Mathematics in the discovery of physical laws

Yu.P.Khapachev, A.A.Kairova (Nedova)

"Vita brevis, ars vero longa, occasio autem praeceps,
experientia fallax, judicium difficile"
Hippocrates
Life is short, the path of art is long, opportunity is fleeting, experience is deceptive, and judgment is difficult. " Hippocrates

Vladimir Igorevich Arnold (June 12, 1937 — June 3, 2010) argued that "mathematics is needed to discover new laws of nature, and not to "strictly" substantiate obvious things"

"There is practically no evidence in physics. Every statement in physics can be shown. This is a weaker statement than to prove, although it is also meaningful." S.P.Kapitsa [1].
In this article, using some examples, we will how  , thanks to mathematics, showing into the Proof of the Law of Nature.

 Philosophers believe that facts give rise to ideas,
and in a sense this is true. But I find the following
in the history of natural science: in order to understand
the facts, it is necessary to have certain ideas in your head,
and that you may not see with your eyes what your mind sees.
J. von Liebig
 
Electricity and magnetism were recognized as a single field and a law of nature with the advent of James Clerk Maxwell's equations in the 19th century. Maxwell's equations combined previously established experimental facts of electric and magnetic phenomena and gave them a clear mathematical form. Maxwell's equations are an example of a fundamental physical law, clearly GUESSED, and not "deduced", in the rigorous sense of the word, from experimental data. Electromagnetic waves were predicted by Maxwell and only discovered 25 years later in Hertz's experiments. It is noteworthy that the inclusion of the famous additional term, the displacement current, in the equations was not absolutely necessary: neither the facts known at that time, nor the prevailing physical ideas, nor the requirements of the mathematical consistency of the theoretical apparatus [2].

Zero law of thermodynamics. For each thermodynamic (TD) system there is a state of thermodynamic equilibrium which, under fixed external conditions, spontaneously reaches in time. This property is specific to TD systems and is required for them without exception. TD equilibrium as a concept is a state where the macroscopic parameters of the system do not change with time and when there are no flows of matter and energy. In macroscopic theory the zero law is a generalization of everyday experience and observations of TD systems. However, from a microscopic point of view, this statement is far from obvious. Poincar; in 1890 proved the socalled recurrence theorem. This theorem states that the mechanical state of an isolated system does not in any way turn into a certain stable equilibrium state with time, but it is reproduced with a given accuracy after a finite period of time. However, this period of time for a system consisting of one mole of a substance is by the roughest estimates 10 ; ( N is the Avogadro number ) It is clear that the age of the Universe in comparison with this quantity is only a moment. In addition, the states recorded using macroscopic devices no longer represent purely mechanical states. However, this problem of recurrence certainly has a fundamental interest.
The Poincar; return time does not show the time of an exact return to the previous state of the system, but the time of a return to a statistical state. I.e. a burnt match will not become whole on its own and dead people will not be resurrected. Simply the entropy of the system will return to the previous smaller value. And this is "a little different". The Poincar; return time is as colossally large as the distance to the nearest copy of our Universe. And with this "copy" too, it is not so simple (this is also "a little different")[3,4, 5].

Max Planck. Next, the Report of February 14, 1900, "On the emissivity of a black body."For the first time, Planck introduced a constant on it that determines the value of the minimum action h = 1.05 10^-34 J s. What is the meaning of this fundamental value? The fact is that in classical physics, quantities such as, for example, momentum – p, energy – E, action (there is also such a quantity, its dimension is energy ; time), can take on any arbitrarily small values. However, as soon as we "enter the microcosm", i.e. we are interested in objects whose dimensions are ~ 10^-7 cm, the situation radically changes. For example, an action can no longer be arbitrarily small. Equal to zero, please, but the first, its smallest value turns out to be equal to this particular Planck constant. The next largest action value will be 2h, then 3h, etc. Other physical quantities, such as energy, find themselves in a similar position. Thus, it turns out that not only matter is discrete, but also a number of physical characteristics describing it. Despite the fact that Planck's formula immediately received experimental confirmation, the idea of energy discreteness did not immediately begin to take on the character of a law, as this contradicted the idea that had developed by the beginning of the twentieth century, and therefore required detailed analysis. Henri Poincare in 1911-1912, having conducted a mathematical study of this issue, showed that the quantum hypothesis (i.e., the exact discreteness of resonator energy E =nhf, where f is the frequency of radiation) is the only fundamental hypothesis that leads to Planck's law. If the discreteness is slightly violated, i.e. If n is not equal to a natural number, then there will be no Planck formula, and in general, a whole class of problems in the theory of radiation simply cannot be solved. In these works, Poincare actually showed that the quantum hypothesis is not only NECESSARY, but also SUFFICIENT not only for the derivation of Planck's formula, but also for the construction of the entire subsequent quantum theory. Withdrawal! The quantum hypothesis turned out to be not just a convenient model, but its mathematical justification, made by Poincare, fixed the quantum nature of action as a fundamental LAW of physics [6].

