Mechanics

[from Greek mechanike (téchne) - the science of machines, the art of building machines], the science of the mechanical movement of material bodies and the interactions between bodies that occur during this process. Mechanical motion is understood as a change in the relative position of bodies or their particles in space over time. Examples of such movements studied by methods of mathematics are: in nature - the movements of celestial bodies, vibrations earth's crust, air and sea ​​currents, thermal movement molecules, etc., and in technology - various movements aircraft And Vehicle, parts of all kinds of engines, machines and mechanisms, deformation of elements of various structures and structures, movement of liquids and gases, and many others.

The interactions considered in mathematics are those actions of bodies on each other, the result of which is changes in the mechanical movement of these bodies. Their examples can be the attraction of bodies according to the law universal gravity, mutual pressures of contacting bodies, the effects of liquid or gas particles on each other and on bodies moving in them, etc. Usually, M. is understood as the so-called. classical mechanics, which is based on Newton’s laws of mechanics and the subject of which is the study of the motion of any material bodies (except elementary particles), performed at speeds small compared to the speed of light. The movement of bodies with speeds on the order of the speed of light is considered in relativity theory (See Relativity theory), and intra-atomic phenomena and the movement of elementary particles are studied in quantum mechanics (See Quantum mechanics).

When studying the movement of material bodies, a number of abstract concepts are introduced into mathematics that reflect certain properties of real bodies; are as follows: 1) A material point is an object of negligible size that has mass; this concept is applicable if, in the motion being studied, the size of the body can be neglected in comparison with the distances traversed by its points. 2) An absolutely rigid body is a body the distance between any two points of which always remains unchanged; this concept is applicable when the deformation of the body can be neglected. 3) Continuous changeable environment; this concept is applicable when, when studying the movement of a variable medium (deformable body, liquid, gas), the molecular structure of the medium can be neglected.

When studying continuous media, they resort to the following abstractions, which reflect, under given conditions, the most essential properties of the corresponding real bodies: ideally elastic body, plastic body, ideal liquid, viscous liquid, ideal gas etc. In accordance with this, M. is divided into: M. material point, M. of a system of material points, M. of an absolutely rigid body, and M. of a continuous medium; the latter, in turn, is divided into the theory of elasticity, the theory of plasticity, hydromechanics, aeromechanics, gas dynamics, etc. In each of these sections, in accordance with the nature of the problems being solved, the following are distinguished: statics - the study of the equilibrium of bodies under the influence of forces, kinematics - the study of geometric properties of the movement of bodies and dynamics - the study of the movement of bodies under the influence of forces. In dynamics, 2 main tasks are considered: finding the forces under the influence of which a given movement of a body can occur, and determining the movement of a body when the forces acting on it are known.

To solve mathematical problems, all kinds of mathematical methods are widely used, many of which owe their very origin and development to mathematics. Study of the basic laws and principles that govern mechanical movement bodies, and the general theorems and equations arising from these laws and principles constitute the content of the so-called. general, or theoretical, mathematics. Sections of mathematics that have important independent significance are also the theory of oscillations (See Oscillations), the theory of equilibrium stability (See Equilibrium Stability) and motion stability (See Motion Stability), the theory of Gyroscope, and Mechanics. bodies of variable mass, theory of automatic control (see Automatic control), theory of Impact a. An important place in mathematics, especially in the mathematics of continuous media, is occupied by experimental studies, carried out using a variety of mechanical, optical, electrical, etc. physical methods and instruments.

Mathematics is closely related to many other branches of physics. A number of concepts and methods of mathematics, with appropriate generalizations, find application in optics, statistical physics, quantum mathematics, electrodynamics, relativity theory, etc. (see, for example, Action, Lagrange function, Lagrange equations of mechanics, Canonical mechanics equations, Principle of least action ). In addition, when solving a number of problems of gas dynamics (See Gas dynamics), explosion theory, heat transfer in moving liquids and gases, aerodynamics of rarefied gases (See Aerodynamics of rarefied gases), magnetic hydrodynamics (See Magnetic hydrodynamics), etc. simultaneously methods and equations of both theoretical mathematics and, respectively, thermodynamics, molecular physics, the theory of electricity, etc. are used. Mathematics is important for many branches of astronomy (See Astronomy), especially for celestial mechanics (See Celestial Mechanics).

The part of mathematics directly related to technology consists of numerous general technical and special disciplines, such as hydraulics, strength of materials, kinematics of mechanisms, dynamics of machines and mechanisms, theory of gyroscopic devices (See Gyroscopic devices), external ballistics, dynamics of rockets, theory of motion various land, sea, and air vehicles, the theory of regulation and control of the movement of various objects, construction mechanics, a number of branches of technology, and much more. All these disciplines use the equations and methods of theoretical mathematics; mechanics is one of the scientific foundations many areas of modern technology.

Basic concepts and methods of mechanics. The main kinematic measures of motion in mathematics are: for a point - its speed and acceleration, and for a rigid body - the speed and acceleration of translational motion and the angular velocity and angular acceleration of the rotational motion of the body. The kinematic state of a deformable solid is characterized by relative elongations and shifts of its particles; the totality of these quantities determines the so-called. strain tensor. For liquids and gases, the kinematic state is characterized by the strain rate tensor; In addition, when studying the velocity field of a moving fluid, they use the concept of a vortex, which characterizes the rotation of a particle.

The main measure of the mechanical interaction of material bodies in metal is Force. At the same time, the concept of moment of force (see Moment of force) relative to a point and relative to an axis is widely used in mathematics. In continuum mathematics, forces are specified by their surface or volumetric distribution, that is, the ratio of the magnitude of the force to the surface area (for surface forces) or to the volume (for mass forces) on which the corresponding force acts. Internal stresses arising in a continuous medium are characterized at each point of the medium by tangential and normal stresses, the totality of which represents a quantity called the stress tensor (See Stress). The arithmetic mean of three normal stresses, taken with the opposite sign, determines the value called Pressure m at a given point in the medium.

In addition to the acting forces, the movement of a body depends on the degree of its inertia, i.e., on how quickly it changes its movement under the influence of applied forces. For a material point, the measure of inertia is a quantity called mass (See Mass) of the point. The inertia of a material body depends not only on its total mass, but also on the distribution of masses in the body, which is characterized by the position of the center of mass and quantities called axial and centrifugal moments of inertia (See Moment of Inertia); the totality of these quantities determines the so-called. inertia tensor. The inertness of a liquid or gas is characterized by its Density.

M. is based on Newton's laws. The first two are true in relation to the so-called. inertial reference system (See Inertial reference system). The second law gives the basic equations for solving problems of the dynamics of a point, and together with the third - for solving problems of the dynamics of a system of material points. In the mathematics of a continuous medium, in addition to Newton’s laws, laws are also used that reflect the properties of a given medium and establish for it a connection between the stress tensor and the strain or strain rate tensors. This is Hooke's law for a linearly elastic body and Newton's law for a viscous fluid (see Viscosity). For the laws governing other media, see Plasticity theory and Rheology.

Important for solving problems of mathematics are the concepts of dynamic measures of motion, which are momentum, angular momentum (or kinetic momentum), and kinetic energy, and about measures of the action of force, which are impulse of force and work. The relationship between measures of motion and measures of force is given by theorems on changes in momentum, angular momentum and kinetic energy, called general theorems of dynamics. These theorems and the laws of conservation of momentum, angular momentum and mechanical energy that follow from them express the properties of motion of any system of material points and a continuous medium.

Effective methods for studying the equilibrium and motion of a non-free system of material points, i.e. a system on the movement of which are imposed in advance restrictions called mechanical constraints (See Mechanical constraints), are provided by the Variational principles of mechanics, in particular the principle of possible displacements, the principle of least action and etc., as well as D'Alembert's principle. When solving problems of mathematics, the differential equations of motion of a material point, a rigid body, and a system of material points arising from its laws or principles are widely used, in particular the Lagrange equations, canonical equations, the Hamilton-Jacobi equation, etc. ., and in the mathematics of a continuous medium - the corresponding equations of equilibrium or motion of this medium, the equation of continuity (continuity) of the medium and the equation of energy.

