously increases over module in its working period TΣ,
with other words,
(393)
|ΓΣN|--->∞, t--->∞, t ª ÒΣ,
loses its efficiency.
Proof:
The module |F(iΩ)| of the resistant force f(t) according the
axiom E18 remains constant. As soon as the GTC of the system continuously
increases, that signs that is possible to permit in some of its composed
periods Ti the impedance ΓNi to rises sufficiently
great so that the velocity v(t) of the system to obtain the spectrum:
(394)
V(iΩ) = F(iΩ)/ΓNi(iΩ),
whose module | V(iΩ)|
continuously will decrease inclining to zero as soon as the module |ΓΣN|
continues to increase. The system is disintegrated trough stopping its work.
The theorem is
proved.
Theorem
T60: System for that is valid the theorem T59 controls an object
decreasing the consummation of energy strives to be transformed in an ideal
insulator.
Proof:
As soon as the module |V(iΩ)| of the spectrum of the velocity
v(t) of the system inclines to zero the consumed from him full energy E,
defined from the scalar produce (see the equation (35)):
(395)
<V(iê), F(iê)> = Å
also will incline
to zero, striving to be transformed in an ideal insulator.
The theorem is
proved.
Theorem
T61: Object transformed in an ideal insulator don’t consumes any power
from the source of the system.
Proof:
The ideal insulator according the theorem T60 don’t consumes
energy. Than the derivative dE/dt of the scalar produce (395) that defines the
consumed from the insulator full power S, with other words:
(396)
dÅ/dt = d(<V(iê), F(iê)>)/dt = S,
also will be zero.
The theorem is
proved.
Theorem
T62: System whose GTC ΓΣN (see the equation (391))
continuously decreases over module in its working period TΣ,
with other words:
(397)
|ΓΣN|--->0, t--->∞, t ª ÒΣ,
is overloaded by
work and it is transformed in a consumer of infinite power.
Proof:
It is the opposite consequence from the proof of the theorems T59,
T60 and T61.
As soon as the
module of the impedance |ΓΣ| continuously decreases over
the sense of the transition (397), the module
V(iΩ)| of the spectrum of the velocity of the system will incline
to infinity over the sense of the equation (394). From other side, according
the theorem T61 follows that and the derivative of the scalar produce (396),
defining the consumed from the system power, also will incline to infinity over
module.
The theorem is
proved.
Axiom
E19. The energetic source of the system don’t depend from the behavior
(respectively, the characteristics) of the controlled object.
Theorem
T63: The GTC of the system is a measure for existence of the controlled
object by its opposition against of the energetic source.
Proof:
Following the logics of the diagram (189) the multitude of
impedances ΓΣN, making the GTC of the system, will defines
single sign for its working period TΣ the consumed from the
controlled object energy E over the diagram:
α
(398)
À:{V(iΩ), ΓΣN}--->E,
when with α
we mark the α-algebra transforming the characteristic multitude A of the
system in the energetic multitude E.
By a transitive
mode the multitude from impedances ΓΣN will belong over
the force of the theorem T51 of the linear impedance space Γ(2) and the
consumed energy E – of the linear energetic space E(2) (see the theorem T22).
The both spaces over the theorem T52 are covariant dual. With other words,
α α-1
(399)
Γ(2)--->Å(2)---->Γ(2)
The diagram (399)
speaks that the consumed from the controlled object energy E in transitive mode
is defined isomorphicaly from the GTC ΓΣN of the system.
From other side the consumed energy E is opposite of the generated from the
source energy. With other words, the existence of an impedance (impedance space
Γ(2)) generates the existence of a consumed energy (energetic space E(2)),
that opposites of the generated from the source energy.
The theorem is
proved.
Theorem
T64: System whose GTC is subordinated if the transition (397) is
disintegrated trough a destruction of the controlled object.
Proof:
It is consequence from the proof of the theorem T63.
As soon as the
impedance multitude (391) inclines to zero, that according the theorem T63
signs that decreases the resistant possibilities of the controlled object
against the energetic source of the system. When the multitude (391) stands
zero the controlled object stops to exist. The energetic source according the
axiom E19 is survived, but it destroyed the object.
The theorem is
proved.
With the proof of
the theorems T55,…,T64 is indicated the existence the limits of the
invariability. Also there is indicated the limits between the efficiency and
the disintegrate of the system. The conditions by that these two sorts of
limits don’t can be pass are indicated in the next part of the theory.
STABILITY
Definition
D38: The property of the system to retain its efficient after a
transition or, with other words, to retain its capability to make the
transition:
(400)
lim(v(t) - vo) = 0, t-->∞, vo ≠
∞,
when with vo
we mark the selected invariant value of the velocity v(t) that we desire to
achieve after the transition in the selected moment to is named
stability. The system possessing this property is stabile.
