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Physics Essa ys, volume 15, number 3, 2002

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Conjugate Parameters of Physical Processes and Physical Time
George P. Shpenkov and Leonid G. Kreidik Inst itute of Mathemat ics & Phys ics, UTA, Bydgoszcz, Poland; max@atr.bydgoszcz.pl

Abstract
All physical pheno mena run in nature with the cont inuous transformat ion of the kinet ic field into the potential field, and vise versa, i.e., they have the wave character. For this reaso n, the structure of expressio ns represent ing these fields (both real physical space and physical t ime space) must be potential-kinet ic. Contemporary physics sat isfactory describes the kinet ic field, but the co mplete descript ion of the potential subfield is absent. In this paper we partially fill in this gap, beginning fro m the descript ion of harmonic oscillat ions (of a material po int ) and the phys ical t ime wave field-space. Key words: potential-kinet ic fields, oscillat ions, waves, physical t ime

RÈsumÈ
Tout phÈnomÕne physique se produit dans la nature par une transformation continuell e du champ cinÈtique en un champ de potentiel et vice versa, i.e., ils ont la caractÈristique ondulatoire. Pour cette raison, la structure d'expressions, reprÈsentant ces champs (a la fois, espace physique rÈels et espace- temps physique) doit Étre aussi potentiel-cinÈtique. La physique contemporaine dÈcrit de faÃon satisfaisante les champs cinÈtiques, mais la description complÕte du sous-champ potentiel est absente. Dans cet Ècrit, nous comblerons cette lacune partielle, en commenÃant par la description des oscillations harmoniques d'un point matÈriel et l'onde espace-champ du temps physique.


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1. Introduction
1.1. Overview The subject ion o f the basic parameters of physical systems to general principles o f philosophy and logic evidences their co mpleteness. However, not many know that. We have analyzed the phys ical parameters used for the descr ipt ion of harmonic oscillat ions and found their incompleteness with respect to the aforement ioned subject ion. Let us elucidate this statement. Since the wave exchange o f matter-space and rest-mot ion (matter-space-time for brevit y) is in the nature of all pheno mena, the possibility of rest and motion generates, at the definit e condit ions, potential-kinet ic fields o f reality, where rest (potential field) and motion (kinetic field) are inseparable linked in one potential-kinetic field [1]. The logical triad matter-spacetime expresses an indissoluble bond of matter, space, and t ime. The logical pair motion-rest presents an indisso luble bond of motion and rest, etc. The possibility of potential-kinet ic processes must be est imated by the co rresponding probability of rest-motion. Then, obviously, states of rest -motion, which are represented through the potential-kinetic field, should be described on the basis o f the potential-kinetic probability. If the potential-kinet ic field o f possibility and, corresponding to it, the field o f reality, are wave fields, then the potential-kinetic probabilit y also must have the wave character. But in order to introduce the wave probabilistic measure of rest-motion, it is necessary to solve at first a so-far-unso lved problem of the complete potential-kinetic descript ion of elementary harmonic o scillat ions. At all levels o f the Universe, rest and mot ion are, mainly, in a dynamic equilibriu m presented in the form of eternal wave motion. However, classical mechanics (rest ing upon formal logic: only Yes or only No) was/is unable to develop a series o f principal notions related to the state of rest, conjugate to the corresponding notions o f the st ate of motion. Moreover, it describes rest and motion separately (independent ly). In Galilean relat ivit y the equivalence of the notion that rest and (uniform) mot ion concerns the contradictory abso luterelat ive character of mot ion and rest (not considered here) and relates to the subjective cho ice of the frame of reference. The introduction of the conjugate parameters makes the phys ics more symmetrical and complete. Indeed, there is the notion of the displacement of an object fro m the state of equilibrium; it is the kinetic displacement. However, there is not the notion the potential displacement. If we recognize that rest and motion are the two sides of the same single process, then the last notion has right away an existence without any doubt . This fo llows fro m the basic laws of dialectical philosophy and dialectical logic (dialectics). Let us recall, in brief, what dialectics is. Dialect ics is highly necessary for understanding the phys ica l questions touched on in this paper, especially since most physicists are not familiar wit h philosophy, in general, and with dialect ical philo sophy, in particular. 1.2. Dialectics Dialectics is an integral part of the foundat ion of world philo sophy. The word "philosophy", in its original broad sense meaning "the love of wisdom", derives fro m the Greek compound philosophia, where the word sophia is ordinarily translated into English as


