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**Ivan Vsevolodovich Meshchersky (1859-1935)**

Ivan Vsevolodovich Meshchersky - one of the largest mechanics of the late XIX and early XX centuries - devoted his life to creating the foundations of the mechanics of bodies of variable mass. A particular task of body mechanics of variable mass is the theory of motion of jet vehicles in which the change in mass during movement is caused by the ejection (expiration) of particles of the burned fuel supply. At the end of the XIX century. I.V. Meshchersky published two works that are still the best in the entire world literature on jet propulsion methods. The mortar guards, armed with special mortars, glorified during the Patriotic War of the Soviet people with the German fascist hordes, have firing tables compiled on the basis of the equations of I.V. Meshchersky. Its general equations for a point of variable mass and some special cases of these equations already after their publication by I.V. Meshchersky were "discovered" in the 20th century. again by many scientists in Western Europe and America (Goddard, Obert, Esno-Peltri, Levi-Civita, etc.).

The field of practical application of mechanics of bodies of variable mass is far from being limited to jet devices and rocketry. Cases of the motion of bodies, when their mass changes, can be indicated in various fields of industry. It is easy to understand, for example, that a rotating spindle, on which a thread is wound, changes its mass during movement. A roll of paper, when it is unwound on the shaft of a printing press, also gives us an example of a body whose mass decreases over time. Numerous examples of the motion of bodies, the mass of which changes over time, we can observe in nature. For example, the mass of the Earth increases due to the fall of meteorites on it. The mass of the falling meteorite moving in the atmosphere decreases because the meteorite particles come off or burn out. A floating ice floe is an example of a body whose mass decreases due to melting or increases due to freezing. The mass of the Sun increases from the sticking of "cosmic dust" and decreases from radiation, etc. In general, a change in the mass of moving bodies can occur due to combustion, evaporation, dissolution, freezing, sticking, radiation, etc.

The mechanics of bodies of variable mass is of great importance for the correct description of the motion of planets and especially the moon. Comparing previous observations of the Moon with their own and the observations of contemporaries, Halley found that the period of the moon's revolution around the Earth is decreasing. This decrease means an increase in the average speed of its movement in orbit. The influence of the acceleration of the moon's movement on its position in orbit increases over time (in proportion to the square of time), and thus, even if it is small, it can be relatively easily detected after large periods of time. The decrease in the period of revolution of the Moon around the Earth is about half a second in 2000 years. Partially, as Laplace showed, the magnitude of the acceleration can be explained by a decrease in the eccentricity of the earth's orbit. The second part of secular acceleration depends on changes in the mass of the Earth and the Moon caused by meteorites falling on them. It turns out that the agreement between observations and calculations is good, if we assume that the radius of the Earth increases from the mass of incident meteorites by 0.5 millimeters per century.

For an accurate study of the phenomena of motion of bodies with varying masses, delivered in large numbers by both technology and nature, it is necessary first of all to establish the basic equation of motion for a point of variable mass, since any body of variable mass can be represented as a system of points. Knowing the equation of motion of a point of variable mass, one can obtain, using simple methods, the basic equations of motion of any body. The fundamental equation of the dynamics of bodies of variable mass was established in the master's thesis of I.V. Meshchersky, published in 1897.

Ivan Vsevolodovich Meshchersky was born on July 29, 1859 in the city of Arkhangelsk, where he received his secondary education. In 1878, he entered the Physics and Mathematics Department of St. Petersburg University. His outstanding abilities drew the attention of the famous Russian mechanic D.K. Bobylev. At the end of the university in 1882, I.V. Meshchersky was left at the department of D. K. Bobylev to prepare for the professorship. In 1890, I.V. Meshchersky became a privat-docent of the Department of Mechanics at St. Petersburg University. He lectured on graphostatics, the integration of equations of mechanics, and conducted exercises on the general course of mechanics. In those same years, I.V. Meshchersky began to study the theory of motion of bodies of variable mass.

A moving body with a change in mass is generally exposed to a reactive force, unless the relative velocity of the separated particles is non-zero. However, I.V. Meshchersky began developing the question from the particular case when the reactive force would not be included in the calculations. The theoretical results of the study of motion under this assumption were reported by I.V. Meshchersky to the St. Petersburg Mathematical Society on January 15, 1893. Moreover, from particular problems of this type he solved one problem of celestial mechanics devoted to the study of the motion of two bodies of variable mass. In 1893, the main conclusions of this study were published in a special astronomical journal.