Theorem
T65 (criterion of Lagrange): Stabile is every system in whose neutral point
(see the axioms A1 and A2) really is not consumed any energy. With other words:
(401)
lim[ÅG(t) - E(t)] = 0, t-->∞,
when with ÅG(t)
we mark the generated from the source and with E(t) – the consumed from the
controlled object energy of the system.
Proof:
The respect of the authenticity of the transition (401) over the
sense of the equation (395) demands:
(402)
lim[<V(iΩ), FG(iΩ)> - <V(iΩ), F(iΩ)>]
= 0, t-->∞
when with FG(iΩ)
we mark the spectrum of the acting by the source over the object force fG(t).
According the
third low of
(403) lim[<V(iΩ), FG(iΩ)>
- <Vo(iΩ), F(iΩ)>] = 0, t-->∞
when with Vo(iΩ)
we mark the selected invariant value of the velocity v(t) that we desire to
achieve after the transition in the selected moment to. And because
according the axiom E19 the energetic source don’t depend from the behavior of
the consumer, the spectrum FG(iΩ) can be accepted like constant
and the transition (403) equivalent of (400).
The theorem is
proved.
Theorem
T66: The criterion of Lagrange is the general criterion for stability
of every system and all other criteria (for example, of Raus – Hurvitz,
Nyquist, Popov etc.) are practically true, so far as they are consequences from
him.
Proof:
The transition (401) defines the aspiration of the system to
achieve an energetic balance. And since every other air of balance – like the balance (400) – according the
theory of the invariability is achieved over an universal energetic measure for
balance, there follows that all criteria, in this number and the material
abstract (mathematical) criteria of Raus – Hurvitz, Nyquist, Popov etc. will be
true only if by conditions that can make they consequences from the criterion
of Lagrange. Because of it the application of the last criteria is possible
only practically for private cases subordinated of the mentioned conditions.
The theorem is
proved.
Theorem
T67: Invariant system possess an eternal stability. With other words, according
the criteria of Lagrange:
(404)
lim[<V(iΩ), FG(iΩ) > - <V(iΩ), F(iΩ)>]
= 0,
t ª T = (-∞, ∞)
Proof:
Foreseeing its constant impedance:
(405)
Γ(iΩ) = const. -∞
≤ t ≤ ∞,
the invariant
system will permit under the action of the force fG(t) of the source
the velocity v(t) with the spectrum:
(406) V(iΩ) = F(iΩ)/Γ(iΩ), -∞ ≤ t ≤ ∞,
that will achieve
after every transitive moment to the constant module |V(iΩ)|o.
From other side, according the third low of
The theorem is
proved.
Theorem
T68: Elastic system (see the definition D31) that has GTC ΓΣ
over the force of the theorem T58 possess a stability only in the limits of its
working period TΣ (see the union (390)). With other words,
according the criterion of Lagrange:
(407)
lim[<V(iΩ), FG(iΩ)> - <V(iΩ), F(iΩ)>]
= 0,
t ª ÒΣ
= (to, ti)
Proof:
As soon as the GTC ΓΣ of the system presents
a row from the impedances ΓNi, with other words,
(408)
ΓNi = const.,, ti-1
≤ t ≤ ti
and foreseeing the
fact that the interval:
(409)
(to, ti) ª (-∞, ∞),
over the force of
the theorem T67 the transition (407) will be true for the period TΣ.
The theorem is
proved.
Theorem
T69: Elastic system (see the definition D31) that not has GTC ΓΣ
over the force of the theorem T58 possess stability only in the surrounding
δti on the limits of its working period TΣ.
With other words, according the criterion of Lagrange:
(407)
lim[<V(iΩ), FG(iΩ)> - <V(iΩ), F(iΩ)>]
= 0,
t ª ÒΣ
= δto, δt1, δt2,...,δti,...
Proof:
As soon as the GTC ΓΣ of the system presents
a row from the impedances ΓNi, that are accepted constant only
for every initial or end interval δti from every working period
Ti, with other words,
(408)
ΓNi=const., t ª ÒΣ = δto,
δt1, δt2, ...,δti,...
and foreseeing the
fact, that the multitude from intervals:
(409)
ÒΣ ª (to, ti),
over the force of
the theorem T68 the transition (407) will be true for the moments TΣ.
The theorem is
proved.
Definition
D39: Maximum characteristic period TΣM of a system we
name the maximum interval from time (0, tM) in that it can be
characterized over the force of the theorem T58.
Definition
D40: Guarantee characteristic period TΣG of a system we
name the guaranteed from the company – producer interval from time (0, tG)
in that it can be characterized over the force of the theorem T58.
