A search for equations of motion as a task of temporology*
Alexander P. Levich
Biological Faculty of M.V. Lomonosov Moscow State University,
Web-Institute of Time Nature Explorations, http://www.chronos.msu.ru
1. What does a dynamical theory begin with?
A task of the scientific approach in natural science can be formulated as the ability to forecast the variety and variability of natural objects.
In my opinion, the basic motivation in studying the time phenomenon is a hope to discover the ways of finding the laws of variability. One of the basic objects of science is to obtain these laws, otherwise it becomes impossible to fulfil the forecasting function of the cognition. Apparently it is impossible to find the laws of variability without having the correct causal and parametric descriptions of time.
A dynamic theory describing any fragment of the reality inevitably includes a number of components whose development forms, deliberately or, more frequently, implicitly, the stages in the process of creating a theory [1, 2]:
- The O component is a description of an idealized structure of the theory's elementary object.
- The S component lists the possible states of the objects. In other words, the S component is called the space of states of the studied system.
- The C component fixes the ways of object variability and corrects superfluous idealization connected with object selection, since there are no objects in the real world, there exist processes whose abstraction leads to the notion of objects. The C component inserts processes, variability, i.e., the "pre-time", to the theory.
Instead of rigorous definitions, I would like to give some examples of elementary objects and their variability.
In classical mechanics the elementary objects are material points with their positions and velocities in the physical space. For instance, the planets of the Solar system. The variability is determined by trajectories of the points. The space of states is the six-dimensional phase space, the product of the three-dimensional Euclidean space and the three-dimensional space of velocities.
In quantum mechanics the elementary objects are the probability amplitudes of the states of microobjects (for instance, the energy states of an atom). Variability in the space of states is determined by trajectories of vectors in the infinite-dimensional Hilbert space.
In nuclear theory the elementary objects are nucleons and some other elementary particles with their specific sets of quantum numbers. The variability consists in mutual transformations of particles and radiation. The space of states is confined to the combinations of quantum numbers for the transforming particles which satisfy the conservation laws.
In embriology the role of an elementary object is played by a living cell, while the variability is the process of cell fission. The space of states is described by archetype morphological indications in zoological taxonomies.
In ecology of communities an elementary object is a population of organisms. The variability consists of births and deaths of individuals. The space of states can be described as a set of all possible vectors (n1, n2, ..., nk) where n is the number of individuals of the i-th species – member of the community. The numbers are limited by the available resources of the environment.
- The T component of a theory consists in the introduction of clocks and parametric time into object description. The parametric time can be understood as an image of changing objects when the variability process is mapped into a linearly ordered set with a metric (generally a number set). The variability of a selected object is usually taken as a standard to be used in the measurements of other variabilities. A clock is just a natural object whose variability is a standard and an operational way of the above mapping.
Traditionally in natural science the clocks are based on physical processes, such as constructions with an elastic or gravitational pendulum, and on astronomic devices fixing the Earth's rotation around its own axis or around the Sun. Modern clocks use caesium or other sources of electromagnetic oscillations; from recent years the pulsar standard of hyperstable periods is widely discussed; radioactive decay can also be used. Here is A.A. Friedmann's [3, p. 50-53] description of the emergence of physical clocks: "Let us relate... a certain basic motion to each physical point and define a clock of the given point M as an instrument showing the length t of an arc covered by the material point in its basic motion along the trajectory... Let us call the quantity t the local physical time of the point M. Consider first the stellar time... Define the basic motion to be the motion of the end of an arrow of a specified length, directed from the center of the Earth to a selected star. The stellar time t will be measured by the length of the arc stroke by the above arrow end. The stellar time t will be the same in all the points of the space, it will be a universal time... Now let us consider another time, called for brevity the gravitational time... Let a material point fall in a constant gravitational field and let us take this motion to be the basic one; the clock shows the length of the path, t , passed by the point. This quantity is the one to be called the gravitational time. The stars move non-uniformly with respect to the gravitational time... Let us introduce also... the pendulum time. Let us build a large amount of equal clocks with a pendulum and define the basic motion to be that of the end of the second arrow of the clock at each point. Denote its path from a fixed initial position by t which will be called the pendulum time. Unlike the universal stellar and gravitational times, the pendulum time will be local and different at different latitudes".
Variability parametrization using physical clocks pierces through nearly all the human existence controlled by consciousness, including science, culture and everyday life... However, the changes occurring in the world, cannot be reduced to mechanical motions: there exist, for instance, chemical transformations of matter, the geological history of the Earth, the development and death of living organisms and whole communities, non-stationarity of the universe and social genesis... Would it be incorrect to recognize that the clocks which we pose in different frames of reference to describe the variability of natural objects, can be different? Can one assert in these conditions that some of the clocks, for instance, physical, are "good" while the others are "bad" ?