In 1902, I.V. Meshchersky was invited by the head of the Department of Theoretical Mechanics to the newly founded Petersburg Polytechnic Institute. Here his main pedagogical and scientific work proceeded.

On January 7, 1935, Ivan Vsevolodovich Meshchersky died at the age of 76.

The basic equation of motion of a variable mass point for any law of mass change and for any relative velocity of the emitted particles was obtained and studied by I.V. Meshchersky in his dissertation in 1897

This equation can be formulated in the following form: when a variable mass point moves at any time, the product of the mass of the point and its acceleration will be the sum of all external forces plus the surplus or reactive force due to the outflow of particles. The magnitude of the reactive force is equal to the second mass flow rate multiplied by the relative velocity of the ejected particles. The greater the relative velocity of the ejected particles, the greater the magnitude of the reactive force. The greater the mass of particles ejected per unit time, the greater the reactive force. To clarify the above, we can refer to the following well-known fact. A small charge of gunpowder gives a low initial speed of the shot to escape from the barrel of a hunting rifle, and the hunter's shoulder experiences a small “recoil” (that is, a small reactive force). An increase in the charge of gunpowder causes an increase in the rate of departure of shot and powder gases, which leads to an increase in reactive force. Artillery guns, where the masses of powder ejected during a shot from a projectile are much larger than the masses ejected from a hunting rifle, have reactive forces so great that it is necessary to construct special devices in order to perceive “recoil” without breaking the gun itself.

The basic physical laws that determine the mechanical motion of bodies of constant mass can be used to derive the equation of motion of a point of variable mass. To do this, it is enough to consider the point of variable mass and all particles that, in the process of movement, were separated from it, as a single mechanical system. Obviously, the mass of such an aggregate system of particles will be constant, since in the processes of mechanical motion the law of conservation of matter holds. The laws of Newtonian mechanics can be applied to a system of particles of constant mass. If the motion and position of all discarded particles is known for each moment of time, then we can find the motion of a point of variable mass that ejects these particles due to some physical process (combustion, radiation, etc.).

However, such a method of studying the motion of a point of variable mass is practically impossible because the motion of particles ejected by the point is unknown.

The outstanding merit of I.V. Meshchersky is that by focusing on the motion of a point ejecting particles, he was able to obtain much simpler equations that are independent of the entire history of motion of discarded particles.

The Meshchersky equations are the simplest initial equations for a new chapter in mechanics studying the motion of bodies of variable mass. The form of the equations of I.V. Meshchersky changes depending on the speed (absolute or relative) of the discarded particles. If the absolute velocity of the particles discarded by the point is zero, then from the Meshchersky equation the following law can be obtained: the change in the momentum (i.e., the product of mass and speed) of a variable mass point per unit time is equal to the sum of all external forces. This law, which follows from the Meshchersky equation as a special case, was rediscovered by the Italian scientist Levi-Civita in 1928. For particular missile problems, this particular case does not matter much. However, for celestial mechanics, consideration of such examples is important and essential.

In his dissertation, I.V. Meshchersky subjected to especially careful analysis the case of motion when the relative velocity of the emitted particles is zero. Tasks of this type arise, for example, in the textile industry when studying the movement of spindles. The original equation for the motion of the point in this case will coincide in form with Newton's law. If we still assume that the external force is proportional to the mass of the moving point, then we will find that the acceleration of the point with variable mass does not depend on the change in mass. Thus, under the action of a force proportional to the mass of a point, a point of variable mass, no matter how its mass changes, moves as a point of constant mass moves under the action of the same forces and with the same initial data.

I.V. Meshchersky solved a large number of particular problems on the motion of a point of variable mass. In particular, he subjected a very thorough study to the motion of a point of variable mass under the action of a central force (i.e., a force directed all the time toward one fixed point), thereby laying the foundations of celestial mechanics of bodies of variable mass. The assumptions of I.V. Meshchersky about the nature of the change in the mass of celestial bodies made back in the works of 1897 and 1902 were thoroughly studied by the largest astronomers, and now these hypotheses are called "Meshchersky laws" in the literature. I.V. Meshchersky also investigated some problems of the movement of comets.