Theory of Relativity.At the end of the 19th century, the predilection for Galileo's "obvious" transformations (along with the rate of addition of velocities) was so strong that no one wanted to seriously pay attention, and therefore analyze from other perspectives, the fact that Maxwell's equations do not remain unchanged under these transformations. This discrepancy was eventually resolved by A. Poincare.So in 1898, in the work "Measuring Time" he. He argued that absolute time and absolute simultaneity do not exist in nature. The fact that some events are considered to occur simultaneously is just a convention, in fact, each participant in the events may have their own time. Poincare further argued that if we accept as a postulate the constancy of the speed of light in the void, then the definition of simultaneity is automatically obtained., In addition, Poincare formulated the following postulate in 1889, the so–called principle of relativity, that all physical phenomena under the same initial conditions proceed in the same way in all inertial frames of reference. The first axiom, about the constancy of the speed of light, actually reflects the observance of a fundamental physical principle – the principle of causality.
Both axioms were later used by A. Einstein as the basis for the special (originally private) theory of relativity (SRT), which led to a profound rethinking of the concepts of space and time. Poincare's works were published either in a philosophical journal or in mathematical journals. This is probably why they were not given due attention and subsequently almost never referred to. Einstein sent his work to a well-known German magazine, and it immediately became available to the general scientific community. We have already said above that since Maxwell's equations do not remain unchanged under Galilean transformations. then the question arises. what should be other, "non-obvious" transformations in order for Maxwell's equations to remain unchanged? Poincare derived these transformations and named them after Lorentz Lorentz transformations. In these new transformations (according to the first axiom), the universal constant c ; 3·10^8 m/s appeared – this is the speed of light in a vacuum. Two, at first glance, completely "absurd" results followed directly from the Lorentz transformations. It turns out that the linear dimensions of a body along the direction of motion are reduced compared to those for a stationary body, and time in a moving system slows down. These Lorentzian results (reduction of distance and time dilation) were a blatant contradiction to the ideas about the properties of space and time that had developed in science by the beginning of the 20th century.
 However, no further conceptual conclusions were immediately drawn. There was too much predilection for the paradigm of G. Galileo and I. Newton – space and time are absolute categories, exist by themselves and do not depend on external circumstances. Taking into account the results of Poincare and Lorentz on the study of the symmetry of Maxwell's equations and after publication in 1905 Einstein's seminal work, his teacher G. Minkovsky, proposed in 1908 a geometric interpretation of the results of this new theory, called the special theory of relativity (SRT). A four-dimensional space-time interval was introduced in the SRT (an idea also first proposed by Poincare). It took Poincare's mathematical genius and Einstein's physical understanding of his ideas to fully realize this connection and understand that space and time do not exist independently of each other, they are inextricably linked through a certain symmetry.
This Lorentz–Poincare symmetry is not just abstract mathematics, it occurs in the real world, being realized through movement. It is now clear that the existence of a four-dimensional space-time continuum is a consequence of the finiteness of the speed of any interaction, which is limited from above by the speed of light. One of the fundamental achievements of the SRT was the famous formula relating mass and energy: E =mc^2. Surprisingly, O. Heaviside obtained this formula independently of A. Poincare and 15 years before A. Einstein. However, this is far from the only result of O. Heaviside, which was much ahead of its time, and which was obtained by him for reasons unknown to us. Let's pay special attention to the fact that the lesson taught by Lorenz and Poincare is that mathematical research, in this case based on symmetry analysis, can become a source of outstanding achievements in science. Even if mathematical symmetry cannot be visualized, it can point the way to revealing new fundamental principles of nature. Here is what academician V.I.Arnold says about the participation of mathematics in the problems of SRT and GRT: "By the way, few people know that three years before Hilbert's problems and ten years before Einstein, Poincare formulated the main problem left by the 19th century in the twentieth century in the field of mathematics. In Poincare's formulation, the main task is to build a mathematical theory for relativistic and quantum phenomena. Well, by the way, he did this for the relativistic case, although for some strange reason Einstein forgot to refer to it until 1945. In '45, he mentioned that Minkowski had advised him to read what Minkowski's friend Poincare had printed ten years before Einstein."[7]Watch an excerpt of the program "The Obvious is incredible," in which Arnold V.I. talks about the role of Henri Poincare in the discovery of the special theory of relativity.