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"wisdom". According to Diogenes LaÊrtius (probably lived in the early part of the third century), Pythagoras (c. 570-500 B.C., an Ionian Greek born on the is land o f Samos) was the first to begin to call philosophy as philosophia (i.e., the love o f wisdo m) and himself a philosopher (a wisdo m-lo ver). By his words, only God can be the sage, but not a man..., and a philo sopher (a wisdo m-lover) is merely o ne who feels drawn to wisdo m. Sages (and poets as well) were also called sophists (philosophizers) ... Philosophy had two origins: one wit h Anaximander (610 -546 B.C., born in Miletus), and the other with Pyt hagoras. Anaximander was the "pupil and suc cessor of Thales", and he is regarded as the founder of Greek astronomy and natural philosophy. Thales of Miletus (c. 625-547 B.C.) is the first Milesian philosopher, the founder of the ant ique and generally European philosophy and sc ience, and the founder of the Ionian school of natural philo sophy. He proposed a simple doctrine on the origin o f the world: he asserted that all variet y o f things and pheno mena originated fro m a single element ­ water... The first philosophy was called the Ionian philo sophy; the second was called the Italian philosophy because Pyt hagoras was occupied wit h philo sophy mainly in Italy. Some philosophers were called physicists because they studied nature; others were referred to as ethicists, owing to their reasoning on morals and manners; a third group o f philosophers were called dialecticians because o f discussio ns on the justificat ion of speech. Physics, ethics, and dialectics are three parts of philosophy. Physics teaches about the world and all that is in it. Ethics is devoted t o the life and behavior of humans. Dialectics is concerned with arguments for both phys ics and ethics. Unt il Archytas of Tarentum (a bright representative of the second-generation Pyt hagoreans who lived in southern Italy during the first half o f the fourth century B.C.), a pupil o f Anaxagoras of Clazo menae, physics was the only kind o f philosophy. Anaxagoras of Clazomenae (who lived approximately during 500428 B.C. and spent his most active years mainly at Athens) first taught philo sophy professio nally; he first advanced mind as the init iator of the physical world. Dialectics originates wit h Zeno of Elea (c. 490-430 B.C.) [4]. Negating the cognit io n o f the sensit ive being, he showed in his famous paradoxes the contradictoriness o f motion. Another of the founders of dialectics was also Socrates (c. 469-399 B.C.). The word dialectics meant, on the one hand, the search for truth by conversations, which were carried out through the formulat ion of quest ions and the methodical search for answers to them. On the other hand, dialectics means the capabilit y of visio n and reflect ion by means of notions the opposite facets of nature. In the wide sense of this word, dialectics is a skill of many-sided descript ion of an object of thought and a logic formation o f the predict ion of necessary and possible events. Thus, dialectics is regarded as the logic of philosophy and all sciences, i.e., as the logic of cognition on the whole. Dialectics represents a synthesis of the best achievements of both materia lism and idealism and it is the ground for understanding o f the material-ideal essence o f the world. The main postulates of the dialectical philosophy are the fo llowing. 1. The Postulate of Existence ^ The material-ideal World ( M ) exists. Symbo lically, the material-ideal essence o f the world can be briefly presented by the logical bino minal
^ M M iR ,


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where M and iR are, correspondingly, material and ideal components of the world and the + sign expresses their mutual bond. 2. The Postulate of Dialectical Contradictoriness of Evolution Any object or relation A in any instant is in a state of evo lut ion, i.e., it simultaneous equa ls and does not equal to itself: ( A A) ( A A) , where is the sign of logical conjunct ion. The fo llo wing logical antino my corresponds to the aforement ioned bino mial judgment :

(Yes Yes) (Yes Yes) .
Dialectics states that "A is A" and "A is not A" simultaneously. For example, Smit h as a child, youngster, man and o ld man is, on the one hand, the realization of the logical forma l formula "A is A", i.e., Smit h is Smit h. On the other hand, the child -youngster-man-old man series is the manifestation o f non-tautology "A is not A", i.e., Smit h cont inuously changed and Smith as a child is not equal to Smit h as a youngster. At ever y instant he is he, "A is A", and simultaneously he is not he, "A is not A". When we consider fast-changing physical processes at the mo lecular level or deeper, the truth of this postulate beco mes yet more obvious. The logical bino minal o f evo lut ion "A is equal to A and, simultaneously, A is not equal to A" is beyo nd the bounds o f formal Aristotelian rules. Aristotle (384-322 B.C.), who laid the foundat ion of metaphysics and formal logic, was an opp onent of dialect ics. He wrote [5], "There are however people which, as we po inted to, themselves speak that the same can exist and non-exist together and assert that it is impossible to hold this po int of view. Many amo ng explorers o f nature turn to this thesis." Accor ding to metaphys ics, two formally logica l judgments A (Yes) and A (Yes) are always assumed to be only equal through the law of identity: A=A (Yes = Yes). This tautology excludes any possibilit y of mot ion and analysis, and if humans fo llowed this rule in fa ct, the development of human thought would be impossible. 3. The postulate of Affirmation of Dialectical Logic (a) A brief dialect ical judgment about an object of thought is presented, in a ge by the symmetrically asymmetric logic structure Yes-No or No-Yes. (b) Relat ively objects are expressed by the logical structure Yes-Yes, or briefly Yes, and asymmetric by the structure No-No, or briefly No. (c) In a general case, a logical judgment L is the funct ion of the elementary judgments Yes and No, neral case, symmetric relat ively dialect ica l