I.V. Meshchersky first posed and partially solved problems of the following type: to find the law of change in the mass of a point at which it would describe a given trajectory under the influence of these external forces. He calls these tasks inverse, since they need to find the law of change in the mass of a point by some given properties of its motion for given forces.

I.V. Meshchersky was the first to give a rigorous equation for the vertical motion of a rocket and to show in which particular cases this equation can be studied up to a numerical result. He also solved the problem of oscillations of a pendulum of variable mass in a resisting medium.

In 1904, I.V. Meshchersky gave a detailed study of the motion of a body of variable mass in the case when particles are attached and separated at the same time. An example of such a change in the mass of a moving body, in which some reasons cause an increase in mass, while others at the same time its decrease, is a flying airplane. Air from the atmosphere is sucked into the engine cylinders and the cooling system, and at the same time gases are emitted from the pipes and the radiator tunnel. We have examples of the same kind when one chain is wound from a rotating shaft and another is wound at the same time, when water pours into one floating vessel through some openings, and pours out through others, etc.

Scientific research by I.V. Meshchersky on the theory of motion of bodies of variable mass are of great importance for the future development of technology and industry. Now it is quite clear to the vast majority of scientists and engineers. At the end of the XIX and the beginning of the XX centuries, the value of scientific work on this issue did not seem significant. The study of the motion of bodies of variable mass was done by individuals. There was no technical basis for the development of experiments, there was no means for creating prototypes, jet propulsion methods did not yet become an urgent need for industrial development. The scientific foresight of I.V. Meshchersky, his consciously directed, purposeful creative searches in an area that was considered fantastic and of little relevance, make his personality somehow especially charming and powerful. To see the future development of technology for decades to come, even in some small industry, few are given. It was very difficult to insist on the need for new ways of technical development for 40 years and until the end of my life to get decisive evidence of the importance of my theoretical work. This lack of understanding by scientists of the progressiveness of scientific research by I.V. Meshchersky made him unusually restrained and punctual. Restraint is the main quality of his scientific style. Everything is in the close framework of formal logical constructions, everywhere the impassive tone of a person of high mathematical culture. In the presentation of the work, everything comes from the mind; no arguments or appeals to the feeling of the reader. There are no hypotheses, dreams, fantasies even in popular reports. Polemic remarks are substantiated with extraordinary skill, and impeccable accuracy is observed in relation to the insignificant formulations of opponents.

.V. Meshchersky was not only a scientist, but also an outstanding teacher of the Russian higher technical school. He attached exceptional importance to the staging of teaching a course in theoretical mechanics. He believed that in higher technical school the course of theoretical mechanics should be closely connected with courses in applied mechanics. When choosing tasks, special attention should be paid to the fact that they have a specific form. Students, solving these problems, must acquire the ability and skills to apply general theorems and methods to specific issues of applied value.

"A collection of problems in theoretical mechanics," compiled by a group of lecturers of the Petersburg Polytechnic Institute under the direction of I.V. Meshchersky, best meets the goal. This is one of the best problem books for the higher technical school. In 1938, this book was translated into English and accepted as the main manual in American higher technical educational institutions. With us he has already stood 13 editions. In addition, I.V. Meshchersky wrote "The Course of Theoretical Mechanics", which also withstood several editions.

I.V. Meshchersky believed that the training of a highly qualified and widely educated engineer requires a good general education. “Mathematics, mechanics, physics and chemistry,” he wrote, “in a known volume that can be established, form the basis of any technical education; when starting to study a technical specialty, a future engineer should already own these subjects in the specified volume.” The influence of I.V. Meshchersky's ideas on the formulation of the teaching of mechanics in higher technical educational institutions can be clearly seen in almost all modern textbooks.

The main merit of I.V. Meshchersky before science is that he was the first to create theoretical methods for solving the diverse problems of the motion of bodies, the mass of which varies over time. I.V. Meshchersky is the creator and pioneer of a new chapter in mechanics, the methods of which will solve the most important technical problems of the present and future.

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