SPIN. Almost a century ago, one of the greatest theorists in the history of science discovered one of the main formulas in physics of the 20th century. The authority of the scientist is universally recognized, the importance of his formula is indisputable, but at the same time, the true value of the Dirac Equation for understanding the structure of the world has still not been comprehended by science. We follow further [8]. In the days of 1995, when the British nation finally immortalized the memory of its great scientist, a scientific memorial conference was held in London - small in scale, but very representative in speakers.The last of the lectures published in this collection is important to us, "The Dirac Equation and Geometry", authored by the great English mathematician Michael Atiyah, Then he was president of the world's most prestigious Scientific Society, The Royal Society of London ("The Dirac equation and geometry," by Michael F. Atiyah).
The main thing is that the Dirac equation made its way into the field of pure mathematics thanks to the joint work of Michael Atiyah and Isadore Singer. In the 1960s, they proved the now famous "index theorem", or the Atiyah-Singer theorem. It is now considered one of the most important achievements in the field of mathematics in the second half of the 20th century. But what is the most interesting thing! It turned out that the unification of mathematical fields far from each other was possible due to a specific design. It turned out to be extremely unexpected!. In physics, this mathematical construction has been known for over a quarter of a century and is called the relativistic Dirac equation.! Thus, Atiyah and Singer rediscovered the Dirac equation along a completely different trajectory, and discovered that the underlying "Dirac operator" is, in a certain sense, the "generator" of all their mathematics. Because all the main results of their theorem can be expressed in terms of this operator.…
You can say it another way. The fundamental Dirac equation, which unites space, time, and matter, turns out to be fundamentally important for combining significantly different areas of mathematics. Alternatively, the Dirac operator, which compactly reconciles the wave nature of matter, the phenomenon of particle spin, and the effects of space-time curvature, also works as a "generative generator" in the fundamental foundations of all abstract mathematics.… Here is a quote from Dirac from his first "philosophical" lecture in 1939. "The Relation between Mathematics and Physics" (The Relation between Mathematics and Physics):
«Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. … Possibly, the two subjects will ultimately unify, every branch of pure mathematics then having its physical application, its importance in physics being proportional to its interest in mathematics»
2028 will mark the centenary of Dirac's discovery of the wave relativistic Equation. If a century later, world science does not finally figure out the true meaning and scope of his great formula, then it will be, to put it a disgrace. …