L f (Yes, No) .
Cognit ion of the World proceeds on the first approximat ion any element o f sides o f comparison. This requires us to judgments of the kind Yes-No. The last the basis o f comparison a a state or a pheno menon describe A by dialectical presents the symmetrical nd through compari son. In of nature has at least two symmetrically asymmetric pair o f judgments Yes and


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No, which are in essence the opposite judgments, so that in this sense both these judgments are asymmetric ones. In a general case, Yes and No are natural judgments about an object of study. They expres s quantitative and qualitative measures o f the object. Here are some examples of po lar-opposite notions: rest-motion, potential-kinetic, continuous-discontinuous, abso lute-relat ive, existencenonexistence, material-ideal, form-contents, basis-superstructure, qualitat ive-quant itative, cause-effect, object ive-subject ive, past -future, necessary-causal, finite-infinite, realimaginary, wave-quantum, particle-ant ipart icle. Chuang Tzu (c. 369-286 B.C., an outstanding representat ive of Tao ism) has written (Ref. [2], p. 215), "In the World, everything denies itself through the other thing, which is it s opposit ion. Every thing states itself through itself. It is impossible to discern (in the one separately taken thing) it s opposit ion, because it is possible to perceive a thing only immediately. This is why, they say: "Negation issues from affirmation and affirmat ion exist s only owing to negation." Such is the doctrine on the condit ional character of negat ion and affirmat ion. If this is so, then all dies already being born and all is born already dying; all is possible already being impossible and all is impossible already being possible. Truth is only inso much as, inasmuch as lie exists, and lie is only inso much as, inasmuch as truth exists. The above stated is not an invent ion of a sage, but it is the fact that is o bserved in nature...". Another Chinese philo sopher Ch'eng Hao (1032-1085) has said (Ref. [3], p. 327): "The highest principle for all things in heaven and on the Earth is that there is not one single thing that is independent, because, it is obligatory, there is its opposite...". In other words, all things do not represent a single whole, but these exist in the form of opposites. His brother Ch'eng I (1033-1107) has stated: "Everyt hing in the space between heaven and the Earth has opposites; if there is the Dark Beginning then the Light One also is; if well is, hence evil is as well", etc. For the descript ion of the opposite properties o f object ive realit y it is convenient to use complex numbers, as the numbers with po lar opposite algebraic properties [6]. The transformation of the kinet ic field into the potential one, or the electric field into the magnetic one, means (in the language o f complex numbers) the transformat ion o f the "real" numerica l field into the "imaginary" one, and vice versa. Thus, as fo llows fro m t he basic law o f dialect ics Yes-No (the law of symmetry and asymmetry Yes and No o f the polar judgments), motion-rest must be described by the conjugate symmetrical parameters. Disregard of the law leads, to put it mildly, to disagreeable consequences for science (see, e.g., Ref. [7]). Correspondingly, the kinetic speed (the first time derivat ive of kinet ic displacement) as the speed o f change o f mot ion must be conjugate with the potential speed o f change o f rest. This supposes the supplementation o f the kinetic momentum wit h the potential momentum. We must operate also with the potential and kinetic force and potential and kinetic work, along with the already-existed potential and kinetic energy. Contemporary phys ics did not develop the notion of the potential-kinetic wave field, which could be regarded as a generalized image of any real physical field (electromagnet ic, for example). It is natural, the above problems also concern the descript ion o f the field of physical (real) time (an ideal field-space o f the Universe), which enters in the triad o f matter-spacetime and differs from the reference (mathematical) time used everywhere. The goal of this paper is an introduction of the above-ment ioned missing conjugate notions (parameters) analyzing harmonic oscillat ions o f a material po int.


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2. Potential-kinetic parameters of harmonic oscillations
2.1. Displacement In dialect ical logic and philosophy, consequent ly, in phys ics as well, the judgment Yes is the qualitat ive measure o f affirmat ion, as such. Concerning its quant itative measure, the last is defined by the measures of studying processes and objects. The implicit dialect ical symbo l Yes is represented by the symbo l of the physical quant ity, which the symbo l Yes expresses logically. Since properties o f the processes and objects, expressed by the judgment Yes, in a genera l case are variable ones, t he dialect ical judgment Yes is a variable quantit y, represented by a funct ion of its arguments. For example, if Yes expresses so me displacement of a materia l point, then the value Yes is equal to the value of the displacement itself. Let a kinetic displacement of a material po int Yes be its displacement fro m the state of equilibrium and defined as
Yes a cos t .

(2.1)

Fo llowing the requirement of symmetry, condit ioned by the dialect ical law Yes-No, one should introduce the notion that will be opposite to the notion of the kinetic displacement, Yes. It is natural to term it the potential displacement, No. The displacement No, as the negation of the kinet ic displacement Yes, can be described by the sine funct ion, since sine is the negation of cosine, just as cosine is the negation of sine. It is natural to accept amplitude of the potential displacement as equal to the amplitude of the kinetic displacement. Apart fro m this, we will present the potential displacement as the negat ion o f the kinet ic one by the ideal number. Thus, in the capacit y o f the potential displacement, we accept the fo llowing measure:
No ia sin t .