Historical sketch. M. is one of the most ancient sciences. Its emergence and development are inextricably linked with the development of the productive forces of society and the needs of practice. Earlier than other sections of M., under the influence of requests mainly from construction equipment, statics began to develop. It can be assumed that elementary information about statics (the properties of the simplest machines) was known several thousand years BC. e., as indirectly evidenced by the remains of ancient Babylonian and Egyptian buildings; but direct evidence of this has not survived. To the first treatises on M. that have come down to us, appearing in Ancient Greece, include the natural philosophical works of Aristotle (See Aristotle) ​​(4th century BC), who introduced the term “M.” into science. From these works it follows that at that time the laws of addition and balancing of forces applied at one point and acting along the same straight line, the properties of the simplest machines and the law of balance of a lever were known. The scientific foundations of statics were developed by Archimedes (3rd century BC).

His works contain a strict theory of the lever, the concept of static moment, the rule of addition of parallel forces, the doctrine of the equilibrium of suspended bodies and the center of gravity, and the principles of hydrostatics. Further significant contributions to research in statics, which led to the establishment of the parallelogram rule of forces and the development of the concept of moment of force, were made by I. Nemorarius (around the 13th century), Leonardo da Vinci (15th century), the Dutch scientist Stevin (16th century) and especially the French scientist P. Varignon (17th century), who completed these studies with the construction of statics based on the rules of addition and expansion of forces and the theorem he proved about the moment of the resultant. The last stage in the development of geometric statics was the development by the French scientist L. Poinsot of the theory of pairs of forces and the construction of statics on its basis (1804). Dr. the direction in statics, based on the principle of possible movements, developed in close connection with the doctrine of movement.

The problem of studying movement also arose in ancient times. Solutions to the simplest kinematic problems on the addition of motions are already contained in the works of Aristotle and in the astronomical theories of the ancient Greeks, especially in the theory of epicycles, completed by Ptolemy (See Ptolemy) (2nd century AD). However, the dynamic teaching of Aristotle, which prevailed almost until the 17th century, was based on the erroneous ideas that a moving body is always under the influence of some force (for an thrown body, for example, this is the pushing force of air, striving to take the place vacated by the body; the possibility of the existence of a vacuum at the same time it was denied) that the speed of a falling body is proportional to its weight, etc.

The period of creation of the scientific foundations of dynamics, and with it the whole of mathematics, was the 17th century. Already in the 15-16 centuries. In the countries of Western and Central Europe, bourgeois relations began to develop, which led to a significant development of crafts, merchant shipping and military affairs (improvement of firearms). This posed a number of important problems for science: the study of the flight of projectiles, the impact of bodies, the strength large ships, oscillations of the pendulum (in connection with the creation of clocks), etc. But finding their solution, which required the development of dynamics, was possible only by destroying the erroneous provisions of the teachings of Aristotle, which continued to dominate. The first important step in this direction was taken by N. Copernicus (16th century). The next step was the experimental discovery by I. Kepler of the kinematic laws of planetary motion (early 17th century). The erroneous positions of Aristotelian dynamics were finally refuted by G. Galileo, who laid the scientific foundations of modern mathematics. He gave the first correct solution to the problem of the motion of a body under the influence of force, having experimentally found the law of uniformly accelerated fall of bodies in a vacuum. Galileo established two basic principles of mathematics - the principle of relativity of classical mathematics and the law of inertia, which he, however, expressed only for the case of motion along a horizontal plane, but applied in his studies in full generality. He was the first to find that in a vacuum the trajectory of a body thrown at an angle to the horizon is a parabola, using the idea of ​​adding movements: horizontal (by inertia) and vertical (accelerated). Having discovered the isochronism of small oscillations of a pendulum, he laid the foundation for the theory of oscillations. Investigating the equilibrium conditions of simple machines and solving some problems of hydrostatics, Galileo uses the so-called formula that he formulated in general terms. golden rule of statics - initial form principle of possible movements. He was the first to study the strength of beams, which laid the foundation for the science of strength of materials. An important merit of Galileo was the systematic introduction of scientific experimentation into mathematics.

The credit for the final formulation of the fundamental laws of mathematics belongs to I. Newton (1687). Having completed the research of his predecessors, Newton generalized the concept of force and introduced the concept of mass into mathematics. The fundamental (second) law of gravity that he formulated allowed Newton to successfully solve a large number of problems related mainly to celestial mathematics, which was based on the law of universal gravitation that he discovered. He also formulates the third of the basic laws of mathematics - the law of equality of action and reaction, which underlies the mathematics of the system of material points. Newton's research completed the creation of the foundations of classical mathematics. The establishment of two initial positions of continuum mathematics dates back to the same period. Newton, who studied the resistance of a liquid by bodies moving in it, discovered the basic law of internal friction in liquids and gases, and the English scientist R. Hooke experimentally established a law expressing the relationship between stresses and deformations in an elastic body.

In the 18th century General analytical methods for solving problems of mathematics for a material point, a system of points, and a rigid body, as well as celestial mathematics, were intensively developed, based on the use of infinitesimal calculus discovered by Newton and G. W. Leibniz. The main merit in the application of this calculus to solve problems of mathematics belongs to L. Euler. He developed analytical methods for solving problems of the dynamics of a material point, developed the theory of moments of inertia, and laid the foundations of solid body mechanics. He also carried out the first studies on the theory of ships, the theory of stability of elastic rods, the theory of turbines and the solution of a number of applied problems of kinematics. A contribution to the development of applied mechanics was the establishment of experimental laws of friction by the French scientists G. Amonton and C. Coulomb.

An important stage in the development of M. was the creation of the dynamics of unfree mechanical systems. The starting point for solving this problem was the principle of possible movements, expressing the general condition of equilibrium of a mechanical system, the development and generalization of which in the 18th century. The studies of I. Bernoulli, L. Carnot, J. Fourier, J. L. Lagrange and others were devoted to the research, and the principle expressed in the most general form by J. D'Alembert (See D'Alembert) and bearing his name. Using these two principles, Lagrange completed the development of analytical methods for solving problems of the dynamics of free and non-free mechanical systems and obtained the equations of motion of the system in generalized coordinates, named after him. He also developed the foundations of the modern theory of oscillations. Another direction in solving problems of mechanics came from the principle of least action in its form, which was expressed for one point by P. Maupertuis and developed by Euler, and generalized to the case of a mechanical system by Lagrange. Celestial mechanics received significant development thanks to the works of Euler, d'Alembert, Lagrange, and especially P. Laplace.

The application of analytical methods to continuum microscopy led to the development theoretical foundations hydrodynamics of an ideal fluid. The fundamental works here were the works of Euler, as well as D. Bernoulli, Lagrange, and D’Alembert. The law of conservation of matter discovered by M. V. Lomonosov was of great importance for the continuum of matter.

In the 19th century Intensive development of all branches of mathematics continued. In rigid body dynamics, the classical results of Euler and Lagrange, and then S. V. Kovalevskaya, continued by other researchers, served as the basis for the theory of the gyroscope, which acquired especially great practical significance in the 20th century. The fundamental works of M. V. Ostrogradsky (See Ostrogradsky), W. Hamilton, K. Jacobi, G. Hertz, and others were devoted to the further development of the principles of mathematics.

In solving the fundamental problem of mathematics and all natural science - the stability of equilibrium and motion, a number of important results were obtained by Lagrange, English. scientist E. Rous and N. E. Zhukovsky. Rigorous formulation of the problem of motion stability and development of the most common methods its solutions belong to A. M. Lyapunov. In connection with the demands of machine technology, research continued on the theory of oscillations and the problem of regulating the speed of machines. The foundations of the modern theory of automatic control were developed by I. A. Vyshnegradsky (See Vyshnegradsky).