Dirac's discovery.
V.I. Arnold said: "But I've been following Dirac's recipe all my life, which taught me how to create New Physics in the following words: "First of all," Dirac said, "we need to discard all so—called 'physical representations,' because they are nothing more than a term for the outdated prejudices of previous generations." According to him, one should start with a beautiful mathematical theory. "If she's really beautiful," says Dirac, "then she's bound to turn out to be an excellent model of important physical phenomena. So we need to look for these phenomena, develop applications of beautiful mathematical theory and interpret them as predictions of new laws of physics," according to Dirac, this is how all new physics, both relativistic and quantum, is built.
 Even less well known is that Dirac's relativistic electronic equations originated from the ancient mathematical theory of braids. Namely, Dirac noticed, based on the topology of the family of elliptic curves in algebraic geometry, that there is a second-order element in the group of spherical braids of four strands, and interpreted this discovery as the theory of electron spin, which has 2 values (this means that in order for the particle to return to its previous position, it needs turn not 360 degrees, but 720 ). It was not clear to anyone, and therefore they did not believe him. To convince physicists of the validity of the corresponding strange mathematical theorem (stating that the fundamental group of the group SO(3) of rotations of three-dimensional space consists of two elements), Dirac demonstrated the corresponding experiment by physically manufacturing his spherical scythe of the second order. This braid is done like this: a sphere and another smaller sphere, concentric with it, are taken and connected by four ropes.
Four nails are driven into the outer sphere, four into the inner one, and four ropes connect them, but so that these ropes do not go along the radius, but intertwine with each other. The second, exactly the same, braid (this is called a "spherical braid")— a second spit, arranged in exactly the same way, connects a smaller sphere with an even smaller one. And now, the second—order element is what it is. This means that if you remove the middle sphere, you get four ropes connecting the largest to the smallest. So, they turned out to be untangled, they were entangled between large and medium, entangled between medium and small in the same way. And if you remove the middle one, then they can be dragged between the large and small ones by continuous transformation to the radial ones that are not entangled. It turns out a trivial braid.
This is the mathematical theorem in question, which Dirac proved. Dirac made these spheres and burned the middle one. The spheres turned out to be connected by untethered ropes, and physicists believed in the theory of spin. That's how he proved it. By the way, neither physicists nor mathematicians know this anymore. Maybe I read one of Dirac's books about how it's done and how he came up with it. And physicists believe in spin, because it's proclaimed there, and they give Nobel prizes for it, which means that everyone already knows it, that it's a famous, great thing. And everyone believes, just because it's proclaimed that it is. Well, that's it. In fact, Dirac's discovery of spin theory was based on an experiment THAT PROVED A MATHEMATICAL THEOREM.[9]

N.B.The memoirs of V.I.Arnold. [10] Spin .
The "Dirac d-function" is the simplest special case of generalized functions, the theory of which was constructed by N. M. Gunther in 1916 under the name "theory of functions of domains": these "generalized functions" are not determined by their values at points, but are determined by their integrals over various domains. Gunther built this theory to prove the existence (and uniqueness) theorems of solutions to the equations of hydrodynamics, Navier—Stokes. With the onset of the revolution, Gunther began to be accused of the aristocracy of his "anti-proletarian" noble science. To protect himself, he organized a seminar for communists and Komsomol members. His participant, Gunther's student S. L. Sobolev, investigated simplified solutions of the linear wave equation using his method (where discontinuous, generalized solutions are needed by the proletariat, for example, in seismology). Sobolev's works were translated (from French) into American by L. Schwartz, who built his "theory of distributions" in this way, which was awarded the Fields Medal. In 1965, Laurent Schwartz told me that "they gave him the Fields Medal for correcting mistakes in Sobolev's wonderful work." I read this work and did not see any errors, so I asked him to point them out. Schwartz replied: "Sobolev published his results in a language that no one understood, in a city where no one was interested in science, and even in a journal that no one read." Although I knew where Sobolev's article was published, I hid it and asked for the name of the language, city and magazine, and Schwartz replied: "In French, in Paris, in the Reports of the Paris Academy of Sciences" (Comptes Rendus de l'Academie des Sciences). When I returned to Moscow in 1966, I pulled Sergei Lvovich Sobolev out of the pit into which he had driven his car near the Zvenigorod market (after going to get milk), and told him about Schwartz's theory. Sergey Lvovich replied: "Laurent is a wonderful man, and treats us both very well, but he lied to you: in his Fields Medal—winning work, he not only translated my article, but also added his theorems on Fourier transforms of my generalized solutions, which I did not even know!" The question of the relationship between Schwartz's work and Sobolev was then decided by Hadamard, who traveled to Moscow to consult with Sergei Lvovich. He failed to do this, since S. L. Sobolev was Kurchatov's deputy in Los Arzamas (Sarov) at that time. Hadamard turned to Kolmogorov for advice, and he preferred the "true author" Gunther to both of them (whose work on "functions of domains" led Kolmogorov to his theory of cohomology).
Dirac introduced his d-function around 1930. When constructing the theory of electron spins, Dirac encountered the following difficulty: physicists could not understand why these spins allowed two values (+1/2 and -1/2), although they described the same "electron rotation". The essence of the matter here lies in a meaningful topological theorem: the fundamental group of the group of rotations of three-dimensional space consists of two elements, i.e. (SO(3)) ; Z2. This means that a 360° rotation does not return the corresponding physical characteristic of rotation to its place. In order for it to return to its original state, it is necessary to continue rotating so that the rotation angle is not 360° , but 720° . This difficult theorem was incomprehensible to physicists and caused distrust in spin theory. Then Dirac found its (also not obvious) consequence in the mathematical theory of braids: he constructed a "spherical braid of four hairs", delivering a second-order element in a group of spherical braids. Ordinary braids are "flat", their group is the fundamental group of the space of configurations of n different points on the plane (for braids of n strands). There are no elements of finite order in this group of flat braids: by tying a second one at the end of the braid, we will not untie it. And for spherical braids, Dirac was able to demonstrate this untying to physicists in an experiment (his threads connected 3 concentric spheres and came undone when he burned the middle one). To come up with this physical experiment, Dirac used a mathematical theory of elliptic functions that he understood (beautiful and nontrivial). Namely, let's consider four (different) points on the Riemann sphere CP1. The two-layered cover of the sphere branching at these points is a two-dimensional torus (the Riemann surface of the function y = (x4 + ax2 + bx)1/2, i.e. an elliptical curve). This circumstance determines the representation of the group of spherical braids of 4 strands into the automorphism group of the Z2 homology group of an elliptic curve. Calculating these automorphisms, Dirac found a spherical braid of 4 strands that provides a second-order element in the braid group. If Dirac had not loved this mathematics, physicists would not have received the theory of electron spins.

