The article is devoted to the formation of mathematical physics as a science, as well as the prerequisites for its emergence and go math first grade. Mathematical physics is concerned with the construction and study of mathematical models of physical phenomena. It is interesting that the distant ancestor of this science was the mythology of various peoples and cultures. The article also mentions the names of scientists who contributed to the development of this science.
Mathematical physics is the theory of mathematical models of physical phenomena. This science is included in the mathematical sciences because it has mathematical proof as a criterion of truth. But it does differ from other mathematical sciences. The difference consists in the fact that its problems are physical problems solved with the help of mathematical methods. The final results of such problems have a physical interpretation and are given in the form of theorems, tables or graphs. Based on such an understanding of such a science as mathematical physics, it should
include such sections of mechanics as theoretical mechanics, hydrodynamics, and elasticity theory. As an independent science, mathematical physics emerged around the end of the eighteenth century. The famous French mathematician Poincaré, who had studied theoretical physics for a long time, owned the following statement: “Mathematical physics, including go math 6th grade as we know, came from celestial mechanics, which gave birth to it in the late 18th century at a time when it had just reached full development”. From this we must conclude about the “mechanical” origin of mathematical physics, which still had a considerable influence on it for a long time. According to the
general definition, mathematical physics is the science of constructing and analyzing mathematical models of physical phenomena. It is interesting that the ways of studying natural phenomena with the help of certain models appeared very long ago.
Long before the emer
gence of mathematical physics and go math 2 grade, people tried to explain such phenomena with the help of mythology. It was myths that had the function of acting as models to explain phenomena unexplained at the time. The romantic myth of the kidnapping of Persephone, the daughter of Demeter, the goddess of fertility, known even to children, told of the change of seasons: “…every year Persephone leaves her mother and every time Demeter sinks into sorrow and is clothed in dark clothes again. All of nature grieves for her passing: the leaves on the trees turn yellow…”. According to the myths, various natural phenomena occur depending on the behavior of the gods. The gods are beings who act as constituents of the model. The motives for their actions are mostly human: love, hatred, revenge, the desire to help some and harm others. Such motives in myths act as “model axioms” that do not require any explanation. In mythology, observable phenomena are explained precisely by the activities of the gods. Based on the accepted model of phenomena, people built their way of life and sought to influence nature – they prayed, arranged a sacrifice, imposed some vows on themselves. Later on, attempts began to be made to justify natural phenomena scientifically. Such scholars as Thales, Anaximenes, Heraclitus, and Democritus wrote works on this subject. According to historical information, the earliest attempts of this kind were undertaken by the ancient Greek philosopher Aristotle, who lived in 384-322 B.C. Aristotle presented a number of postulates, with the help of which he tried to explain some natural phenomena. Some of these postulates corresponded to everyday experience; some were mystical and even looked childish compared to the conclusions of Aristotle’s predecessors. In his postulates the philosopher, by means of the deductive method, drew corollaries which, in the simplest cases, did not contradict the observations available to everyone. Aristotle’s assertions, which seemed scientific at the time, are now known even to a preschooler. For example, it was the claim that heavier bodies fall faster than lighter ones, and that any body can only be moved if force is applied. But in spite of all the naivety of Aristotle’s theories, his great merit lies not in providing specific information on the description of natural phenomena, but in the careful and constant development and application of the deductive method, which is popular in modern science, one might say, in its pristine form. Aristotle did not attempt to describe natural phenomena quantitatively and represented experiment as the criterion for the truth of theory. This can be explained by the complete unpreparedness of science to solve such problems at the time.
In later times, many ancient thinkers talked about the need for experimentation in the study of natural phenomena in their works: the architect Vitruvius, the scientist, artist and philosopher Leonardo da Vinci. However, the first scientist to systematically conduct experiments was the Italian physicist and philosopher Galileo Galilei. He was an advocate of in-depth critical examination and analysis of the results of observations made, and in studying the motion of bodies under the action of gravity, he tried to quantitatively combine theoretical and experimental data.
The importance of the method of experiments in the study of phenomena contributed to the emergence in mathematical physics of a separate scientific field – computational experiment.