(2.2)

Both displacements, reflect ing the indisso luble bond of rest and motion, constitute the ^ potential-kinetic displacement , which we present in the fo llo wing form:
^ Yes No .

(2.3)

If we denote the kinetic displacement Yes as xk and the potential displacement No as ix p , we will o btain the fo llowing dialect ical expression for the potential-kinet ic displacement (Fig. 2.1):

^ xk ix

p

or

^ a cos t ia sin t .

(2.4)


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Fig. 2.1. A graph o f the potential-kinet ic displacement Yes ­ No. The kinetic displacement is the possibilit y o f the potential displacement , and, conversely, the potential displacement is the possibilit y of the kinet ic displac ement. When a material po int passes through the equilibrium state, its motion is more intensive (the maximum o f mot io n takes place). After passing equ ilibrium, the intensit y of mot ion falls and, simultaneously, it increases the extent of rest, expressed through the growing value o f the potentia l displacement. Using Euler's equat ions, we present the potential-kinetic harmo nic displacement as
^ ae
it

.

(2.4a)

The constant component of the potential-kinetic displacement is expressed by the amplitude a, and the variable co mponent is expressed by the ideal exponent ial function. The ideal exponent ial funct ion e it is also the relative measure of displacement , and it s fundamental quantum of qualitat ive changes is

e

it



^ . a

(2.5)

And, because the relation (2.5) is valid for all harmonic potential-kinet ic measures, all these measures have (in the capacit y of a relat ive measure) the ideal exponent ial funct ion. In this sense, their relat ive measures are turned out to be equal between themselves. 2.2. Speed and acceleration The potential-kinet ic displacement defines the potent ial-kinetic speed

^
where

^ d k i p , dt
p

(2.6) (2.6a)

k i ix p x

is the kinet ic speed, i.e., the speed of change o f motion, and


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i p i x

k

(2.6b)

is the potential speed, i.e., the speed of change of rest . Amplitude, or a module of speed, as the total speed, is the constant equal to
a .

(2.6c)

As fo llows fro m the formulas (2.6a) and (2.6b), the kinet ic speed is co nnected with the potential displacement, whereas the potential speed is defined by the kinet ic displacement. The potential-kinet ic speed defines the potential-kinet ic accelerat ion

^ w
where

^ d 2 ( xk ix p ) wk iwp , dt
wk 2 x
k

(2.7) (2.7a)

is the kinet ic acceleration, i.e., the speed of change of the k inet ic speed, and

iwp 2 ix

p

(2.7b)

is the potential acceleration, i.e., the speed of change of the potential speed. 2.3. State

^ ^ In the potential-kinet ic field, a displacement characterizes a potential-kinetic state S of a material point. We define this state through the product of its mass and the displacement:

^ ^ S m sk is p ,
where

(2.8) (2.8a)

sk mxk
are, correspondingly, mot ion. The state of a ^ displacement , i.e., as matter-space). The forms

and

is p mx

p

the kinet ic and potential states of a material po int in the harmo nic
^ material po int S expresses the indisso lubilit y o f its mass m and the indisso lubilit y of matter and space (which is reflected in its writ ing potential-kinet ic harmo nic state can also be presented in the fo llowing

^ ^ S meit a (m cost im sin t )a (mk im p )a ma sk is p , (2.9)
where


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^ m meit mk im

p

(2.9a)

is the kinematic potential-kinetic mass of a material po int in the harmonic oscillat ion. 2.4. Charge and current
^ The measure o f the speed o f change of the potent ial-kinet ic state of mass m in the ^ oscillat ing process is call the kinematic charge Q . According t o this definit ion, the potential^ ^ kinet ic mass m and the kinematic potential-kinetic charge Q are related as

and, modulo, as

^ ^ dm im ^ Q dt
q m .

(2.10)

(2.10a)

The kinemat ic potential-kinet ic charge defines the kinematic potential-kinetic current
2 ^ ^ ^ ^ dQ d m iQ 2 m ^ I 2 dt dt

(2.11)

with the amplitude

I q 2 m .

(2.11a)

The amplitude of the kinemat ic current (2.11a) is called the elasticity coefficient k. This name relates the amplitude of kinemat ic current wit h the biological sensation of exchange o f mot ion-rest. It is analogous to such terms as heat, force, and "fluid" (once used in physics and, actually, related to the mo lecular level of exchanges o f motion-rest). Notions o f dialect ical phys ics are the notions o f exchange of matter-space and motion-rest. We will denote the amplitude of the kinematic current (2.11a) by the symbo l k as well.

2.5. Momentum and force
^ The potential-kinet ic state S defines the field of the potential-kinetic momentum Yes-No:

^ ^ dS m m( i ) p ip , ^ P k p k p dt

(2.12)

where pk and ip p are the kinet ic and potential mo menta. The mo mentum Yes is the kinet ic mo mentum pk mk miix p mx p , (2.12a)


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whereas the mo mentum No is the potential mo mentum
ip p mi p mixk .