Definition
D41: Additive security (degree of security) of a system presents the
difference:
(410) DS = ÒΣM - ÒΣG,
and the quotient
(ratio):
(411) RS = ÒΣM / ÒΣG
-
multiplicative
security.
Definition
D42: Really sure is every system for that:
(412)
DS ≥ 0 and RS ≥ 1
and back to front
– non really sure (non sure) is every system for that the inequalities (412)
are not in force.
Theorem
T70: Plastic system is stabile only
in its working period TΣ when this period is smaller or equal
of the maximum characteristic period TΣM of the system. With
other words, when:
(413)
ÒΣ ≤ ÒΣM.
Proof:
If the inequality (413) is respected over the force of the
theorems T68 and T69 the conditions for stability of an elastic system will
coincide with these of a plastic system. Therefore, the system will be stabile
while according the inequalities (412) is really sure. Out from this security
the system is liable of a repair or is disintegrated by stopping or destruction
(see the theorems T59 and T64).
The theorem is
proved.
With the proof of
the theorems T65,…,T70 are indicate the conditions by that the system can by
stabile. By these condition the system retains its efficiency without to come
its non-reversible disintegration trough stopping or destruction. There was
also indicated that a stability exists or if the system works in the conditions
of a allowed non invariability that decreases its working period, or in the
conditions of invariability. Let generalize by airs, systems and theorems the
conditions for existence of stability in the table (414) on the next page.
(414) CONDITIONS
AND PERIODS OF STABILITY
|
Air of system |
Conditions over theorem |
Period |
|
Invariant |
T67 |
unlimited |
|
Elastic |
T68 |
continuous
intervals |
|
Elastic |
T69 |
short
intervals |
|
Plastic |
T69, T70 |
continuous
intervals |
Out from the
periods of stability the system is non controlled and it will be seen in the
next part of the theory.
AUTOMATIC CONTROL
Definition
D43: The preliminary registration for the work vo of the
system in the time t trough its working period T from the air:
(415)
vo = vo(t), t ª T,
we name working
(production) program.
Definition
D44: Control that has the purpose to respect the working program of the
system we name programmed control.
Definition
D45: Control that has the purpose to respect the stability of the
system we name regulation.
Definition
D46: The variation in the time t of the really necessary for a
consumption from the object power we name controlling the system action
(influence).
Definition
D47: When the controlling action is made from a person (human hand,
foot, voice etc.) the control is in hand mode and when the system make self it
– in automatic mode.
Definition
D48: The possible combinations from the airs of control according its
purpose and the manner of introduction of the program defines the next (see the
next page):
(416) AIRS OF SYSTEMS OVER PURPOSE OF THE
CONTROL AND MANNER OF INTRODUCTION OF THE PROGRAMS
|
System |
Regulation |
Programme
control |
|
Hand |
hand |
hand |
|
Programme |
hand |
automatic |
|
Automatic |
automatic |
hand |
|
Automatic
programme |
automatic |
automatic |
Definition
D49: When the controlling action is an enumerated (discreet) multitude
from values, the control is discreet (pointy, positional, protecting, numeric
etc.), when it is continuous (analogical) multitude - continuous (analogical)
and when it is an union from the both airs multitudes – control from common
air.
Axiom
E20: The concept “discreet controlling action” is conditional and it is
used only for simplicity of the reasoning, when some function of the time t
make transitions toward the zero, absolute or relative.
Definition
D50: The airs of control according the mathematical air of the
controlling action defines discreet, analogical and from common air controlling
systems.
Definition
D51: The transformation of the non evident variations of a physical
value in evident we name informational action for the value and the achieved by
the information result (record of the variations for a period T from the time
t) – information.
Definition
D52: The device transforming the non evident value in evident creates
information and it is an information source (transmitter) and the device
accepting the information – information receiver (observer, consumer).
Definition
D53: The device transferring the information from the source to the
receiver we name informational line (network, canal).
Definition
D54: The energy (electrical, luminous, mechanical etc.) transferring
the information over the information line we name informational carrier.
Definition
D55: The value from the characteristic multitude of the informational
carrier that changes in a function from the non evident (according the
definition D51) value by the creation or by the accepting of information for
him we name informational signal.
Definition
D56: When the function of the signal is covariant with the non evident
(according the definition D51) value, the signal is positively true (true),
when is contra-variant – negatively true
(contra-true) and when is not co- or contra-variant – non true (non
informational).
Axiom
E21: When the non informational signal is a result from a premeditated
action the truth or the contra-truth is protecting changed (coded), when it is
a result from an accidental action the truth or the contra-truth is mixed with
noises over the informational line.
Definition
D57: The time trough that the signal pass from the source to the
receiver of the information without to come a functional difference between
created and the accepted signal we name real time.
Definition
D58: The value for that the information is created and accepted over
the working period of the system, with other words, by t ª T we name free
variable (variable) and this for that it comes outside the working period of
the system – parametrical variable (parameter).