In parallel with the dynamics in the 19th century. Kinematics also developed and became increasingly important in its own right. Franz. the scientist G. Coriolis proved the theorem about the components of acceleration, which was the basis of M. relative motion. Instead of the terms “accelerating forces”, etc., the purely kinematic term “acceleration” appeared (J. Poncelet, A. Rezal). Poinsot gave a number of visual geometric interpretations of the motion of a rigid body. The importance of applied research on the kinematics of mechanisms has increased, to which P. L. Chebyshev made an important contribution. In the 2nd half of the 19th century. kinematics became an independent section of M.

Significant development in the 19th century. M. of continuous medium also received. Through the works of L. Navier and O. Cauchy, the general equations of the theory of elasticity were established. Further fundamental results in this area were obtained by J. Green, S. Poisson, A. Saint-Venant, M. V. Ostrogradsky, G. Lame, W. Thomson, G. Kirchhoff and others. Research by Navier and J. Stokes led to the establishment of differential equations movement of viscous fluid. Significant contributions to the further development of the dynamics of ideal and viscous fluids were made by Helmholtz (the study of vortices), Kirchhoff and Zhukovsky (separated flow around bodies), O. Reynolds (the beginning of the study of turbulent flows), L. Prandtl (boundary layer theory), and others. N. P. Petrov created the hydrodynamic theory of friction during lubrication, further developed by Reynolds, Zhukovsky together with S. A. Chaplygin and others. Saint-Venant proposed the first mathematical theory plastic flow of metal.

In the 20th century the development of a number of new sections of mathematics begins. Problems put forward by electrical and radio engineering, problems of automatic control, etc., gave rise to the emergence of a new field of science - the theory of nonlinear oscillations, the foundations of which were laid by the works of Lyapunov and A. Poincaré. Another branch of mathematics on which the theory of jet propulsion is based is the dynamics of bodies of variable mass; its foundations were created at the end of the 19th century. through the works of I.V. Meshchersky (See Meshchersky). The initial research on the theory of rocket motion belongs to K. E. Tsiolkovsky (See Tsiolkovsky).

Two important new sections appear in continuum mathematics: aerodynamics, the foundations of which, like all aviation science, were created by Zhukovsky, and gas dynamics, the foundations of which were laid by Chaplygin. The works of Zhukovsky and Chaplygin had great value for the development of all modern hydroaerodynamics.

Modern problems of mechanics. Among the important problems of modern mathematics are the already noted problems of the theory of oscillations (especially nonlinear ones), the dynamics of a rigid body, the theory of stability of motion, as well as the mathematics of bodies of variable mass and the dynamics of space flights. In all areas of mathematics, problems in which instead of “deterministic”, that is, previously known, quantities (for example, acting forces or laws of motion of individual objects) have to be considered, are becoming increasingly important, with “probabilistic” quantities, i.e., quantities for which only the probability that they can have certain values ​​is known. In continuum mathematics, the problem of studying the behavior of macroparticles when their shape changes is very relevant, which is associated with the development of a more rigorous theory of turbulent flows of liquids, the solution of problems of plasticity and creep, and the creation of a well-founded theory of the strength and destruction of solids.

A large range of questions in magnetophysics are also associated with the study of the motion of plasma in a magnetic field (magnetic hydrodynamics), i.e., with the solution of one of the most pressing problems of modern physics - the implementation of controlled thermonuclear reaction. In hydrodynamics, a number of the most important problems are associated with problems of high speeds in aviation, ballistics, turbine construction and engine building. Many new problems arise at the intersection of mathematics and other fields of science. These include problems of hydrothermochemistry (i.e., studies of mechanical processes in liquids and gases entering chemical reactions), the study of the forces causing cell division, the mechanism of muscle force formation, etc.

Electronic computers and analog machines are widely used to solve many problems in mathematics. At the same time, the development of methods for solving new problems of machining (especially the machining of continuous media) using these machines is also a very pressing problem.

Research in different areas Mechanics are conducted at universities and higher technical educational institutions of the country, at the Institute of Mechanical Problems of the USSR Academy of Sciences, and also at many other research institutes both in the USSR and abroad.

For coordination scientific research International congresses on theoretical and applied mathematics and conferences devoted to individual areas of mathematics are periodically held in mathematics, organized by the International Union of Theoretical and Applied Medicine (IUTAM), where the USSR is represented by the USSR National Committee on Theoretical and Applied Medicine. The same committee together with other scientific institutions, periodically organizes all-Union congresses and conferences dedicated to research in various areas M.

GYMNASIUM No. 1534

RESEARCH

IN PHYSICS

“HISTORY OF MECHANICS DEVELOPMENT”

Completed by: student of grade 11 “A”

Sorokina A. A.

Checked by: Gorkina T.B.

Moscow 2003

1. INTRODUCTION
4. HISTORY OF THE DEVELOPMENT OF MECHANICS
The era before the establishment of the foundations of mechanics
The period of creation of the fundamentals of mechanics
Development of mechanical methods in the 18th century.
Mechanics of the 19th and early 20th centuries.
Mechanics in Russia and the USSR
5. PROBLEMS OF MODERN MECHANICS
6. CONCLUSION
7. LIST OF REFERENCES USED
8. APPENDIX

1. INTRODUCTION

For every person there are two worlds: internal and external; The mediators between these two worlds are the senses. The outside world has the ability to influence the senses, cause special kinds of changes in them, or, as they say, arouse irritation in them. The inner world of a person is determined by the totality of those phenomena that absolutely cannot be accessible to the direct observation of another person.

The irritation in the sense organ caused by the external world is transmitted to the internal world and, for its part, causes in it subjective feeling, the appearance of which requires the presence of consciousness.

Perceived inner world subjective sensation is objectified, i.e. transferred to external space as something belonging to a certain place and a certain time. In other words, through such objectification we transfer our sensations to the outside world, with space and time serving as the background on which these objective sensations are located. In those places in space where they are located, we involuntarily assume the cause that generates them.

A person has the ability to compare perceived sensations with each other, to judge their similarity or dissimilarity and, in the second case, to distinguish qualitative and quantitative dissimilarities, and quantitative dissimilarity can relate either to tension (intensity), or to extension (extensiveness), or, finally, to duration of the irritating objective reason.

Since the inferences accompanying any objectification are exclusively based on the perceived sensation, the complete identity of these sensations will certainly entail the identity of objective causes, and this identity, in addition to, and even against our will, is preserved in those cases when other senses indisputably testify us about the diversity of reasons. Here lies one of the main sources of undoubtedly erroneous conclusions, leading to the so-called illusions of vision, hearing, etc. Another source is the lack of skill in dealing with new sensations.

Perception in space and time of sensory impressions that we compare with each other and to which we attach meaning objective reality existing outside of our consciousness is called an external phenomenon. Changes in the color of bodies depending on lighting, the same level of water in vessels, the swing of a pendulum are external phenomena.

One of the powerful levers that moves humanity along the path of its development is curiosity, which has the final, unattainable goal - knowledge of the essence of our being, the true relationship of our internal world to the external world. The result of curiosity was an acquaintance with very a large number the most diverse phenomena that form the subject of a number of sciences, among which physics occupies one of the first places, due to the vastness of the field it processes and the importance that it has for almost all other sciences.

2. DEFINITION OF MECHANICS; ITS PLACE AMONG OTHER SCIENCES; MECHANICAL DIVISIONS

Mechanics (from the Greek m h c a n i c h - skill related to machines; the science of machines) is the science of the simplest form of the movement of matter - mechanical movement, representing change over time spatial arrangement bodies, and the interactions between them associated with the movement of bodies. Mechanics studies the general laws connecting mechanical movements and interactions, accepting for the interactions themselves laws obtained experimentally and substantiated in physics. Mechanics methods are widely used in various fields of natural science and technology.