(2.12b)

Thus the kinet ic mo mentum is related to the potential displacement and the potential ^ mo mentum to the kinetic displacement. The field of the P -mo mentum is the field of motion^ rest of the first level with respect to the S -state. The field of potential-kinet ic mo mentum defines the field o f the potential-kinetic rate of ^ exchange of momentum F (force):

^ ^ dP f if mw m( w iw ) I , ^ ^ F k p k p dt
where

(2.13)

fk

dpk mwk kxk Ix dt

k

(2.13a)

is the kinet ic rate of exchange of mot ion, expressed by the kinetic mo mentum, and

if p

d ip dt

p

miwk kixp Iix

p

(2.13b)

is the potential rate of exchange of rest, defined by the potential mo mentum. ^ The rate of exchange F is the field of motion-rest of the second level with respect to the ^ S -state and, at the same time; it is the state of exchange, defined by the kinemat ic current. 2.6. Energy As fo llows (2.13),
I ^ F , ^
2t

^ F ^ . I

(2.14)

The integral
^ ^ ^ I A Fd 2 0
t



0

I 2 I02 2 2

(2.15)

^ ^ defines the kinematic work A , and the kinematic energy E is defined by:
2 2 2 2 ^ I k ( xk ix p ) kxk kx p ikx x . E kp 2 2 2 2

(2.16)


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The first and second components of energy (2.16) are the kinetic and potential energies
2 2 pk m2 kxp k , Ek 2m 2 2 2 kxk . Ep 2m 2 2

p

2 p

m

2 p

(2.16a)

The third component is the sum o f potential-kinetic and kinetic-potential energies:

E

pk



kixp xk 2

,

Ekp

kxk ix 2

p

.

(2.16b)

Thus, fo llowing dialect ics, the kinet ic energy is represented by four co mponents: the kinetic energy Yes-Yes, the potential energy No-No, the potential-kinetic energy No-Yes, and the kinetic-potential energy Yes-No. These components logically represent the majo r quaternion of dialect ical judgments/laws: Yes-Yes, Yes-No, No-Yes, and No-No. The potential displacement ix p ia sin t defines the kinet ic energy and the kinet ic displacement xk a cost defines the potential energy. Thus the potential displacement, as the potential displacement, is simult aneously the kinet ic displacement in the sense that it defines the kinetic energy and the extremum o f the state of mot ion. Just so, the kinetic displacement, as the kinet ic displacement, is simultaneously the potential displacement in the sense that it defines the potential energy and the extremum o f the state of rest. There is direct evidence o f the dialectical contradiction, expressed by the law Yes-No. For this reason we can rename the potential displacement as the kinet ic displacement and denote it as ixk ia sin t , and, similarly, the kinet ic displacement as the potential displacement and denote it as x p a cos t . At such definit io ns o f displacements the formulas o f kinet ic and potential displacements, speeds, and energies will take the fo llowing form:

^ x p ixk a cost ia sin t ,

k

dx

p

dt

a sin t iixk ,

i p

dixk ia cos t ix p , dt
p
2 p

(2.17)

2 p 2 m2 kxk k , Ek k 2m 2 2

Ep

2m



m 2

2 p



kx

2 p

2

.

As we see, it is impossible to avoid dialect ics of the law Yes-No by changing the names of the measures into opposite ones. Now the kinetic speed o f mot ion is the derivat ive o f the potential displacement and, conversely, the potential speed is the derivat ive o f the kinet ic displacement. For this reason, if it is necessary to dist inguish rest or motion, we will use the conjugated kinet ic or potential terms. At the circular motion-rest, the energy on the basis of vector measures [1] is


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or

^2 ^2 ^ Fdr mvdv Irdr kr mv , ^^ ^^ ^^ E 2 2

(2.18)

2 2 2 2 ^ mv k mv p 2mv k v p co s m m(i) 2mv k v p co s( / 2) 0 , E 2 2 2 2 2 2

(2.19)

where (see Fig. 2.2)

^ v

^ dr ^ ^ v k v p in dt

or

^ v

^ dr r irn , dt

(2.20)

^ ^ ^ ^ r and v k dr p / dt is the kinet ic tangent ial velocit y, v p drk / dt in is the

potential normal velocit y.

Fig. 2.2. The kinemat ics o f mot ion-rest along a circumference: a) the tangent ial and norma l n units vectors; b) v p ian in is the potential velocit y, v k a is the kinetic velocit y; c) p in is the potential specific velocit y, k is the kinet ic specific velocit y. According to the above and the theory of oscillat ions o f a string and the theory o f circular mot ion [1], the energetic measures o f rest a nd motio n are represented by the opposite, in sign, but equal in value, kinetic and potential energies. Because an insignificant part of an arbitrar y trajectory is equivalent to a small part of a straight line, any wave motion of an arbitrar y microparticle (and, to an equal degree, a macro - and megaobject) is characterized by the kinetic and potential energies, also equal in value and opposite in sign:

m2 k , Ek 2

Ep

m(i) 2

2 p



m 2

2 p

.