Axiom
E22: For every created from the man system there exists a probability
for static or dynamic ageing (see the definition D30).
Axiom
E23: The GTC ΓΣ of the system is defined
(measured, controlled) outside from its working period and the work v(t) of the
system – over the time t of the working period T (t ª T).
Theorem
T71: For preserving the stability of the system (see the definition
D45) over the time t of its working period T (t ª T) there is necessary a
continuous observation of its work v(t).
Proof:
It is a consequence from the axiom E23 according that over the
working time t is follows the function v(t).
If we allows that
the GTC ΓΣ of the system changes over the time of its
working period T, it – according the definition for ΓΣ
(see the definition D36) substituted in the equation (406) – will provoke a
variation in the function v(t) over the equation:
(417)
v(t) = F-1[F(iΩ)/ΓΣ], t ª T,
when F-1
is the opposite Fourier transform of the spectrum V(iΩ) of the velocity
(work) v(t).
According the
third low of
(417)-1 (417)
(418)
ΓΣ -------->v(t)-----> ΓΣ
and it signs that
observing v(t) also and ΓΣ is observed. With other words,
the state of non stability is provoked from the degree of the non invariability
and, back to front – the state of non invariability is provoked the non
stability.
The theorem is
proved.
Theorem
T72: For preserving the stability of the system (see the definition
D45) in the time t of its working period T (t ª T) there is necessary to be
exercised over the system a controlling action in a functional dependence from its work v(t).
Proof:
According the definition D46 the controlling action U presents the
second derivative over the time t of the really consumed from the object
energy, that according the axiom E19 always exists (the energetic source of the
system is independent from the controlled object). The controlling action U has
the air:
(419)
U = d2/dt2<V(iΩ), V(iΩ)∙ΓΣ>
Or with other
words, the change of the ΓΣ leads isomorphically to a
change of U over the diagram:
(419)-1
(419)
(420)
ΓΣ -------->U------> ΓΣ
From other side,
comparing the diagrams (418) and (420) we reaches to the conclusion that there
exists the isomorphisme:
(419) (417)-1 (417)◦(419)-1
(421)
U-------> ΓΣ
-------->v(t)-------------->U,
from that follows that
the exercise of the controlling action U over the system really preserves its
stability if U is in functional dependence with the work v(t) of the system.
The theorem is
proved.
Theorem
T73: For preserving the stability (see the definition D45) and the
production program (see the definition D43) of the system trough the time t of
its working period T (t ª T) there is necessary and sufficiently a continuous
observation over its work v(t) and a exercise over the system a controlling
action UG from the air:
(422) UG = - A∙dv/dt,
when A is a
normalizing value measured in Watts∙seconds/meter (Ws/m) or
Proof:
As soon as according the theorem T72 is necessary the controlling
over the system action U to be in a functional dependence from the observed
work v(t) of the system, in signs that a control over the system without any
observation can’t exist. From that there follows that the condition (422)
really is necessary. For it to be also and sufficient, it must coerce the
system to remove every diversion from its program vo, that signs to
make the transition:
(423)
lim(v - vo)--->0, t ª T.
t-->to
From this
transition follows that the differences v - vo and t - to
must simultaneously incline to zero. That, from other side, signs that the
transition:
(424)
lim[(v - vo)/(t - to)]--->0, t ª T, t-->to.
And that, from
other side signs that the derivative:
(425)
dv/dt-->0, t-->to.
And because for
the realization of the transition (425) there is necessary to be leaded a
variation in the action U of the energetic source, it follows that the
consequence (419) is accepted to be generated from the cause:
(426) UG = d2/dt2<V(iΩ),
FG(iΩ)> = d2/dt2<v(t), fG(t)>,
when with fG(t)
we mark the acting over the object force of the source and with FG(iΩ)
– its spectrum. By this condition to be realized a balance in the system there
is necessary the both action over the system UG and U to be
reciprocally neutralized over the rule:
(427)
UG + U = 0 or UG = -U,
with other words,
on the action UG to opposite the contra-action U and back to front.
Let overseeing the
axioms E17 and E19 we make a double derivate of the end result from (426).
There will achieve:
T
⌠
(428)
d2/dt2<v(t), fG(t)> = d2/dt2[⌡v(t)fGdt]
=
o
= fG(dv/dt)
= UG,
that signs that
the condition (422) is true.
From other side,
according the third low of
(429)
F-1[V(iΩ)∙ΓΣ] = - fG,
from that after a
comparison of the equations (428) and (419) follows that by a exercise of the
action UG the rule (427) will be respected and the system will
strive to a balance. With other words, the necessary condition is also and
sufficient.
The theorem is
proved.