Non-integrated systems.
Evidence of a revolutionary change in the understanding of the
determinism of mechanical systems is a statement made by the president of
The International Union for Theoretical and Applied Mechanics, Sir
Michael James Lighthill:
Here I have to pause, and to speak once again on behalf of the broad global fraternity opractitioners of mechanics. We are all deeply conscious today that the enthusiasm of our forebears for the marvellous achievements of Newtonian mechanics led them to make generalizations in this area of predictability which, indeed, we may have generally tended to believe before 1960, but which we now recognize were false. We collectively wish to apologize for having misled the general educated public by spreading ideas about the determinism of systems satisfying Newton's laws of motion that, after 1960, were to be proved incorrect. In this lecture, I am trying to make belated amends by explaining both the very different picture that we nowdiscern, and the reasonsfor it having been uncovered so late. (Lighthill 1986,38) [11].
This recognition is caused by the exponential spread of trajectories of highly unstable chaotic systems, described by positive Lyapunov exponents. However, this is not all that caused such an unusual recognition. We will now focus on one of the aspects of this problem related to self-organization. The main problem in dynamics is the problem of integration. Since we have Newton's or Hamilton's equations of motion, naturally we would like to have explicit analytical expressions for variables, i.e. coordinates or velocities, as functions of time. At the end of the 19th century, A. Poincare showed that not all dynamical systems are similar to each other, as was previously thought. It turns out that there are two types of systems: integrable and non-integrable. For the former, we can eliminate interaction and reduce the problem to a problem of free movement. For the second, non–integrable ones, it is necessary to abandon the description in terms of trajectories (i.e., in fact, from determinism) and switch to a probabilistic description. Let's see how this is obtained in the framework of Hamiltonian dynamics, where the central, fundamental quantity is the Hamilton function H = E + U, or the Hamiltonian equal to the sum of kinetic E and potential U energies. For conservative systems, where the Hamiltonian H is clearly independent of time [12], it is expressed in terms of generalized pi momenta and ri coordinates as follows: H = E(p1 , ..., pN) + U(r1,...., rN). This notation is Hamiltonian in the so-called canonical variables, where kinetic energy depends only on the momenta, and potential energy depends only on the coordinates of the particles. The canonical representation of the equations of motion is rightfully considered the apotheosis of classical dynamics, since in this representation they are expressed in terms of a single quantity – the Hamiltonian. To understand what an integrated system is, we use the simplest example given in every textbook on theoretical mechanics. This is a one-dimensional harmonic oscillator. For him, the Hamiltonian has the form: P = p^2/2m + kr^2/2, where k is a certain elastic constant, m is the mass. For this system, it turns out that there is a so–called canonical transformation in which the Hamiltonian takes the form: H = w·J, where J is the action variable, and w is defined in terms of the angular variable U as follows: U = wt + const. Thus, motion is now expressed in terms of cyclic variables J and W. This result is very characteristic. In the new variables, the action is an angle, and the Hamiltonian depends only on the new momentum– the action variable. As a result, dJ/dt = -dH/dU=0. That is, the action variable J is an invariant of motion. A similar result is obtained for a free particle when dp/dt =-dH/dr=0. As we can see, in this case, the Hamilton equations are easily integrated, since there is no potential energy. The ability to eliminate potential energy by canonical transformation to new cyclic variables is the main characteristic of integrable dynamical systems in the sense of Henri Poincare. This means that for integrable systems, after the transformation of the Hamiltonian into the corresponding form, there is no term with potential energy, i.e., interaction between particles is virtually eliminated. Before 1889 It was assumed (though tacitly) that all dynamic systems are integrable, and the problems associated with the problem of three or more bodies are purely technical and computational. However, A. Poincare showed in 1889 [4] that in the general case it is impossible to obtain a canonical transformation preserving the form of Hamiltonian equations, which would lead to cyclic variables, and most systems are just nonintegrable. What is the meaning of such a strong mathematical statement? What would have happened if Poincare had proved the integrability of all dynamical systems? This would mean that all dynamical systems with any number of particles, without exception, are essentially isomorphic to the motion of free particles that do not interact with each other in any way. This would mean that these particles can never act as a collective, that is, coherently! And this means that there can be no self-organization in principle! It means that life cannot arise in an integrated world! However, this is not enough. A. Poincare not only proved the non-integrability, but also pointed out the reason for the non-integrability of the systems. This is the existence of resonances between degrees of freedom and the emergence of the problem of the so-called "small denominators". I must say that this problem was known in astronomy before A. Poincare. But it was his theorem that showed that the main difficulty associated with divergence (small denominators tend to zero, and their reciprocal tends to infinity) in solving problems of dynamics cannot be eliminated and makes it impossible to introduce cyclic variables for most dynamical systems, starting with a three-body system. This is how M. Born assessed this problem at the time: "It would be very strange if Nature hid from further progress of knowledge behind the analytical difficulties of the problem of many bodies." With the appearance of the works of A.N. Kolmogorov, continued by V.I. Arnold and Yu. Moser and with the advent of the Kolmogorov–Arnold–Moser theory, the problem of nonintegrability and small denominators began to be considered as the starting point for a new development of dynamics, including the dynamics of both coherent and chaotic movements. KAM theory considers the influence of resonances on trajectories. Resonances exist at different points in the phase space of the dynamical system, while they do not exist in others. Resonances correspond to rational ratios between frequencies. Since (this is a classic result of number theory due to the fact that the set of rational numbers is countable, and the set of irrational numbers has a continuum power) the measure of rational numbers is zero compared to the measure of irrational numbers, resonances are extremely rare, most points in the phase space are non-resonant. Resonances lead to periodic motions, and the absence of resonances leads to quasi–periodic motion. Consequently, periodic movements are usually an exception to the general case of movements of a more complex type. The main result of KAM theory is that there are two fundamentally different types of trajectories. The first are slightly changed quasi–periodic trajectories. KAM theory considers the influence of resonances on trajectories. Resonances exist at different points in the phase space of the dynamical system, while they do not exist in others. Resonances correspond to rational ratios between frequencies. Since (this is a classic result of number theory due to the fact that the set of rational numbers is countable, and the set of irrational numbers has a continuum power) the measure of rational numbers is zero compared to the measure of irrational numbers, resonances are extremely rare, most points in the phase space are non-resonant. Resonances lead to periodic motions, and the absence of resonances leads to quasi–periodic motion. Consequently, periodic movements are usually an exception to the general case of movements of a more complex type. The main result of KAM theory is that there are two fundamentally different types of trajectories. The first are slightly changed quasi–periodic trajectories. The second is stochastic trajectories that occur when resonances are destroyed. KAM theory does not lead to dynamic chaos theory, but it shows that for small values of a certain parameter, an intermediate regime is obtained in which two types of trajectories coexist – regular and stochastic.