(2.21)

Therefore the total potential-kinet ic energy o f any object in the Universe is equal to zero:


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E Ek E p 0 ,

(2.22)

and its amplitude is equal to the difference in kinetic and potential energies:

Em Ek E p m2 .

(2.23)

Under the motion along a circumference (as in particular takes place with the electron in the H atom), the potential-kinetic vector energy of a material point is equal to zero. By virtue of this, the circular motion is the optimal (equilibrium) state of the field of rest -motion, where "attraction" and "repulsion" are mutually balanced, which, in turn, provide for the steadiness of orbital motion in the micro- and macroworld. The quantitative equality of "attraction" and "repulsio n" is acco mpanied, simultaneously, by the qualitative inequality of the directions o f fields of rest and motion, which generates the eternal circular wave mot ion. In order to break such a mot ion, it is necessary to destroy this system ent irely. However, in this case, a vast number of new circular wave mot ions o f more disperse levels will appear as a result.

3. Physical time
^ The wave function,
^ ^ ^ ^ ^ R(r)()()T (t ) (r , , )T (t ) ,

satisfying the ordinary wave equat ion
2^ ^ 1 , 2 2 0 t

describing arbitrary periodic processes running in space and t ime, is the mathemat ica l expressio n of the indisso luble bond of the fields o f material space and physical t ime. The t ime ^ funct ion T (t ) (its simplest solut ion is T e it ) expresses the alternating physical time field by means o f the variable t, which represents the ideal mathematical time of the imaginar y abso lute uniform motion. The real t imes o f natural processes are compared wit h this mathemat ical t ime. We call mathemat ical t ime the absolute or reference time. The real (physical) time, as the measure of pure motion-rest, must also be potentialkinetic. Let us show this. By analogy wit h the absolute time


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t

l ,

the physical time of harmonic oscillations t^ is defined as the ratio o f the potential-kinetic ^ displacement to the module of potential-kinet ic speed :
it ^ ^ ae t m e t a it

t k it p .

(3.1)

In this expressio n, tm 1 / T /(2) is the module of the potential-kinet ic t ime. The kinetic and potential times,

t k t m cost

and

it p it m sin t

(3.2)

are funct ions of the uniform mathematical t ime t. In the capacit y of the basic unit of phys ica l time, we accept the second of the abso lute time. The physical time allows the more co mplete description o f the dialectically contradictory potential-kinetic processes. The physical t ime is the t ime of the logical structure Yes-No. As fo llo ws fro m the definit io n of physical t ime,
^ ^ t ,

xk t k ,

ix p it p .

(3.3)

The physical t ime repeats the form o f the potential-kinet ic displacement. The equat ions of displacements (3.3), defined by the physical t ime, are similar, in form, to the equation o f displacement l in the uniform mot ion on the basis of reference time t:
l t .

(3.4)

By analogy wit h the relat ions between contents and fo rm, we express the relat ions between the extensio n of space and the duration of time through the speed as

0 r ,
where 0 1 cm / s is the abso lute unit speed and r is the relat ive speed. We also introduce the "inverse speed" according to the equalit y

(3.5)



1 0 r ,

(3.6)

where 0 1 s / cm is the abso lute unit o f the inverse speed and r is the relative inverse speed.


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Rest ing upon (3.5) and (3.6), we can rewrite the equation o f displacement (3.4) in two ways:

l 0 r t ,

t 0 r l .

(3.7)

^ Analogously, we express the relat ion between the displacement and time t^ :

^ ^ 0 r t ,

^ ^ t 0 r .

(3.8)

The phys ical potential-kinet ic time o f harmonic oscillat ions in wave pro cesses is the wave time field. It also is the ideal space of matter. Just this wave potential-kinetic time field enters in the dialectical triad matter-space-time. The physical t ime o f harmo nic oscillations ^ t tmeit tk it p runs nonuniformly with the time potential-kinetic speed
^ ^ dt ie dt
it

k i p ,

(3.9)

Where

k
and

dtk t m sin t sin t dt

(3.9a)

i p

d it

p

dt

it m cos t i cos t

(3.9b)

are the kinetic and potential time speeds, correspondingly. ^ The derivative of any funct ion (describing an arbitrary phys ical fie ld) with respect to ^ ^ some argument defines a new field d / d . This field is the field of negation of the ^ initial field. Correspondingly, the derivat ive d / d defines the field o f negat ion o f the field

^ ^ ^ ^ , etc. Thus, the field of the second derivat ive d 2 / d 2 of -funct ion is the field of ^ ^ negation of negation of the field, or the field of double negation . ^ In such a case the field, defined by the derivative dt^ / dt , represents by it self the field of negation of the field of physical time. This new field is the time field of potential-kinetic motion of time. As such, it is the quantitative-qualitative field of the Universe, because quantity and quality exist objectively in it. Its subject ive image is the dialect ical numerica l field of affirmat ion-negat ion Yes-No [6]. The quant itative-qualitative field o f change o f the physical t ime is simultaneously the material-ideal field, because quant ity and qualit y are in the same relation, as material and ideal. The space and time speeds are related by the fo llo wing equalit ies:

^ ^ m ,

k m k ,

i p i m p .