Theorem
T74: For preserving the stability (see the definition D45) and the
production program (see the definition D43) of the system trough the time t of
its working period T (t ª T) there is necessary and sufficiently it to be
automatically and programmed controlled over the scheme:
Figure 14.
On the scheme with
(1) is marked the controlled object, with (2) – the energetic source, with (3)
– the programmer of the provided work vo(t) of the system, with (4)
– algebraic adding device for comparison of vo(t) with the really
made work v(t) and making of a signal ε(t) for error in the work, with (5)
– proportional, integral and differential (PID) regulator for making a
correcting signal δx(t) against the error ε(t), with (6) – hand
regulating device for set the regulator according the components (parameters)
μ, Cl and m (see the equations (94), (119) and (116)) of the
GTC ΓΣ over the rule:
(430)
P = μ, I = Cl and D = m,
with (7) –
servo-motor changing trough the chuck device – (8) – the power of the source SG
to really consumed from the object power S.
The devices (3),
(4), (5),…,(8) makes an automatic controlling – regulating circle (loop) or a system
for automatic control and regulation. Over this manner according the
definitions (416) the object, the energetic source and the controlling –
regulating circle, from its side, makes an automatic programmed system. It is
subordinated of the next common regularities:
a) really consumed
from the object power S is proportional of the walk h of the servo-motor (7),
measured in meters (m) over the rule:
(431) S = cSh,
when cS
is a normalizing constant (m/W),
b) the walk h of
the servo-motor is proportional of the outlet signal y of the regulator (5)
over the rule:
(432) h =
chy,
when ch
is a normalizing constant (m/Iy). Here with Iy we mark a
unit measure of the informative signal y of the regulator that can be Amperes
(A), Volts (V), Pascals (Pa) etc. depending from the physical value – carrier
of the information for the function y(t),
c) the outlet of
the regulator y is a function of the inlet values x(t) and z(T) over the rule:
t
⌠
(433) y = Px + (1/I)⌡xdt + D(dx/dt)
o
Here with T we
mark the working period of the system.
d) the parametric
inlet z of the regulator is functionally depended from its working period T.
Once set from a human hand trough the hand regulating device (6) it remains
constant trough entire working period T of the system, respecting the rule:
(434) z = z(P, I, D), P = const., I = const., D
= const., t ª T.
e) the variable
inlet x of the regulator that is also an outlet of the adding device (4) is
functionally depended of the inlet values v(t) and vo(t) to same
device over the rule:
(435) x = xo - δx,
when xo
is the proportional of the program vo(t) and normalized with the
coefficient cx (sIx) signal:
(436)
xo = ñõvo,
and the function
δx, subordinated of the rule:
(437)
δx = ñõε = ñõ(v - vo)
is a signal for
command – correction to the regulator, when the work v(t) of the system allows
the error:
(438) ε = v - vo,
with other words,
when the work of the system v(t) increases toward the provident programmed
value vo(t) over the rule:
(439)
v = vo + ε.
f) the controlling
– regulating circle imports the correction δS in the power S before the
error ε in the work v to be achieved a new value. With other words, the
system for automatic control and regulation is a working in real time
information – commanding system.
Proof:
The object and the energetic source are created from a man.
Therefore, according the axiom E22 it is possible they to age and the GTC
ΓΣ of the system to change that according the theorems T71
and T72 signs that there is necessary to be observed
the work v(t) of
the system and an exercise of a controlling action U over it.
According the
scheme on the figure 14 the work v(t) of the system is observed, the
information from the observation is transmit of the controlling circle that
make the controlling walk h(t) of the servo-motor (7). In result, the walk h(t)
provokes same controlling action dS(t)/dt over the source of energy so, that to
change the consumed power S. That according the theorems T71 and T72 signs that
the necessity for preserve the stability and the production program of the system
exists. Whether is it sufficient?