Discoveries come unexpectedly.
Let's focus on one more important issue. It is well known that in the field of natural sciences, the most fundamental discoveries occur unexpectedly. So in some cases, there is a scientific search for a certain "A", and as a result they find a completely unexpected –B". The more unexpected this "B" is, the more significant the new result. In the field of mathematical discoveries, the situation is similar. As an example, here is an excerpt from an article by V.I. Arnold devoted to A.N. According to Kolmogorov [13], "Andrey Nikolaevich noticed that in "integrable" problems, the phase definition on the torus changes evenly over time. He also asked himself the question: is this so if the system on the torus is not integrable, but only has an integral invariant? He solved this problem in his 1953 work on systems on a torus, the first where small denominators appear. A.N.'s conclusion is that it is almost always possible to introduce phases that change evenly over time, but sometimes mixing is possible. The remark about mixing, referring to a pathological case, does not seem particularly important. But that's exactly it ("thanks to") and it became the source of the famous work of A.N. Kolmogorov on small denominators, published in 1954, where the conservation of invariant tori with a small change in the Hamilton function was proved. The arguments of A.N. Kolmogorov, mentioned by him in a report at the International Mathematical Congress in Amsterdam in 1954, were as follows.
In integrable systems, motion along invariant tori is always conditionally periodic. Consequently, mixing does not occur in integrable systems (which means that there can be no self-organization). To find out whether the phenomenon he discovered has mechanical applications, A.N. Kolmogorov decided to find motion along tori in non-integrable systems, where mixing could be observed in principle. It is natural to start with perturbation theory, considering a system that is close to integrable. Various versions of perturbation theory have been discussed many times in celestial mechanics, and then in early quantum mechanics. But all these perturbation theories lead to divergent series. Kolmogorov realized that the divergence can be overcome if, instead of decompositions by degrees of a small parameter, Newton's method is used in a functional space. Thus, the "accelerated convergence method" by A.N. Kolmogorov was invented not at all for the sake of (but "contrary to") those remarkable applications in classical problems of mechanics to which it leads, but for the sake of exploring the possibility of implementing a special set-theoretic pathology in systems on a two-dimensional torus. He set himself the task of implementing mixing on weakly perturbed invariant tori. Kolmogorov did not solve it ("A" was not found), since his method automatically builds angular coordinates that change uniformly with the movement of the phase point on the tori he found. The question of mixing, from which the entire work of the scientist grew, remains unresolved today. The significance of this technical issue (the search for "A") is negligible compared to the results obtained (the unknown "B" was found). Now no one remembers about him anymore, but new mathematics arose when clarifying small technical details of previous works. It is already clear from this that planning fundamental research is bureaucratic nonsense, and often just a deception."