Mechanics studies the movements of material bodies using the following abstractions:

1) A material point is like a body of negligible size, but of finite mass. The role of a material point can be played by the center of inertia of a system of material points, in which the mass of the entire system is considered concentrated;

2) An absolutely rigid body, a collection of material points located at constant distances from each other. This abstraction is applicable if the deformation of the body can be neglected;

3) Continuous medium. With this abstraction, change is allowed relative position elementary volumes. In contrast to a rigid body, innumerable parameters are required to specify the motion of a continuous medium. Continuous media include solid, liquid and gaseous bodies, reflected in the following abstract concepts: ideal elastic body, plastic body, ideal liquid, viscous liquid, ideal gas and others. These abstract ideas about the material body reflect the actual properties of real bodies that are significant under given conditions.

Accordingly, mechanics are divided into:

• mechanics of a material point;
• mechanics of a system of material points;
• absolutely mechanical solid;
• continuum mechanics.

The latter, in turn, is subdivided into the theory of elasticity, fluid mechanics, aeromechanics, gas mechanics and others (see Appendix).

The term “theoretical mechanics” usually denotes the part of mechanics that deals with the study of the most general laws of motion, its formulation general provisions and theorems, as well as the application of mechanics methods to the study of the motion of a material point, a system of a finite number of material points and an absolutely rigid body.

In each of these sections, first of all, statics is highlighted, combining issues related to the study of the conditions of equilibrium of forces. There are statics of a solid body and statics of a continuous medium: statics of an elastic body, hydrostatics and aerostatics (see Appendix). The movement of bodies in abstraction from the interaction between them is studied by kinematics (see Appendix). An essential feature of the kinematics of continuous media is the need to determine for each moment in time the distribution in space of displacements and velocities. The subject of dynamics is the mechanical movements of material bodies in connection with their interactions.

Significant applications of mechanics are in the field of technology. The tasks posed by technology to mechanics are very diverse; These are issues of the movement of machines and mechanisms, the mechanics of vehicles on land, at sea and in the air, structural mechanics, various departments of technology and many others. In connection with the need to satisfy the demands of technology, special ones were allocated from mechanics Technical science. Kinematics of mechanisms, dynamics of machines, theory of gyroscopes, external ballistics (see Appendix) represent technical sciences using absolutely rigid body methods. Strength of materials and hydraulics (see Appendix), related to elasticity theory and hydrodynamics general basics, develop calculation methods for practice, corrected by experimental data. All branches of mechanics have developed and continue to develop in close connection with the needs of practice, in the course of solving technical problems.

Mechanics as a branch of physics developed in close connection with its other branches - optics, thermodynamics and others. The foundations of so-called classical mechanics were summarized at the beginning of the 20th century. in connection with the discovery of physical fields and laws of motion of microparticles. The content of the mechanics of fast-moving particles and systems (with velocities on the order of the speed of light) is set out in the theory of relativity, and the mechanics of micro-motions - in quantum mechanics.

3. BASIC CONCEPTS AND METHODS OF MECHANICS

The laws of classical mechanics are valid in relation to the so-called inertial, or Galilean, frames of reference (see Appendix). To the extent that Newtonian mechanics is valid, time can be considered independently of space. The time intervals are practically the same in all reporting systems, whatever their mutual motion, if their relative speed is small compared to the speed of light.

The main kinematic measures of movement are speed, which has a vector character, since it determines not only the speed of change of the path over time, but also the direction of movement, and acceleration - a vector, which is a measure of the velocity vector in time. Measures of the rotational motion of a rigid body are the vectors of angular velocity and angular acceleration. In the statics of an elastic body, the displacement vector and the corresponding deformation tensor, which includes the concepts of relative elongations and shears, are of primary importance.

The main measure of the interaction of bodies, characterizing the change in time of the mechanical movement of a body, is force. Sets of magnitude (intensity)

force, expressed in certain units, the direction of the force (line of action) and the point of application quite unambiguously determine the force as a vector.

Mechanics is based on the following Newton's laws. The first law, or the law of inertia, characterizes the movement of bodies in conditions of isolation from other bodies, or when external influences are balanced. This law states: every body maintains a state of rest or uniform and rectilinear movement until the applied forces force it to change this state. The first law can serve to define inertial frames of reference. The second law, which establishes a quantitative relationship between a force applied to a point and the change in momentum caused by this force, states: the change in motion occurs in proportion to the applied force and occurs in the direction of the line of action of this force. According to this law, the acceleration of a material point is proportional to the force applied to it: this force F causes less acceleration A body, the greater its inertia. The measure of inertia is mass. According to Newton's second law, force is proportional to the product of the mass of a material point and its acceleration; with proper choice of the unit of force, the latter can be expressed as the product of the mass of a point m for acceleration A :

This vector equality represents the basic equation of the dynamics of a material point. Newton's third law states: an action is always accompanied by an equal and oppositely directed reaction, that is, the action of two bodies on each other is always equal and directed along the same straight line in opposite directions. While Newton's first two laws apply to one material point, the third law is fundamental for a system of points. Along with these three basic laws of dynamics, there is a law of independence of the action of forces, which is formulated as follows: if several forces act on a material point, then the acceleration of the point is the sum of those accelerations that the point would have under the action of each force separately. The law of independent action of forces leads to the rule of parallelogram of forces.

In addition to the previously mentioned concepts, other measures of motion and action are used in mechanics. The most important are the measures of motion: vector - momentum p = mv, equal to the product of mass by the velocity vector, and scalar - kinetic energy E k = 1 / 2 mv 2, equal to half the product of mass by the square of the velocity. In the case of rotational motion of a rigid body, its inertial properties are specified by the inertia tensor, which determines at each point of the body the moments of inertia and centrifugal moments about three axes passing through this point. The measure of the rotational motion of a rigid body is the angular momentum vector, equal to the product of the moment of inertia and the angular velocity. Measures of the action of forces are: vector - elementary impulse of force F dt(the product of force and the time element of its action), and scalar - elementary work F*dr(scalar product of force vectors and elementary displacement of the position point); During rotational motion, the measure of impact is the moment of force.

The main measures of motion in the dynamics of a continuous medium are continuously distributed quantities and, accordingly, are specified by their distribution functions. Thus, density determines the distribution of mass; forces are given by their surface or volumetric distribution. The movement of a continuous medium, caused by external forces applied to it, leads to the emergence of a stressed state in the medium, characterized at each point by a set of normal and tangential stresses, represented by a single physical quantity - the stress tensor. The arithmetic mean of three normal stresses at a given point, taken with the opposite sign, determines the pressure (see Appendix).

The study of equilibrium and motion of a continuous medium is based on the laws of connection between the stress tensor and the strain tensor or strain rates. This is Hooke's law in the statics of a linear elastic body and Newton's law in the dynamics of a viscous fluid (see Appendix). These laws are the simplest; Other relationships have been established that more accurately characterize the phenomena occurring in real bodies. There are theories that take into account the previous history of movement and stress of the body, theories of creep, relaxation and others (see Appendix).

The relationships between the measures of motion of a material point or system of material points and the measures of the action of forces are contained in the general theorems of dynamics:

momentum, angular momentum and kinetic energy. These theorems express the properties of motions of both a discrete system of material points and a continuous medium. When considering the equilibrium and motion of a non-free system of material points, i.e. a system subject to predetermined restrictions - mechanical connections (see Appendix), the application of the general principles of mechanics - the principle of possible displacements and D'Alembert's principle - is important. When applied to a system of material points, the principle of possible displacements is as follows: for the equilibrium of a system of material points with stationary and ideal connections, it is necessary and sufficient that the sum of the elementary works of all active forces acting on the system for any possible movement of the system is equal to zero (for non-liberating connections) or was equal to zero or less than zero (for liberating connections). D'Alembert's principle for a free material point states: at any moment of time, the forces applied to the point can be balanced by adding the force of inertia to them.