Examined in
balanced and non balanced state the regularities (431),…,(439) generates the
next:
(440) DYNAMIC CHARACTERISIZATION OF THE SYSTEM
|
Observed
value |
Balanced
state |
Non
balanced state |
|
Work,
v(t) |
vo(t) |
v(t) |
|
Program,
vo(t) |
vo(t) |
vo(t) |
|
Error of
the work, ε(t) = v(t) -vo(t) |
0 |
v(t) - vo(t) |
|
Inlet to
the regulator, x(t) |
xo(t) |
x(t) |
|
Corection
to the regulator, δx = xo(t) - x(t) |
0 |
xo(t) - x(t) |
|
Outlet
from the regulator, y(t) |
yo(t) |
y(t) |
|
Corection
after the regulator, δy = yo(t) - y(t) |
0 |
yo(t) - y(t) |
|
Walk of
the servo-motor, h(t) |
ho(t) |
h(t) |
|
Corection
of the servo-motor, δh = ho(t) - h(t) |
0 |
ho(t) - h(t) |
|
Power,
S(t) |
So(t) |
S(t) |
|
Corection
of the power, δh = So(t) - S(t) |
0 |
So(t) - S(t) |
|
Controlling
action, U |
dSo(t)/dt |
dS(t)/dt |
According the
dynamic characterization results that if the work v of the system increases
toward its programmed value vo with the error ε, the
controlling circle will provoke a decrease of the power S to the object toward
the programmed power So with the correction δS over the chain
from devices (4), (5), (7) and (8) (see figure 14) following the sequence from
actions:
(4) (5) (7)
(441)
(vo + ε)--->(xo - δõ)--->(yo
- δy)--->
(7) (8)
--->(ho
- δh)--->So - δS,
By this action of
the controlling circle the derivative of the work:
(442)
dv/dt = lim[(v-vo)/(t-to)] > 0, v > vo,
t-->to
and the derivative
of the power (the controlling action U):
(443)
U = dS/dt = lim[(S-So)/(t-to)] < 0, S < So. t-->to
With other words,
the derivatives (442) and (443) are opposite over algebraic sign, that
according the theorem T73 signs that the provided over the figure 14 necessity
is also and sufficient.
The theorem is
proved.
Definition
D59: Fast-action fs (s-1) of a controlling –
regulating circle (automatic or hand) we name the value:
(444) fs = 1/τ,
when with τ
we mark the time from the beginning of the non balanced state (the appearance
of the error ε) of the system to the beginning of the controlling action U
or the contra-action UG.
Definition
D60: Non exactness θ-1 (W/s) of a controlling –
regulating circle (automatic or hand) we name the difference:
(445)
θ-1= |dS/dt| - |dSG/dt|
between the
necessary for the system controlling action U = dS/dt and the really made from
the source contra-action:
(446)
UG = dSG/dt
Definition
D61: Non intelligence (disinformation effect) φ-1 of a
controlling – regulating circle (automatic or hand) we name the value:
(447)
φ-1 = y(x) - Y(X),
when with X we
mark a disinformation signal on the inlet of the regulator (5) from the figure
14 that over the rule (433) generates the function:
t
⌠
(448) Y = PX + (1/I)⌡Xdt +
D(dX/dt),
o
on the outlet of
the regulator.
Definition
D62: The values θ and φ that are reciprocal of the non
exactness θ-1 and the non intelligence (disinformation effect)
φ-1 of the controlling – regulating circle (automatic or hand),
with other words,
(449)
θ = 1/θ-1 è φ = 1/φ-1,
we name,
respectively, exactness and intelligence of the controlling – regulating
circle.
Axiom
E24: Every created by the man controlling – regulating circle possess
greater fast-action and greater exactness, but smaller intelligence from the
man.
Definition
D63: State of the system by that its controlling – regulating circle
loses its intelligence φ in its working period T, with other words,
(450)
φ(t)-->0, t-->to, t ª Ò = (0, to),
we name emergency
(average) and a state, by that the controlling – regulating circle can restore
its intelligence, with other words,
(451)
φ(t)-->∞, t-->to, t ª Ò = (0, to),
we name normal.
Axiom
E25: Every inlet signal to the devices (4), (5), (7) and (8) from controlling
– regulating circle of the system on the figure 14 can be created and set also
and in hand mode (by a man).
Theorem
T75: The non exactness θ-1 of the controlling –
regulating circle of the system increases by decrease of its fast-action fs.
Proof:
If we calculate the controlling over the system action dSτ/dt
accounting the delay τ from the equation (444) of the signal that provoke
him, there will be achieved:
(452)
dSτ/dt = lim[(S - So)/(t + τ- to)],
t-->to
From an other
side, the necessary for balance of the system contra-action dSG/dt
is calculated over the equation:
(453)
dSG/dt = lim[(S - So)/(t - to)],
t-->to
Following the
logics of the definition D60 the non exactness θ-1 of the
controlling – regulating circle of the system will be defined from the
difference:
(454)
θ-1 = |dSτ/dt| - |dSG/dt|.
It will increase
than τ increases, with other words, when the fast-action fs of
the controlling – regulating circle decreases.
The theorem is
proved.
Theorem
T76: The system on the figure 14 can preserve its fast-action fs
and its production program vo if it is controlled automatically in
normal state and in hand mode – by emergency state (average).
Proof:
It is consequence from the axioms E24 and E25 and the theorem T75.
According the
theorem T75 the fast-action fs of the controlling – regulating
circle leads to its exactness θ and according the equation (447) and the
definition D63 the increasing of the intelligence φ decreases the
disinformation effect φ-1 of the controlling signal Y(X) and
from there – and the probability by average of the system.