"One more last the tale"
For a random distribution of k points on an integer circle, two Kolmogorov "stochastic parameters" were determined - K[14] and Arnold - A[15].These parameters were independently introduced by A.N.Kolmogorov in 1933 and V.I. Arnold in 2003. The following is striking. These parameters (obtained from completely different considerations) seem to be independent characteristics of the field of random points, but they become functionally dependent when their values are averaged over small fluctuations of the field points. So, as V.I. Arnold insisted, their dependence is due to some Law of Nature! It is not clear which ones. Arnold made numerous empirical observations (many calculations, almost a million) and established the following. The dependence of A and K turns out to be parabolic in the (AK) plane. This is an empirical discovery. Arnold said that he was afraid and did not call it a theorem, since there is no proof, but only Empirical evidence. That is, (according to S.P. Kapitsa) we have (for now!) not proof, but Evidence. The Kolmogorov criterion is based on calculating the value of a certain stochastic parameter K from a given sequence, the probability of randomness depends on its value. The average value of the Kolmogorov parameter K - K(cp) is approximately =0.87. If the observed value of K is much less or much greater than K(cp), then randomness of the studied sequence is unlikely. The average value of the Arnold parameter A is close to the two of A(cp) =2. Similarly to Kolmogorov's theory, if the observed value of A is much less or much greater than A(cp), then randomness of the studied sequence is unlikely. In Arnold's theory (unlike Kolmogorov's theory), there is one extremely interesting property (the repulsion or attraction of points on a circle). This is due to the fact that the observed value of A is less than or greater than A(cp). Conclusion. In the examples given by Kolmogorov and Arnold, the stochasticity parameter for geometric progression is K=0.7 (which is close to 0.87), and for arithmetic K=0.3 (which is far from 0.87). Therefore, the geometric progression, as shown by the corresponding calculation [15], is about 300 times more random than the arithmetic progression. It's weird, isn't it? But that's the math. It can be the same in many other situations. Where events are described by sets of numbers that look like regular sequences, but in fact (according to the criteria of Kolmogorov and Arnold) may turn out to be, to one degree or another, random. On the other hand, supposedly random sequences may turn out to be natural. Example. "A sequence of quadratic residues, for which the probability of randomness according to the Kolmogorov criterion is extremely small. Quadratic residues are not random. They are chosen according to some principle that no one knows yet. Despite the fact that dozens of papers on the randomness of quadratic residues have been published in number theory, all this is an erroneous theory. They are not random, they behave differently"[15]. All these amazing results allowed us to formulate the Theorem in [16].  Theorem. In some cases, a pattern is an unrecognized accident, and randomness is an unrecognized pattern.