Such an estimate would be understandable if it concerned, e.g., Galilei, who tried to determine the law of motion of the physical pendulum, the temple chandelier, using his "physiological clock", the rhythm of his own heart.
A. Poincaré stressed that "... a way of measuring time which would be more correct than another one, does not exist. The one adopted is just more convenient than the others. Comparing clocks, we cannot say whether one of them operates well or not, we can only say that one of them is preferred" . In the nonphysical branches of natural science more and more frequently there appears the need to use a clock unsynchronized with the physical standards but more convenient and adequate than the latter when unphysical phenomena are to be described.
In embriology the development of different organisms is effectively described using the biological time unit equal to the interval between the same fission phase . The above unit ("a detlaf") depends on the temperature and the species, therefore the laws of development revealed using the description in detlafs, remain undiscovered when the astronomical time is used. The populational time in ecology , ethnography  and genetics  is conveniently measured by the number of changed generations. The chronostratigraphic geological time scale is formed from a sequence of rocks with standard points selected in open-casts with the best preserved boundary layers . For biology-based stratigraphy the geological epoch durations can be measured by the vertical thickness of the layers where the corresponding species are met .
In a psychological time model  the durations of time intervals between the events significant for an individual are measured by the number of connections between the events.
The main difference between possible clock types is the irregularity of their speed . More rigorously, the time intervals which turn out to be equal when mesured by one clock become different when using another clock. Thus, to be able to measure the variability, it is necessary to agree which reference process measures time intervals, taken, by agreement, to be equal.
The necessity of such an agreement is realized by natural scientists : A priori we can take any dynamic phenomenon and use its development to define the time scale. However, a uniform natural scale does not exist, since we cannot say what is meant by the word "uniform" with respect to time; we cannot catch the present minute and put it side by side with the next one. It is sometimes said that a uniform time scale is defined by periodic phenomena. However, allow me to ask a question: can anybody tell us that the two periods, following one another, are equal?
In physics the role of a uniformity agreement is played by Newton's first law: the time intervals during which a body, moving without interaction with other bodies, covers equal distances, are called equal .
The L component of the theory is the formulation of the variability law which selects the real generalized motion of the objects in the space of states among all the possible motions (the term "generalized motion" is used as a synonym of object variability).
In mechanics and field theory such a law usually has the form of the "equations of motion" which are postulated in the theory, for instance, Newton's equations for motions of macroscopic bodies with small velocity and weak fields, or the Schroedinger equations of non-relativistic quantum mechanics, or the Maxwell, Einstein, Dirac equations, etc. The law can be formulated in a form other than equations, such as, for instance, an extremum principle, like the minimum action principle (only those trajectories are real for which the temporal integral of the difference between the kinetic and potential energies is the least). The formulation of the variability law using the equations of motion is equivalent to that of the extremum principle. The "derivations" of the functionals used in the extremum principles often include considerations connected with the invariance properties of the space-time or field variables.
If the action functional of the studied system is known, the dynamic equations, e.g., in quantum mechanics can be obtained using the Feynman path integral method . The least action principle turns out to be a specific case of the Feynman principle.
An unusual method of variability law derivation, in particular, in the form of the Newton and Dirac equations, appears in the physical structure theory and binary geometrophysics [16, 17]. Formally the laws follow from the requirement that a certain specially constructed Gram determinant should be zero.
For many fields of natural science (in particular, for the already mentioned nuclear theory, embriogenesis and ecology) the variability law formulation is the aim of theory construction. This aim cannot be achieved if the classes of problems forming the O, C, S, and T components of the theory, are not solved in a consistent way. In natural science methodology the C and T components are elaborated less than the others. There is a close connection between the choice of these components and the way of obtaining the L component. According to , the law of motion is a description of the variability of the studied object in terms of the variability of the standard clock, therefore the ability to discover the variability law can depend on the adequacy of the standard clock selection to the studied processes. The laws of motion affect the ways of measuring time in the domains where the T and L components agree with each other , for instance, "...simultaneity of two events or their sequential order, and the equality of two durations, must be defined in such a way that the formulations of the natural laws would be as simple as possible" .
It seems that the difficulties in obtaining the equations of motion in many fields of science are connected with the disagreement of the physical methods of time measurement with the unphysical nature of the studied laws.
Finally, the I component of the theory is formed from the set of interpretation procedures. Firstly, that is the procedure of relating the mathematical constructions of the theory, which have, as a rule, a formal character, to the abstract notions of the reality; secondly, those are the rules of how these abstract subject notions are related to the quantities measured in the experiment.