And as soon as
according the axiom E25 the man has more intelligence, but it is more slow and
more non exact from the controlling –
regulating circle of the system and also the man can substitute every device
from the circle, he will controls the system better in emergency state and
worse in normal state.
The theorem is
proved.
With the proof of
the theorem T76 there are proved the conditions by that is possible non
invariant systems to be controlled. They are:
-
continuous observation of their work;
-
exercise of controlling action.
The exact
quantitative criteria for realization of the control are indicated in the
theorem T73 and they are illustrated trough its applications in the theorem
T74. These criteria are in force by respect of the limits (388). Out from them
the non invariant systems are non controlled. Same more, they are menaced from
averages, as indicates the theorem T76, and they must be protected.
How to realize for
this purpose the protecting control so that the controlled object to survive
physically when it is menaced from an average and to be repaired for a new
working state, it will be seen in the next (last) part from this scientific
work.
POSITIONAL CONTROL
AND PROTECTION AGAINST AVERAGES
Definition
D64: Step transition of the function f in the time t we name a
variation of its effective value F according the rule:
(455)
F = Fo = const., -∞ ≤ t < 0,
F = F(t) ≠ const., 0 ≤
t ≤ τ,
F = F1 = const., τ
≤ t < ∞,
by that Fo
≠ F1 and with τ we mark the time of the transition.
Definition
D65: The step transition is positive when F1 > Fo (step
up) and negative when F1 < Fo (step down).
Definition
D66: Impulse transition of the function g in the time t we name the
variation of its effective value G according the rule:
(456)
G = Go = const., -∞ ≤ t < 0,
G = G(t) ≠
const., 0 ≤ t ≤ τ,
G = Go =
const., τ ≤ t < ∞
Definition
D67: The impulse transition is positive when G(t) > Go
and negative when G(t) < Go.
Theorem
T77: The derivative dF/dt of the effective value F of the function F(t)
making a step transition is a function making an impulse transition (impulse)
with effective value from the air:
(457)
dF/dt = 0, -∞ ≤ t < 0,
dF/dt = dF(t)/dt ≠ 0, 0 ≤ t ≤
τ,
dF/dt = 0, τ ≤ t < ∞.
Proof:
The derivation of the equations (455) without any change of theirs
limit conditions will comes to the results (458).
The theorem is
proved.
Definition
D68: System by that (see the figure 15) the controlling action U
contains from alternative impulse transitions that provokes step transitions of
the consumed from the controlled object power is named discreet.
Figure 15.
Definition
D69: When the transitions of the controlling action U are provoked over
a program road like a result from the Boolean function:
(458)
vo = VB(V1, V2,...,Vi,...),
when with V1,
V2,...,Vi,... we mark extreme values of the program
function vo(t) there exists a discreet (positional, pointy etc.) program
control of the system.
Definition
D70: When the transitions of the controlling action U are provoked like
a contra-action of a danger from loss of the stability of the system leading to
a danger from average, with other words, when they are a result from the
Boolean function:
(459)
Γ = ΓB(Γo, Γ1,..., Γi,...),
when with Γo,
Γ1,..., Γi,... we mark extreme values of the
GTC of the controlling object there exists a protecting control (protection) of
the system.
Theorem
T78: If on the graph of the figure 15 we mark the characteristic values
of the discreet controlling system:
-
maximum consumed power Smax = S1,
S2,…,Si+1, Si+2…;
-
minimum consumed power Smin = So,
S3,…,Si, Si+3…;
-
intervals of the maximum consummation
τmax: τ12 = t2 – t1,…,τi+1,i+2,
= ti+2 – ti+1;
-
intervals of the minimum consummation
τmin: τoo = to – 0,…,τ3,i,
= ti – t3…;
-
maximum controlling action Umax:
Uo1,...,Ui,i+1,...;
-
minimum controlling action Umin:
U23,...,Ui+2,i+3,...,
by respect of the
marked in the table (460) correlations between them, there will exist the
follows:
(460) AIRS OF
DISCREET SYSTEMS
|
System |
Controlling
action, U |
Consummed
power, S |
Time for
extreme consumation, τ |
|
Elastic
over T68 |
Umax
= Umin = const. |
Smax
= const.,Smin = const. |
τmax
>> τmin |
|
Elastic
over T69 |
Umax
= Umin = const. |
Smax
= const.,Smin = const. |
τmax
<< τmin |
|
Plastic
over T68 and T70 |
Umax
≠ Umin ≠ const. |
Smax
≠ const., Smin ≠ const. |
τmax
>> τmin |
|
Plastic
over T69 and T70 |
Umax
≠ Umin ≠ const. |
Smax
≠ const., Smin ≠ const. |
τmax
<< τmin |
|
Non
stable to destruction over T64 |
U→-∞,
t→∞ (protecting contra-action) |
S→∞,
t→∞ (spontaneous action) |
None |
|
Non
stable to stopping over T59 |
U→∞,
t→∞ (protecting contra-action) |
S→0,
t→∞ (spontaneous action) |
None |
By that we must
supplement that the plastic system possess the properties:
(461)
S1 ≈ S2,..., Si+1 ≈ Si+2,...