When formulating problems, mechanics proceeds from the basic equations expressing the found laws of nature. To solve these equations, mathematical methods are used, and many of them originated and were developed precisely in connection with problems of mechanics. When setting a problem, it was always necessary to focus attention on those aspects of the phenomenon that seem to be the main ones. In cases where it is necessary to take into account side factors, as well as in cases where the complexity of the phenomenon does not lend itself to mathematical analysis, experimental research is widely used. Experimental methods of mechanics are based on developed techniques of physical experimentation. To record movements, both optical methods and electrical recording methods are used, based on the preliminary conversion of mechanical movement into an electrical signal. To measure forces, various dynamometers and scales are used, equipped with automatic devices and tracking systems. To measure mechanical vibrations, various radio circuits have become widespread. The experiment in continuum mechanics achieved particular success. To measure the voltage, an optical method is used (see Appendix), which consists of observing a loaded transparent model in polarized light. To measure strain great development V last years acquired strain gauging using mechanical and optical strain gauges (see Appendix), as well as resistance strain gauges. To measure velocities and pressures in moving liquids and gases, thermoelectric, capacitive, induction and other methods are successfully used.

4. HISTORY OF THE DEVELOPMENT OF MECHANICS

History of mechanics, as well as others natural sciences, is inextricably linked with the history of the development of society, with the general history of the development of its productive forces. The history of mechanics can be divided into several periods, differing both in the nature of the problems and in the methods for solving them.

The era that preceded the establishment of the foundations of mechanics. The era of the creation of the first tools of production and artificial buildings should be recognized as the beginning of the accumulation of experience, which later served as the basis for the discovery of the basic laws of mechanics. While the geometry and astronomy of the ancient world were already quite developed scientific systems, in the field of mechanics, only individual provisions related to the simplest cases of equilibrium of bodies were known. Statics arose earlier than all branches of mechanics. This section developed in close connection with the building art of the ancient world.

The basic concept of statics - the concept of force - was initially closely associated with muscular effort caused by the pressure of an object on the hand. Around the beginning of the 4th century. BC e. the simplest laws of addition and balancing of forces applied to one point along the same straight line were already known. Of particular interest was the lever problem. The theory of leverage was created by the great ancient scientist Archimedes (III century BC) and outlined in the essay “On Leverages.” He established the rules for the addition and expansion of parallel forces, defined the concept of the center of gravity of a system of two weights suspended from a rod, and clarified the conditions for the equilibrium of such a system. Archimedes is responsible for the discovery of the basic laws of hydrostatics. Their

He applied theoretical knowledge in the field of mechanics to various practical issues of construction and military equipment. The concept of moment of force, which plays a fundamental role in all modern mechanics, is already present in a hidden form in Archimedes’ law. The great Italian scientist Leonardo da Vinci (1452 – 1519) introduced the concept of leverage under the guise of “potential leverage”. The Italian mechanic Guido Ubaldi (1545 – 1607) applied the concept of moment in his theory of blocks, where the concept of a pulley was introduced. Polyspast (Greek p o l u s p a s t o n, from p o l u - a lot and s p a w - I pull) - a system of movable and fixed blocks, bent around a rope, used to gain strength and, less often, to gain speed. Usually, statics also includes the doctrine of the center of gravity of a material body. The development of this purely geometric doctrine (geometry of masses) is closely connected with the name of Archimedes, who indicated, using the famous method of exhaustion, the position of the center of gravity of many regular geometric forms, flat and spatial. General theorems on the centers of gravity of bodies of revolution were given by the Greek mathematician Pappus (3rd century AD) and the Swiss mathematician P. Gulden in the 17th century. Statics owes the development of its geometric methods to the French mathematician P. Varignon (1687); These methods were most fully developed by the French mechanic L. Poinsot, whose treatise “Elements of Statics” was published in 1804. Analytical statics, based on the principle of possible displacements, was created by the famous French scientist J. Lagrange.

With the development of crafts, trade, navigation and military affairs and the associated accumulation of new knowledge, in the XIV and XV centuries. - During the Renaissance, the flourishing of sciences and arts begins. A major event that revolutionized the human worldview was the creation by the great Polish astronomer Nicolaus Copernicus (1473 - 1543) of the doctrine of the heliocentric system of the world, in which the spherical Earth occupies a central stationary position, and around it celestial bodies move in their circular orbits: the Moon, Mercury, Venus , Sun, Mars, Jupiter, Saturn.

Kinematic and dynamic studies of the Renaissance were mainly aimed at clarifying the ideas about the uneven and curvilinear movement of a point. Until this time, the dynamic views of Aristotle, set out in his “Problems of Mechanics,” which did not correspond to reality, were generally accepted. Thus, he believed that in order to maintain uniform and linear motion of a body, it is necessary to constantly apply effective force. This statement seemed to him to agree with everyday experience. Aristotle, of course, knew nothing about the fact that a friction force arises in this case. He also believed that the speed of free fall of bodies depends on their weight: “If half the weight passes so much in some time, then double the weight will travel the same amount in half the time.” Believing that everything consists of four elements - earth, water, air and fire, he writes: “Heavy is everything that is capable of rushing to the middle or center of the world; everything that rushes from the middle or center of the world is easy.” From this he concluded: since heavy bodies fall towards the center of the Earth, this center is the center of the world, and the Earth is motionless. Not yet possessing the concept of acceleration, which was later introduced by Galileo, researchers of this era considered accelerated motion as consisting of separate uniform movements, each interval having its own speed. At the age of 18, Galileo, observing the small damped oscillations of a chandelier during a church service and counting time by pulse beats, established that the period of oscillation of a pendulum does not depend on its swing. Doubting the correctness of Aristotle's statements, Galileo began to carry out experiments with the help of which he, without analyzing the reasons, established the laws of motion of bodies near the earth's surface. By throwing bodies from the tower, he established that the time a body falls does not depend on its weight and is determined by the height of the fall. He was the first to prove that when a body falls in free fall, the distance traveled is proportional to the square of time.

Remarkable experimental studies of the free vertical fall of a heavy body were carried out by Leonardo da Vinci; These were probably the first specially organized experimental studies in the history of mechanics.

The period of creation of the fundamentals of mechanics. Practice (mainly merchant shipping and military affairs) confronts the mechanics of the 16th - 17th centuries. row the most important problems occupied the minds of the best scientists of that time. “... Along with the emergence of cities, large buildings and the development of crafts, mechanics also developed. Soon it also becomes necessary for shipping and military affairs” (Engels F., Dialectics of Nature, 1952, p. 145).

It was necessary to accurately study the flight of projectiles, the strength of large ships, the oscillations of a pendulum, and the impact of a body. Finally, the victory of the Copernican teaching raises the problem of the movement of celestial bodies. Heliocentric worldview by the beginning of the 16th century. created the prerequisites for the establishment of the laws of planetary motion by the German astronomer J. Kepler (1571 - 1630). He formulated the first two laws of planetary motion:

1. All planets move in ellipses, with the Sun at one of the focuses.

2. The radius vector drawn from the Sun to the planet describes equal areas in equal periods of time.

The founder of mechanics is the great Italian scientist G. Galileo (1564 – 1642). He experimentally established the quantitative law of falling bodies in a vacuum, according to which the distances covered by a falling body in equal periods of time are related to each other as successive odd numbers. Galileo established the laws of motion of heavy bodies on an inclined plane, showing that whether heavy bodies fall vertically or along an inclined plane, they always acquire such speeds that must be imparted to them in order to raise them to the height from which they fell. Moving to the limit, he showed that on a horizontal plane a heavy body will be at rest or will move uniformly and in a straight line. Thus he formulated the law of inertia. By adding the horizontal and vertical motions of a body (this is the first addition in the history of mechanics of finite independent motions), he proved that a body thrown at an angle to the horizon describes a parabola, and showed how to calculate the flight length and the maximum height of the trajectory. In all his conclusions, he always emphasized that we're talking about about movement in the absence of resistance. In dialogues about two systems of the world, very figuratively, in the form of an artistic description, he showed that all the movements that can occur in the cabin of a ship do not depend on whether the ship is at rest or moving straight and evenly. With this, he established the principle of relativity of classical mechanics (the so-called Galileo-Newton principle of relativity). In the particular case of the weight force, Galileo closely connected the constancy of weight with the constancy of the acceleration of the fall, but only Newton, by introducing the concept of mass, gave a precise formulation of the relationship between force and acceleration (the second law). By exploring the conditions for the equilibrium of simple machines and the floating of bodies, Galileo essentially applied the principle of possible displacements (albeit in a rudimentary form). Science owes him the first study of the strength of beams and the resistance of fluid to bodies moving in it.