Thus the formalism of quantum mechanics uses the complex-valued wave functions and the operators acting upon them as the above formal objects. A transition to the concepts of the macroscopic reality is carried out by certain postulated rules: the absolute squared value of the wave function is the probability of finding the particle in the specified point of the space at a given time instant, while an eigenvalue of an operator is a numerical value of the corresponding physical quantity. Observation of the probability distributions requires, for instance, interference experiments with particles penetrating through barriers. The energy characteristics of an atom are determined via spectral line spacings in the experiments with radiation emission and absorption by the atoms.
The I component is a necessary ingredient of a theory. It is the interpretation procedure that turns a formal theoretical scheme into a science studying the reality. The possibility of developing the I component depends not only on the advantages of the theoretical scheme, and often not even as much so, but also on the "technology sum" achieved by the whole civilization.
Democritis' conjecture on the atomic structure of matter required a millennium to become a verified theory.
The enormous experience of the X-ray structure analysis has turned out to be necessary to transform the discrete heredity substance conjecture, the one put forward nearly a century before, into a consistent model involving a double spiral of the deoxyribonucleic acid.
The interpretation procedures are extremely ambiguous. Elaboration of the I component often turns out to be the most difficult and the most vulnerable stage in the creation of a working theory.
2. What does it mean - to study time?
Advances in studying time are modest not only on the scale of the twenty-year history of the Russian Inter-Disciplinary Seminar on Temporology (http://www.chronos.msu.ru/seminar/eindex.html), but also on the scale of more than 300 years of history of European science, if we count it from Isaac Newton's "Philosophiae Naturalis Principia Mathematica", and on the scale of more than two thousand years of ancient science if one refers to the achievements of Aristotle's "Physica".
The direct methods of sciences have turned out to be underproductive: the experimental approach, numerous attempts of simulation, embedding into the conceptual framework of science (time is a kind of matter? a field? a coordinate axis? a property of consciousness? a construct of thinking?...) Many difficulties in time studies may turn out to be caused by imperfection of the scientific methodology as well as by insufficiency of the "summa technologiae" available to the modern civilization.
In the modern knowledge, time is an initial and undefinable notion. Science cannot work without such notions, but it does not study them.
Using the idea of time rests on the researcher's intuition, on his unreflected professional experience, on element of his (often unrealized) world outlook, not necessarily scientific. There have been unjustified hopes to introduce a unique instrumental concept of time: clocks can be absolutely different in nature and can be irreducible to each other in the properties of time they create .
To answer the question "what is time?" means to replace the image of time in the conceptual basis of science with some other basic concept able to be a platform for discussing the concept of time itself. Thus, figuratively, the properties of time become "theorems" instead of "axioms". Only being removed from the set of undefinable notions, time can become an object of scientific research.
Time is not an isolated "brick" in the conceptual foundation of the knowledge. The idea of time is deeply intertwined with other initial concepts of space, matter, charges, interaction, energy, development, life, consciousness and many others. S.V. Meien (in his private correspondence) expressed this very artistically: "Each time when I read the words `what is this or that', a question occurs to me: what do these words mean? Which kind of answer is expected? Just a definition? But one cannot give definitions to philosophical categories and natural taxons. I have a strong suspicion that an answer to `what is...' with respect to a consistent notion requires an exposition of a large fragment of the Universe (world attitude etc.) with the object to be characterized embedded in this fragment. Thus, one cannot give definitions to the Moon, a sunflower, the force of gravity etc. It is then necessary to expound certain pieces of astronomy, botany, physics and embed the corresponding notions in these pieces, indicating their place. The same story with time. To answer the question of what is time, one has to expound a piece of his world outlook (general, special, scientific etc.) and to place there the notion of time".
Reconstruction of a foundation by replacing a single brick is impossible. One has to re-build a vast region. One should, in fact, speak of building a new "picture of the World" able to be a basis for new dynamical theories. Creating a picture of the World becomes, for a theorist of natural science, a necessary stage in bringing the initial concepts of the theory in agreement. What is the status of such an activity? Is it science? Philosophy? Metaphysics? Natural philosophy? Art? Fiction? And, if it is a science, what are its name and status? In any case, such a sort of activity balances on a thin edge between positivism and basic methodology, between amateurishness and the work of a professional theorist.
3. Two kinds of physics?
"Two kinds of physics" is a metaphor, which can, however, stress that, in scientific research (physics yields the brightest example) there are at least two kinds of activity from the viewpoint of importance of the above-enumerated components.
The usual activity of a theoretical physicist consists in a search for and interpretation of solutions to a known set of basic equations. (Examples: the Hamiltonian equations in classical mechanics, the Maxwell equations in electrodynamics, the Schroedinger or Dirac equations in quantum mechanics, the Einstein equations in general relativity, the Boltzmann equations in statistical physics... The list may be continued, but it will not be too long.)
Another kind of activity is aimed at a search for (or guessing) the basic equations themselves. Solving such problems necessarily includes an analysis of the basic components of the theory: elementary objects, their space of states, their variability and its measurement.