So ≈
S3,..., Si ≈ Si+3,...
Proof:
It is a consequence from indicated theorems T59, T64, T68, T69 and
T70. The graph on the figure 15, the table (460) and the properties (461)
illustrates the theorems T68, T68 and T70 in the cases when the consumed power
S and the controlling action U are enumerated (discreet) in a stable mode.
Because the unstable mode is a result from natural or working age of the
controlling object it comes spontaneously, in accidental moment, that
practically can be not easy provided. Over this cause the object needs a
protection. Excepting that, as soon as by non stability spontaneously the
action S-->+∞, for a balance coming there must the contra-action:
(462)
dS/dt = U-->-∞, t-->∞
and, back to
front, if the action S spontaneously inclines to zero there must the
contra-action:
(463)
dS/dt = U-->+∞, t-->∞.
The theorem is
proved.
With the proof of
the theorem T78 are proved the conditions for existence of the maximally
popular air of control – the discreet control. Its applications like positional
program and protecting control (regulation) to day are a part from our everyday
way of life. Electrical on and off of luminaries, motors, pumps, fans, switch
over the gear of the velocity of cars, machine tools, protection against short
circuits etc. - who don’t know all that?
Let finish already
some years made scientific work. It don’t exhausts all, but the measure for its
true and completeness can define only the time, naturally, if it be read from
sufficiently great number of competent men.
The author
finished his Emerson mouse-trap. Whether it is better from that of the
neighbor?
Metroproject,
USED LITERATURE
1. Yavorskiy
B. M., A. A. Detlaf, Reference Book of
Physics (Russian text), !977, Naouka,
2. Zlatev
M., Foundations of the Electricity (Bulgarian text), 1964, Technika, Sofia.
3. Nenchev
M., S. Saltiel, Laser Technology (Bulgarian text), 1994, Technika, Sofia.
4. Tyagarajan
K., A. K. Ghatak, Lasers. Theory and Applications, 1981, Plenum Press,
5. Glimm
J., A. Jaffe, Mathematical Methods of Quantum Physics (Russian text), 1984,
Mir,
6. Arguirova
T., Theory of Analytical Functions (Bulgarian text), 1992, Sv. Kliment
Ohridski,
7. Chakalov
L., Introduction in the Theory of Analytical Functions (Bulgarian text), 1957,
Naouka I Izkoustvo, Sofia.
8. Corn
G., T. Corn, Reference Book of Physics
(Russian text), 1973, Naouka,
9.
Maclane G., G. Birghoff, Modern Algebra
(Bulgarian text), 1974, Naouka I izkoustvo,
10.
Stankov M., Theory of Invariant Systems for
Control (Bulgarian manuscript), 2000, Central Technological Library,
Sofia.
INVARIANT SYSTEMS
THEORY
SYNOPSIS
The Theory of
Invariant Control Systems or in
brief Theory of Invariability, is developed as a new viewpoint of
Control. Without ignoring the widely spread metaphysical theory studied in almost all Higher schools in the world, the author is basing on the concept of energy as
an measure for the existence
of the matter and on
this ground builds-up his theory considering that it completely covers all possibilities for the
movement control of all kinds
of material substances.
Basing only on the
methodology introduced by Galilee at the end of
16-th century the
author makes an
assumption for some real evolution of the Theory of Control
and its applications in practice; some of
them used by himself. After more then 30 years of control systems projecting he
couldn’t find such an evolution, neither could apply in practice the theory
that he learned as a student, because it
is no possible to
creat something material if it
does not possess material measure. And
the only measure in control systems could be the energy and the conditions for
its application are conditions for the existence of material invariants.
Under the
material invariant the
author considers the capability of the
material substances to
preserve constantly some characteristic
features (mass, volume,
thermal capacity, electric
conductivity etc.) in static or dynamic
mode. This capability and the conditions
for its existence
pre-determine the possibility
for creating, by the
material substances, an
machine, an engine, an equipment, a vehicle or some kind
of means that could be controlled or managed by
a preliminary program. The conditions for
the invariant existence are prerequisite
for the control of the engine created by the substance.
The author
uses the modern
algebra language in
order to describe and prove his
thesis. For some readers this is unusual for others - inadmissible.
However, everyone having
some knowledge and experience in
the field of
analogue and discrete
(digital) technological process could read the text without any difficulties if he/she is not
preconceived concerning the author
and his way of thinking. And this way of thinking is being used even
by philosophers like B. de Spinoza and I. Kant.