The French geometer and philosopher R. Descartes (1596 – 1650) expressed the fruitful idea of ​​conservation of momentum. He applies mathematics to the analysis of motion and, by introducing variables into it, establishes a correspondence between geometric images and algebraic equations. But he did not notice the essential fact that the quantity of motion is a directional quantity, and added the quantities of motion arithmetically. This led him to erroneous conclusions and reduced the significance of his applications of the law of conservation of momentum, in particular, to the theory of impact of bodies.

A follower of Galileo in the field of mechanics was the Dutch scientist H. Huygens (1629 – 1695). He is responsible for the further development of the concepts of acceleration during curvilinear motion of a point ( centripetal acceleration). Huygens also solved a number of important problems in dynamics - the motion of a body in a circle, the oscillations of a physical pendulum, the laws of elastic impact. He was the first to formulate the concepts of centripetal and centrifugal force, moment of inertia, and the center of oscillation of a physical pendulum. But his main merit lies in the fact that he was the first to apply a principle essentially equivalent to the principle of living forces (the center of gravity of a physical pendulum can only rise to a height equal to the depth of its fall). Using this principle, Huygens solved the problem of the center of oscillation of a pendulum - the first problem of the dynamics of a system of material points. Based on the idea of ​​conservation of momentum, he created a complete theory of the impact of elastic balls.

The credit for formulating the basic laws of dynamics belongs to the great English scientist I. Newton (1643 – 1727). In his treatise “Mathematical Principles of Natural Philosophy,” which was published in its first edition in 1687, Newton summed up the achievements of his predecessors and pointed out the ways for the further development of mechanics for centuries to come. Completing the views of Galileo and Huygens, Newton enriches the concept of force, indicates new types of forces (for example, gravitational forces, environmental resistance forces, viscosity forces and many others), and studies the laws of the dependence of these forces on the position and motion of bodies. The fundamental equation of dynamics, which is an expression of the second law, allowed Newton to successfully solve a large number of problems related mainly to celestial mechanics. In it, he was most interested in the reasons that made him move along elliptical orbits. Also in student year Newton thought about the issues of gravity. The following entry was found in his papers: “From Kepler’s rule that the periods of planets are in one and a half proportion to the distance from the centers of their orbits, I deduced that the forces holding the planets in their orbits must be in the inverse ratio of the squares of their distances from the centers , around which they revolve. From here I compared the force required to keep the Moon in its orbit with the force of gravity on the surface of the Earth and found that they almost correspond to each other.”

In the above passage, Newton does not provide evidence, but I can assume that his reasoning was as follows. If we approximately assume that the planets move uniformly in circular orbits, then according to Kepler’s third law, which Newton refers to, I will get

T 2 2 / T 2 1 = R 3 2 / R 3 1 , (1.1) where T j and R j are the orbital periods and orbital radii of the two planets (j = 1, 2).

When the planets move uniformly in circular orbits with speeds V j, their periods of revolution are determined by the equalities T j = 2 p R j / V j.

Hence,

T 2 / T 1 = 2 p R 2 V 1 / V 2 2 p R 1 = V 1 R 2 / V 2 R 1 .

Now relation (1.1) is reduced to the form

V 2 1 / V 2 2 = R 2 / R 1 . (1.2)

By the years under review, Huygens had already established that centrifugal force is proportional to the square of the speed and inversely proportional to the radius of the circle, i.e. F j = kV 2 j / R j, where k is the proportionality coefficient.

If we now introduce the relation V 2 j = F j R j / k into equality (1.2), then I get

F 1 / F 2 = R 2 2 / R 2 1 , (1.3) which sets inverse proportionality centrifugal forces of planets to the squares of their distances to the Sun.

Newton also studied the resistance of liquids to moving bodies; he established the law of resistance, according to which the resistance of a fluid to the movement of a body in it is proportional to the square of the body’s speed. Newton discovered the fundamental law of internal friction in liquids and gases.

By the end of the 17th century. the fundamentals of mechanics were thoroughly developed. If the ancient centuries are considered the prehistory of mechanics, then the 17th century. can be considered as the period of creation of its foundations.

Development of mechanical methods in the 18th century. In the 18th century. the needs of production - the need to study the most important mechanisms, on the one hand, and the problem of the movement of the Earth and the Moon, put forward by the development of celestial mechanics, on the other - led to the creation of general methods for solving problems in the mechanics of a material point, a system of points of a rigid body, developed in “Analytical Mechanics” (1788) J. Lagrange (1736 – 1813).

In the development of the dynamics of the post-Newtonian period, the main merit belongs to the St. Petersburg academician L. Euler (1707 - 1783). He developed the dynamics of a material point in the direction of applying infinitesimal analysis methods to solving the equations of motion of a point. Euler's treatise “Mechanics, i.e., the science of motion, expounded by the analytical method,” published in St. Petersburg in 1736, contains general uniform methods for the analytical solution of problems of point dynamics.

L. Euler is the founder of solid body mechanics. He owns the generally accepted method of kinematic description of the motion of a rigid body using three Euler angles. A fundamental role in the further development of dynamics and many of its technical applications was played by the basic differential equations established by Euler for the rotational motion of a rigid body around a fixed center. Euler established two integrals: the integral of angular momentum

A 2 w 2 x + B 2 w 2 y + C 2 w 2 z = m

and the integral of living forces (energy integral)

A w 2 x + B w 2 y + C w 2 z = h,

where m and h are arbitrary constants, A, B and C are the main moments of inertia of the body for a fixed point, and w x, w y, w z are the projections of the angular velocity of the body onto the main axes of inertia of the body.

These equations were an analytical expression of the theorem of angular momentum discovered by him, which is a necessary addition to the law of momentum, formulated in general form in Newton’s Principia. In Euler’s “Mechanics”, a formulation of the law of “living forces” close to the modern one was given for the case of rectilinear motion and the presence of such movements of a material point was noted in which the change in living force when the point moves from one position to another does not depend on the shape of the trajectory. This laid the foundation for the concept of potential energy. Euler is the founder of fluid mechanics. They were given the basic equations of the dynamics of an ideal fluid; he is credited with creating the foundations of the theory of the ship and the theory of stability of elastic rods; Euler laid the foundation for the theory of turbine calculations by deriving the turbine equation; in applied mechanics, Euler's name is associated with issues of the kinematics of figured wheels, the calculation of friction between a rope and a pulley, and many others.

Celestial mechanics was largely developed by the French scientist P. Laplace (1749 - 1827), who in his extensive work “Treatise on Celestial Mechanics” combined the results of the research of his predecessors - from Newton to Lagrange - with his own studies of stability solar system, solving the three-body problem, the motion of the Moon and many other issues of celestial mechanics (see Appendix).

One of the most important applications of Newton's theory of gravitation was the question of the equilibrium figures of rotating liquid masses, the particles of which gravitate towards each other, in particular the figure of the Earth. The foundations of the theory of equilibrium of rotating masses were outlined by Newton in the third book of his Elements. The problem of equilibrium figures and stability of a rotating liquid mass played a significant role in the development of mechanics.