Many thousands of researchers are involved in the first type of activity, and only tens in the second type. Among them are those whose names became the names of the equations found.
The first kind of activity is an everyday work of many generations of researchers during hundreds of years since the beginning of science. The second kind comprises short periods of a few years (or maybe decades) in the formation process of each theory.
With the resulting ratio of "man-years", it is not surprising that there has formed an opinion according to which a correct activity in physics is to be able to skillfully solve the known equations and to precisely calculate observable effects on the basis of such solutions. As to the questions of the origin of the equations and of the meaning of basic concepts, they are merely "philology", as expressed by Lev Landau, great physicist and positivist.
Using an industrial terminology, one can say that solving the equations is a methodologically equipped trade, a well-developed scientific technology (which, however, as any other activity, requires talent, insight and good luck). Unlike that, creation of equations is a handicraft, a piece-work on the verge of art of plausible reasoning, half-empirical arguments and intuitive forecast.
The components of scientific theories that precede equation solving are cursorily, or in the form of terms, mentioned in the process of training the researchers (the brightest examples are space, time, interaction and mass). It is implicitly meant that the undefinable notions and a huge set of their empirical prototypes are intuitively known to the students and that, moreover, they are the same for different carriers of the knowledge. Views of such a sort cause the majority of mutual misunderstandings, struggle between scientific schools and difficulties in both intra- and interdisciplinary communication. This is, though marginal, an inherent part of the scientific paradigm.
4. The structure principles of science
Let us discuss in more detail the starting stages of theory creation, namely, the choice of elementary objects and ways of their variability. The corresponding components of the theory have acquired the name of "structure principles" . Here are some examples of structure principles:
The atomistic doctrine.
Material points in the phase space of positions and velocities in classical mechanics.
Planetary model of the atom, the structure of an atomic nucleus.
The world of elementary particles and physical fields.
The physical vacuum concept.
The geo- and heliocentric systems of the near space.
Cosmology of the expanding Universe.
Everett's parallel universes.
The cellular structure of the organism.
The bacterial nature of the infectuos diseases.
The discrete nature of the biological heredity.
The population, trophic and other structures of ecosystems and the Earth's biosphere.
Plate tectonics in geology. Shell structure of the Earth's interior.
The class theory of the society.
The structure principles determine for many years the functioning frameworks of whole sciences. The structure principles are something "self-evident", often unconscious, unreflected as something having an alternative, but they form a necessary part of any knowledge. The status of the principles themselves is quite different, from strict scientific facts to symbols of faith and evident fallacies. Thus, Democritus' atomistic hypothesis is about 2400 years old, but as late as nearly a century ago there was a marathon dispute between the great scientists L. Boltzmann and E. Mach on whether or not the atoms really exist. It required about a century to embody G. Mendel's hypothesis on discrete units of the genetic code in the image of the double spiral of desoxiribonucleic acid. But, as expressed by M. Ichas : "The most difficult point in the `code problem' was to realize that the code does exist".
Among possible prerequisites which lead a researcher to forming structure principles, there are empirical generalizations, fragments of scientific theories, intuitive insights, borrowings from scientific or non-scientific pictures of the World, philosophical elements of a world outlook, artistic images, and so forth. The structure principles are, as a rule, postulates rather than logical inferences, therefore it is not so important which paths have led to them. Of importance is the result: how close to reality are the immediate and remote consequences of our belief in the existence of the principles themselves.
Summarizing, I would note that our knowledge contains a huge stratum of precisely that "philology" which is renounced by researchers with a positivistic mood.
5. It is necessary to "excogitate hypotheses"
It is meaningless to try to explain time without reworking the whole conceptual basis of the knowledge, since otherwise any explanation will rest on this basis which already contains the "bricks of time". In my view, the methodological difficulties, both past and existing now, in time comprehension and in a search for the necessary equations of motion, are connected with the absence of structure principles postulating certain elementary objects and their properties suitable for modelling the World variability. In understanding the nature of time, we are lacking some new essences able to replace time in the conceptual basis of science. Any attempt of conceptual comprehension of time or a search for its equations must begin with introducing appropriate structure principles into the everyday life of science, or - which is the same - with an introduction of a certain fragment of the World picture. These principles may reflect quite different approaches to the search for solutions. It is only important that the "excogitation" stage concerning principles and a consistent picture of the World is mandatory and inevitable. The art of choosing postulates is also science. Maybe, one of the hardest sciences. A way from the structure principles, from a consistent and contradiction-free picture of the World, via a formal theory to reality is, as a rule, a way whose duration is life, and frequently it is life of several generations of researchers, free of timeserving and fearless at unfair criticism.
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* The work is supported by Russian Fund of Basic Research (grant 05-06-80062).