(3.10)


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The kinet ic and potential energies, expressed with use of time speeds, have the form
Ek Em 2 , k

E p Em2 , p

(3.11)

where Em m2 / 2 is the amplitude o f kinemat ic energy. m

4. The wave equation of time field-space
The potential-kinetic parameters of oscillations have the universal character and are applied to any ^ potential-kinetic waves of matter-space-time. Relative measur es r of all potential-kinetic parameters of harmonic oscillations of equal frequencies, expr essed through a mplitudes, are equal to the sa me ideal exponential function

^ r

^ e it . a

(4.1)

In this sense, all these measures are identical. For the description of waves of differ ent nature, the wave spatial vector k, related to the basis of the wave, is used. It is deter mined by the equality

2 1 ,

(4.2)

wher e n is the unit vector dir ected along the wa ve extension, is the length of the spatial wa ve, and is the wave radius. We supplement the k-vector with the analogous wave time vector , conjugated to k,

1 2 . t T m

(4.3)

Comparing the vectors, (4.2) and (4.3), we see that, at the level of the basis of time wa ves, the period T is the time wave conjugated to the space wave . The module of physical potential-kinetic time tm (see (3.1)) is the radius of time circu mfer ence T ( T 2t m ), wher eas in wave processes it is the wave time radius (compare and tm in (4.2) and (4.3)). The vectors k and are connected through the equality:

c c0 cr ,

(4.3a)


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wher e c c0 cr is the basis wave speed, c0 1 cm / s is the unit speed, and c r is the r elative speed. Thus the physical time of unifor m motion, equivalent to the r efer ence time, is contradictory: being the scalar magnitude it is simultaneously the vector magnitude. For the descripti dimensional space of axes Tx, Ty, and Tz. constant amplitude a, on of the p the absolute If a spatial is travelling hysical time field-space we use the r efer ence r ectangular threetime. Na mely, we present it by the frame of refer ence with the time ^ wave bea m of harmonic potential-kinetic oscillations , with a along the x axis, then its equation has the for m

^ ae

i ( t kx )

.

(4.4)

^ The following wa ve bea m of harmonic potential-kinetic oscillations of time field t corresponds to it:
^ t tme
i ( t kx )

.

(4.5)

Harmonic bea ms with arbitrary constant a mplitudes and equal fr equencies are conjugated to the time wa ve bea m of the sa me a mplitude tm. This amplitude is expressed through the a mplitude of its own oscillator y speed, i.e., the speed of superstructure. In view of this the measure of the a mplitude of the time har monic wave does not reflect the measure of the a mplitude of the conjugated spatial wave. In order to ma ke the time a mplitude reflect the measure of the spatial amplitude, we should introduce the relative time amplitude m equal, by the definition, to the ratio of spatial a mplitude a to the unit linear speed-density c0 1 cm / s :

m

a . c0

(4.6)

^ m and the relative measure of the bea m-wave r :

Now we can accept, as the measure of the time wave, the product of the relative time a mplitude

^ ^ m m r meit .
If waves of the kind

(4.7)

^ me

i ( t kr )

(4.8)

arise along the three axes of Cartesian coordinates x, y, z , the following time thr ee-dimensiona l wave field-space is for med

^ T xm e

i ( xt x k x x )

ym e

i ( yt y k y y )

zme

i ( z t z k z z )

,

(4.9)


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wher e x , y , z are components of the time wave vector and t x , t y , t z are components of the vector of absolute time t. Fields-spaces of the structure (4.9) are multiplicative fields-spaces because spatial and time waves (components) in it are multiplicatively linked together. In other words, the principle of multiplicative superposition is valid for such fields. These are spaces -systems, or atomic spaces [1]. The sums of the multiplicative atomic fields-spaces for m complicated fields-spaces, which can be called molecular spaces. These are additive fields-spaces, since the principle of additive superpos ition is valid for them.

^ The wa ve function of the thr ee-dimensional wave field of physica l time is the mathematica l ima ge-measure of the wave three-dimensional time space. The three-dimensional time wave is represented by its multiplicative comp onents-wa ves:

^ x xm e

i ( xt x k x x )

,

^ y ym e

i ( yt y k y y )

,

^ z zme

i ( zt z k z z )

.

(4.10)

^ Because t x t x y t y z t z , the T -ima ge of the thr ee-dimensional wave (4.9) can b e
presented as

^ T xm ym zme

i ( t k x x k y y k z z )

or

^ T Tme

i ( t k x xk y y k z z )

.

(4.11)

In a general case the wave a mplitude is variable a nd, in the wa ve stationary field, depends on coordinates, so we expr ess it in the following wa y

^ T Tm (k x x, k y y, k z z )e

i ( t k x x k y y k z z )

.