The great Russian scientist M.V. Lomonosov (1711 – 1765) highly appreciated the importance of mechanics for natural science, physics and philosophy. He owns a materialistic interpretation of the processes of interaction between two bodies: “when one body accelerates the movement of another and imparts to it part of its movement, it is only in such a way that it itself loses the same part of the movement.” He is one of the founders kinetic theory heat and gases, the author of the law of conservation of energy and motion. Let us quote Lomonosov’s words from a letter to Euler (1748): “All changes that occur in nature take place in such a way that if something is added to something, then the same amount will be taken away from something else. Thus, as much matter is added to one body, the same amount will be taken away from another; no matter how many hours I spend sleeping, I take the same amount away from vigil, etc. Since this law of nature is universal, it even extends to the rules of movement, and a body that encourages another to move loses as much of its movement as it communicates. to another, moved by him.” Lomonosov was the first to predict the existence of absolute zero temperature and expressed the idea of ​​a connection between electrical and light phenomena. As a result of the activities of Lomonosov and Euler, the first works of Russian scientists appeared, who creatively mastered the methods of mechanics and contributed to its further development.

The history of the creation of the dynamics of a non-free system is associated with the development of the principle of possible movements, which expresses the general conditions of equilibrium of the system. This principle was first applied by the Dutch scientist S. Stevin (1548 – 1620) when considering the equilibrium of a block. Galileo formulated the principle in the form of the “golden rule” of mechanics, according to which “what is gained in strength is lost in speed.” The modern formulation of the principle was given in late XVIII V. based on the abstraction of “ideal connections”, reflecting the idea of ​​an “ideal” machine, devoid of internal losses due to harmful resistance in the transmission mechanism. It looks like this: if in an isolated equilibrium position of a conservative system with stationary connections the potential energy has a minimum, then this equilibrium position is stable.

The creation of the principles of dynamics of a non-free system was facilitated by the problem of the movement of a non-free material point. A material point is called non-free if it cannot occupy an arbitrary position in space. In this case, D’Alembert’s principle sounds as follows: the active forces and reactions of connections acting on a moving material point can be balanced at any time by adding the force of inertia to them.

An outstanding contribution to the development of the analytical dynamics of a non-free system was made by Lagrange, who in his fundamental two-volume work “Analytical Mechanics” indicated the analytical expression of D’Alembert’s principle - the “general formula of dynamics”. How did Lagrange get it?

After Lagrange has laid down the various principles of statics, he proceeds to establish “ general formula statics for the balance of any system of forces.” Beginning

with two forces, Lagrange establishes by induction the following general formula for

equilibrium of any system of forces:

Pdp+ Q dq + R dr + … = 0. (2.1)

This equation represents a mathematical representation of the principle of possible movements. In modern notation this principle has the form

å n j=1 F j d r j = 0 (2.2)

Equations (2.1) and (2.2) are practically the same. The main difference, of course, is not in the form of notation, but in the definition of variation: in our days it is an arbitrarily conceivable movement of the point of application of force, compatible with connections, but for Lagrange it is a small movement along the line of action of the force and in the direction of its action.

Lagrange introduces the function P(now called potential energy), defining it by the equality

d P = Pdp + Q dq + R dr+ … , (2.3) in Cartesian coordinates function P(after integration) has the form

P = A + Bx + Сy + Dz + … + Fx 2 + Gxy + Hy 2 + Kxz + Lyz + Mz 2 + … (2.4)

To further prove this, Lagrange invents the famous method of indefinite multipliers. Its essence is as follows. Consider the equilibrium n material points, each of which is acted upon by a force F j. Between the coordinates of the points there is m connections j r= 0, depending only on their coordinates. Considering that d j r= 0, equation (2.2) can immediately be reduced to the following modern form:

å n j=1 F j d r j+ å m r=1 l r d j r= 0, (2.5) where l r– indefinite factors. From this we obtain the following equilibrium equations, called Lagrange equations of the first kind:

X j+ å m r=1 l r ¶ j r / ¶ x j = 0, Y j+ å m r=1 l r ¶ j r / ¶ y j = 0,

Z j+ å m r=1 l r ¶ j r / ¶ z j= 0 (2.6) To these equations we need to add m constraint equations j r = 0 (X j,Y j, Z j– force projections F j).

Let us show how Lagrange uses this method to derive the equilibrium equations for an absolutely flexible and inextensible thread. First of all, related to the unit length of the thread (its dimension is equal to F/L). Communication equation for inextensible the thread looks like ds= const, and therefore d ds= 0. In equation (2.5), the sums turn into integrals over the length of the thread l

ò l 0 F d rds + ò l 0 l d ds= 0. (2.7) Taking into account the equality

(ds) 2 = (dx) 2 + (dy) 2 + (dz) 2 ,

d ds = dx / ds d dx + dy / ds d dy + dz / ds d dz.

ò l 0 l d ds = ò l 0 (l dx / ds d dx + l dy / ds d dy + l dz / ds d dz)

or, rearranging the operations d and d and integrating by parts,

ò l 0 l d ds = (l dx / ds d x + l dy / ds d y + l dz / ds d z) –

- ò l 0 d (l dx / ds) d x + d (l dy / ds) d y + d (l dz / ds) d z.

Assuming that the thread is fixed at the ends, we obtain d x = d y = d z= 0 at s= 0 and s = l, and, therefore, the first term becomes zero. We enter the remaining part into equation (2.7) and expand the scalar product F*dr and group the members:

ò l 0 [ Xds – d (l dx / ds) ] d x + [ Yds – d (l dy / ds) ] d y + [ Zds – d (d dz / ds) ] d z = 0.

Since variations d x, d y and d z are arbitrary and independent, then all square brackets must equal zero, which gives three equilibrium equations for an absolutely flexible inextensible thread:

d / ds (l dx / ds) – X = 0, d / ds (l dy / ds) – Y = 0,

d/ ds (l dz / ds) – Z = 0. (2.8)

Lagrange explains it this way physical meaning multiplier l: “Since the value l d ds may represent a moment of some force l (in modern terminology – “virtual (possible) work”) tending to reduce the length of the element ds, then the term ò l d ds the general equation of equilibrium of the thread will express the sum of the moments of all forces l that we can imagine acting on all elements of the thread. In fact, due to its inextensibility, each element resists the action of external forces, and this resistance is usually considered as an active force, which is called tension. Thus l represents thread tension ”.

Moving on to dynamics, Lagrange, taking bodies as points of mass m, writes that “the values

m d 2 x / dt 2 , m d 2 y / dt 2 , m d 2 z / dt 2(2.9) express the forces applied directly to move the body m parallel to the axes x, y, z" Specified accelerating forces P, Q, R, ..., according to Lagrange, act along the lines p, q, r,..., are proportional to the masses, directed towards the corresponding centers and tend to reduce the distances to these centers. Therefore, variations in action lines will be - d p, - d q, - d r, ..., and the virtual work of the applied forces and forces (2.9) will be respectively equal

å m (d 2 x / dt 2 d x + d 2 y / dt 2 d y + d 2 z / dt 2 d z) , - å (P d p + Q d q + R d r + …) . (2.10)

Equating these expressions and transferring all terms to one side, Lagrange obtains the equation

å m (d 2 x /dt 2 d x + d 2 y / dt 2 d y + d 2 z / dt 2 d z) + å (P d p + Q d q + R d r + …)= 0, (2.11) which he called “the general formula of dynamics for the motion of any system of bodies.” It was this formula that Lagrange used as the basis for all further conclusions - both general theorems of dynamics and theorems of celestial mechanics and dynamics of liquids and gases.

After deriving equation (2.11), Lagrange expands the forces P, Q, R, ... along the axes of rectangular coordinates and reduces this equation to the following form:

å (m d 2 x / dt 2 +X) d x + (m d 2 y / dt 2 + Y) d y + (m d 2 z / dt 2 + Z) d z = 0. (2.12)

Up to signs, equation (2.12) completely coincides with modern form general equation of dynamics:

å j (F j – m j d 2 r j / dt 2) d r j= 0; (2.13) if we expand the scalar product, we obtain equation (2.12) (except for the signs in brackets).