Bibliography
1. "The Obvious-the Incredible. V. Arnold on the formulation of problems" – video from the TV project "The Obvious – the Incredible". Conversation of S.P. Kapitsa with Academician V.I. Arnold on the formulation of problems. https://ya.ru/video/preview/17829931208764422636
2. I. S. Shapiro On the history of the discovery of Maxwell's equations33. UFN 1972. October Volume 108, Issue 2, pp. 319-333.
3. Poincare H. Methodts nouvelles de la mecanique celeste. – Paris: Gauthies Villars, 1882.
4. Poincare A. New Methods of Celestial Mechanics // Selected Works. Vol. 1, 2. – Moscow: Nauka, 1971–1972.
5. Arnold V. I. Mathematical Methods of Classical Mechanics. Ed. 5th stereotype. M.: Editorial URSS, 2003. P. 62. ISBN 5-354-00341-5).
6. Poincare A. Izb. works. M.: Hauka, "Mathematics. Theoretical physics. Analysis of mathematical and natural-science works of Henri Poincare", year of publication - 1974. Vol.3. 769 pages. The edition is published under the editorship of Acad. N. N. Bogolyubov (chief editor), with comments by V. I. Arnold and V. M. Alekseyev.
7.The program "The Obvious is the Incredible".Arnold V.I. about the role of Henri Poincar; in the discovery of the special theory of relativity. https://www.youtube.com/watch?v =AIrt0PVoQys.If you are interested in the entire program (25 minutes), then watch it here: https://www.youtube.com/watch?v=135WkxG57wI
8. Dirac as a premonition. August 9th, 2020. https://kiwibyrd.org/2020/08/09/20h81 /
9. V. I. Arnold in a public lecture "The complexity of finite sequences of zeros and ones and the geometry of finite functional spaces", delivered on May 13, 2006 at the Akademicheskiy Concert Hall at the invitation of the Dynasty Foundation, Vladimir I. Arnold mentioned that he originated relativistic Dirac electronic equations from the ancient mathematical theory of braids. https://elementy.ru/nauchno 10.V. I. Arnold Mathematical Understanding of Nature Essays on Surprising Physical Phenomena and Their Understanding by Mathematicians (with the author's drawings) Third Edition, stereotyped Moscow Publishing House of the Moscow Center for Non-Standard Mathematics and Physics 2011.p.99-102.
11.Sir D. Ligthill.President of the International Union of Theoretical and Applied Mechanics. 1986. Ligthill J. The Recognized Failure of Predictability in Newtonian Dynamics // Proceedings of the Royal Society. – 1986. – P. 35–50.
12. Tabor M. Chaos and Integrability in Nonlinear Mechanics. M.: Editorial URSS. 2001.  320 p.
13. Arnold V.I. On A.N. Kolmogorov . https://ega-math.narod.ru/LSP/ANK.htm.
14. Kolmogoroff A. Sulla determinazione empirica di una legge di distribuzione // Guornale dell’ Instituto Italiano degli Attuari. 1933. V.4. №1. P.83-91. A.N. Kolmogorov "On the Empirical Determination of the Distribution Law".
15.V.I.Arnold.Video. Measurement of the objective degree of randomness of a finite set of points - Vladimir Arnold. Lecture series: Summer school "Modern mathematics", 2009. July 19, 2009 11:15, Dubna.
16. Yu.P.Krapachev. Proza.ru. Randomness – regularity. natural science, 31.08.2025