(4.12)

Assuming that the a mplitude Tm is a constant within the space of a material point, we have

^ 2T ^ k x2T , 2 x

^ 2T 2^ k y T , 2 y

^ 2T ^ k z2T 2 z

and

^ ^ ^ 2T 2T 2T ^ 2 2 k 2T . 2 x y z

This equation can be also presented as

^ ^ ^ 2T 2T 2T ^ T (kx) 2 (ky) 2 (kz) 2

or

^ ^ ^ k k T k T T ,

(4.13)


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wher e

k

kx ky kz

(4.14)

^ is the vector of the negation of the time field and
2 2 2 (kx) 2 (ky) 2 (kz)

kk 2 k k

2

(4.15)

is the operator of negation of negation, or the operator of double negation. In essence, the equation

^ ^ k k T T or

^ ^ k T T ,

(4.16)

is the differential expression of the dialectical law of negation of negation:

Yes Yes ,

or

Yes Yes ,

(4.17)

^ ^ wher e Yes is a judgement about T , i.e., Yes T , and

k .

Thus the wave equation of time field-space (4.16) is one of the forms of the universal law of dialectics the law of negation of negation, or double negation. Since

^ 2T ^ T (t ) 2

or

^ 2T ^ T , 2

wher e t is the relative linear refer ence time, corresponding to the r elative reference distanc e kr , the equation (4.13) can be presented in the following for m

^ ^ ^ ^ 2T 2T 2T 2T 2 2 2. 2 x y z

(4.18)

^ This equation mea ns the equality of time double and spatial double negations of the T -ima ge of the physical wave time field-space. Because


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2

2 2 2 2 2, 2 y z x

the wave equation (4.18) can be written as

^ T

^ 2T . 2

(4.19)

In the language of dialectical logic (4.19) repr esents the la ws of double spatial and double time negations:
2

^ T

2

^ ^ T T ,
2

(4.20)
2

wher e
2

2 ,

2 /

(4.21)

are the logica l operators of double spatial and double time negations, corr espondingly. The physical wave time field-space is inseparable fr om the wa ve field of space of ma tter of the ^ ^ same structure, because the time wa ve T repeats the structure of spatial waves (compare

^ ^ 2 / 2 with (4.19)).

5. Conclusion
1. The kinetic-potential parameters of displacement, speed, acceleration, state, mo mentum, force, energy, charge, and current were first introduced for the description o f harmo nic oscillat ions. These symmetrical binary potential-kinetic parameters give the more complete descript ion of potential-kinet ic fields of any nature. 2. The introduced parameters of oscillat ions have the universal character and are applied to any potential-kinetic waves of matter-space-time. At that, we should ment ion one result especially: it was shown that the total potential-kinet ic energy o f any object in the Universe is equal to zero. 3. The difference between reference (mathemat ical) time and physical (real) t ime has bee n revealed. The phys ical t ime wave fie ld is an ideal fie ld-space o f the Universe. Just physical time enters in the triad of matter-space-time. For its descript ion the notions o f the wave potential-kinetic t ime field and corresponding time potential-kinet ic parameters were introduced. The wave time vector was introduced (conjugated to the wave vector k) to allow us to consider the period T as the time wave conjugated to the spatial wave . 4. The wave funct ion o f the three-dimensio nal wave fie ld of physical t ime, as the mathemat ical image-measure of the wave three-dimensio nal t ime space, sat isfies the wave equation o f t ime field-space. This equat ion reflects the universal law of dialect ics ­ the law o f double negat ion. The physical wave t ime field-space is inseparable fro m the wave field o f


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space of matter of the same structure, because the time wave repeats the structure of spat ia l waves.

References
[1] [2] [3] [4] L. Kreidik and G. Shpenkov, Atomic Structure of Matter-Space, Geo. S., Bydgoszcz, 2001, 584 p. Anthology of the World Philosophy, V.1, Part 1, "Mysl", Moscow, 1969. A History of Chinese Philosophy, "Progress", Moscow, 1989. Zeno of Elea. On the Life, Doctrines, and Dicta of Famous Philosophers, "Mysl", Moscow, 1979, pp. 66-67. [5] J. L. Ackrill, Aristotle: Categories and De Interpretatione, Calendon Press, Oxford, 1963; W. D. Ross, Aristotle's Metaphysics, Calendon Press, Oxford, 1924. [6] L. Kreidik and G. Shpenkov, Philosophy and the Language of Dialectics and the Algebra of Dialectical Judgements, Proceedings of The Twentieth World Congress of Philosophy, Copley Place, Boston, Massachusetts, USA, 10-16 August, 1998; http://www.bu.edu/wcp/Papers/Logi/LogiS hpe.htm [7] L. Kreidik and G. Shpenkov, Important Results of Analyzing Foundations of Quantum Mechanics, Galilean Electrodyna mics & QED-East, Special Issues 2, 13, 23-30, (2002).