In 1687, the scientist Isaac Newton published a work called “Mathematical Beginnings of Natural Philosophy”. This work is known for the fact that it presented the first successful mathematical model of the simplest physical phenomenon (mechanical motion), and the mathematical formulation of one of the most fundamental laws of nature (the law of universal gravitation). Newton based his research on the works of Copernicus, Kepler, Galileo, as well as on the methods of famous ancient Greek mathematicians. The scientist’s research, together with the works of Galileo Galilei, marked the beginning of a new era in natural science. It is also interesting that the beginning of this era coincided with the beginning of the regular and widespread use of mathematical methods in physics. Newton applied a peculiar scheme of reasoning, which later became a model for works on mathematical physics: mathematical clear formulation of a problem, solution of this problem, and interpretation of the solution in terms of a physical model. During the same period, thanks to the research of many scientists inspired by the ideas of Eudoxus and Archimedes, the apparatus of differential and integral calculus emerged. The apparatus they created was initially used in solving problems of mechanics, and in the period in question began its systematic application to the creation of mathematical models of wave propagation in elastic bodies and in the “ether.” “Aether” according to scientists of the time was a hypothetical substance that acted as a carrier of interaction. Also “ether” acted as a kind of reference system, heat propagation, for solving problems of electrostatics, gas dynamics, mechanics of liquids. Most of the famous mathematicians of the time were talented physicists or mechanics. The most famous of these were Gauss, Lagrange, Euler, and Riemann. The increasing number and increasing complexity of mathematical models characterizing physical phenomena was the main prerequisite for the creation of a science that dealt with the study of these models. This science is mathematical physics. Construction of almost all physical models at that time was based on classical mechanics. For a long time, the opinion was firmly held that the full justification of all phenomena could be obtained by reducing these phenomena to mechanical motion (atoms, “ether,” etc.). In 1873, J. K. Maxwell’s “Treatise on Electricity and Magnetism” was published, in which the scientist presented a mathematical model of electrodynamics based on Michael Faraday’s research and a number of previously known laws of interaction of charges and currents. This model is still relevant today. The model is based on the equations, which are now called Maxwell’s equations. At first, these equations were attributed to the description of a special kind of mechanical motions of a certain hypothetical medium (“ether”). But later, scientists came to the conclusion that the equations describe some independent object – the electromagnetic field. The role of Maxwell’s equations in mathematical physics is very great. They, like the equations of hydrodynamics, have long been the source of problems and the cause of great advances in mathematical physics. Many varieties of mathematical models of physical phenomena that occur in solids, liquids and gases are created on the basis of statistical hypotheses about the behavior of the molecules of these substances. In research at the present stage, the statistical approach is used to calculate the parameters of kinetic equations – the equations describing macroscopic flows of particles of matter or radiation on the basis of ideas about the nature of microscopic interaction of particles. Scientists R. Clausius, J. K. Maxwell, and L. Boltzmann in their works laid down the foundations of the statistical approach to construction of mathematical models of physical phenomena, and in 1872 L. Boltzmann systematized and expounded them in his work “Lectures on the Kinetic Theory of Gases” The use of probabilistic theoretical approaches and statistical hypotheses was a specifically new direction in mathematical physics. Statistical ideas had a serious influence on preparation of physicists’ thinking for acceptance of ideas of quantum theory (for example, Planck at the time of creation of quantum hypothesis was influenced by Boltzmann’s ideas). The statistical approach received its maximum scope in the works of J. Gibbs. The methods of mathematical physics owe their development to the talented Russian scientists M.V. Ostrogozhsky, A.M. Lyapunov, and V.A. Steklov. The beginning of the twentieth century was marked by a revolution in theoretical physics, which led to a change in the view of most models studied in mathematical physics and to the definition of more rigid limits to their use. At the same time, circumstances were created that called for a more rigorous approach to the basics. In 1900, along with his famous problems, Hilbert formulated the problem of the axiomatic construction of physics. Research related to axiomatics and the study of the logical structure of various physical theories is an important section of modern mathematical physics.
In 1916, Albert Einstein proposed the equation of gravitation theory, known as the Einstein equation. In 1932, John von Neumann published his book Mathematical Beginnings of Quantum Mechanics, which describes the results of the grandiose discoveries of the founders of quantum theory. In this book quantum mechanics was presented for the first time as a mathematically consistent theory. The current period in the development of mathematical physics is ideologically connected with the works of Gibbs, Einstein, von Neumann and Hilbert. The modern period is characterized by a wide application of the ideas of functional analysis, probability theory, modern geometry, and topology.
Besides differential equations of mathematical physics, integral and integro-differential equations, variational and probability-theoretic methods, potential theory, methods of the theory of functions of a complex variable, and a number of other sections of mathematics are widely used in the study of mathematical models of physics. Because of the rapid development of computational mathematics, direct numerical methods involving the use of computer programs are of specific importance for studying mathematical models of physical phenomena. These include finite-difference methods and other computational algorithms for boundary value problems. The use of computers has made it possible, with the help of methods of mathematical physics, to efficiently find solutions to new problems in gas dynamics, transfer theory, and plasma physics, including the inverse problems of these most important areas of physical research.