Thus, continuing the works of Euler, Lagrange completed the analytical formulation of the dynamics of a free and non-free system of points and gave numerous examples illustrating the practical power of these methods. Based on the “general formula of dynamics,” Lagrange indicated two main forms of differential equations of motion of a non-free system, which now bear his name: “Lagrange equations of the first kind” and equations in generalized coordinates, or “Lagrange equations of the second kind.” What led Lagrange to equations in generalized coordinates? Lagrange, in his works on mechanics, including celestial mechanics, determined the position of a system, in particular, a rigid body various parameters(linear, angular or a combination thereof). For such a brilliant mathematician as Lagrange was, the problem of generalization naturally arose - to move on to arbitrary, non-specific parameters. This led him to differential equations in generalized coordinates. Lagrange called them “differential equations for solving all problems of mechanics”, now we call them Lagrange equations of the second kind:

d / dt ¶ L / ¶ q j - ¶ L / ¶ q j = 0 ( L=T – P).

The overwhelming majority of problems solved in “Analytical Mechanics” reflect the technical problems of that time. From this point of view, it is necessary to highlight a group of the most important problems in dynamics, united by Lagrange under the general name “On small oscillations of any system of bodies.” This section represents the basis of modern vibration theory. Considering small movements, Lagrange showed that any such movement can be represented as the result of simple harmonic oscillations superimposed on each other.

Mechanics of the 19th and early 20th centuries. Lagrange’s “Analytical Mechanics” summed up the achievements of theoretical mechanics in the 18th century. and identified the following main directions of its development:

1) expansion of the concept of connections and generalization of the basic equations of the dynamics of a non-free system for new types of connections;

2) formulation of the variational principles of dynamics and the principle of conservation of mechanical energy;

3) development of methods for integrating dynamic equations.

In parallel with this, new fundamental problems of mechanics were put forward and solved. For the further development of the principles of mechanics, the works of the outstanding Russian scientist M. V. Ostrogradsky (1801 – 1861) were fundamental. He was the first to consider time-dependent connections, introduced a new concept of non-containing connections, i.e. connections expressed analytically using inequalities, and generalized the principle of possible displacements and the general equation of dynamics to the case of such connections. Ostrogradsky also has priority in considering differential connections that impose restrictions on the speeds of points in the system; Analytically, such connections are expressed using non-integrable differential equalities or inequalities.

A natural addition that expands the scope of application of D’Alembert’s principle was the application of the principle proposed by Ostrogradsky to systems subject to the action of instantaneous and impulse forces that arise when the system is subjected to impacts. Of such kind shock phenomena Ostrogradsky considered it as a result of the instant destruction of connections or the instant introduction of new connections into the system.

In the middle of the 19th century. the principle of conservation of energy was formulated: for any physical system you can define a quantity called energy and equal to the sum of kinetic, potential, electrical and other energies and heat, the value of which remains constant regardless of what changes occur in the system. Significantly accelerated towards early XIX V. the process of creating new machines and the desire for their further improvement gave rise to the emergence of applied, or technical, mechanics in the first quarter of the century. In the first treatises on applied mechanics, the concepts of work of forces were finally formalized.

D'Alembert's principle, which contains the most general formulation of the laws of motion of a non-free system, does not exhaust all the possibilities for posing problems of dynamics. In the middle of the 18th century. arose in the 19th century. new ones have been developed general principles dynamics – variational principles. The first variational principle was the principle of least action, put forward in 1744 without any proof, as some general law of nature, by the French scientist P. Maupertuis (1698 - 1756). The principle of least action states “that the path it (the light) follows is the path for which the number of actions will be the least.”

The development of general methods for integrating differential equations of dynamics dates mainly to the middle of the 19th century. The first step in bringing differential equations of dynamics to a system of first-order equations was made in 1809 by the French mathematician S. Poisson (1781 - 1840). The problem of reducing the equations of mechanics to the “canonical” system of first-order equations for the case of time-independent constraints was solved in 1834 by the English mathematician and physicist W. Hamilton (1805 – 1865). Its final completion belongs to Ostrogradsky, who extended these equations to cases of non-stationary connections.

The largest problems of dynamics, the formulation and solution of which relate mainly to the 19th century, are: the motion of a heavy rigid body, the theory of elasticity (see Appendix) of equilibrium and motion, as well as the closely related problem of oscillations material system. The first solution to the problem of the rotation of a heavy rigid body of arbitrary shape around a fixed center in the particular case when the fixed center coincides with the center of gravity belongs to Euler. Kinematic representations of this movement were given in 1834 by L. Poinsot. The case of rotation, when a stationary center that does not coincide with the center of gravity of the body is placed on the axis of symmetry, was considered by Lagrange. The solution to these two classical problems formed the basis for the creation of a rigorous theory of gyroscopic phenomena (a gyroscope is a device for observing rotation). Outstanding research in this area belongs to the French physicist L. Foucault (1819 - 1968), who created a number of gyroscopic devices. Examples of such devices include a gyroscopic compass, artificial horizon, gyroscope and others. These studies indicated the fundamental possibility, without resorting to astronomical observations, establish the daily rotation of the Earth and determine the latitude and longitude of the observation site. After the work of Euler and Lagrange, despite the efforts of a number of outstanding mathematicians, the problem of the rotation of a heavy rigid body around a fixed point did not receive further development for a long time.

Definition

Mechanics is the part of physics that studies the movement and interaction of material bodies. In this case, mechanical movement is considered as a change over time in the relative position of bodies or their parts in space.

The founders of classical mechanics are G. Galileo (1564-1642) and I. Newton (1643-1727). The methods of classical mechanics are used to study the movement of any material bodies (except microparticles) at speeds that are small compared to the speed of light in a vacuum. The movement of microparticles is considered in quantum mechanics, and the movement of bodies with velocities close to the speed of light is considered in relativistic mechanics ( special theory relativity).
Properties of space and time accepted in classical physics Let us define the above definitions.
One-dimensional space - a parametric characteristic in which the position of a point is described by one parameter.
Euclidean space and time means that they themselves are not curved and are described within the framework of Euclidean geometry.
Homogeneity of space means that its properties do not depend on the distance to the observer. Uniformity of time means that it does not stretch or contract, but flows evenly. Isotropy of space means that its properties do not depend on direction. Since time is one-dimensional, there is no need to talk about its isotropy. Time in classical mechanics is considered as an “arrow of time” directed from the past to the future. It is irreversible: you cannot go back to the past and “correct” something there.
Space and time are continuous (from Latin continuum - continuous, continuous), i.e. they can be crushed into smaller and smaller parts for as long as desired. In other words, there are no “gaps” in space and time within which they would be absent. Mechanics is divided into Kinematics and Dynamics

Kinematics studies the movement of bodies as simple movement in space, introducing into consideration the so-called kinematic characteristics of movement: displacement, speed and acceleration.

In this case, the speed of a material point is considered as the speed of its movement in space or, from a mathematical point of view, as a vector quantity equal to the time derivative of its radius vector:

The acceleration of a material point is considered as the rate of change of its speed or, from a mathematical point of view, as a vector quantity equal to the time derivative of its speed or the second time derivative of its radius vector:

Dynamics

Dynamics studies the motion of bodies in connection with the forces acting on them, using the so-called dynamic characteristics of motion: mass, impulse, force, etc.

In this case, the mass of a body is considered as a measure of its inertia, i.e. resistance to a force acting on a given body that tends to change its state (set it in motion or, conversely, stop it, or change the speed of movement). Mass can also be considered as a measure of the gravitational properties of a body, i.e. its ability to interact with other bodies that also have mass and are located at some distance from this body. The momentum of a body is considered as a quantitative measure of its movement, defined as the product of the mass of the body and its speed:

Force is considered as a measure of mechanical action on a given material body from other bodies.

Plan:

Introduction

• 1 Basic Concepts
• 2 Basic laws
  • 2.1 Galileo's principle of relativity
  • 2.2 Newton's laws
  • 2.3 Law of conservation of momentum
  • 2.4 Law of energy conservation
• 3 History
  • 3.1 Ancient times
  • 3.2 New time
    • 3.2.1 17th century
    • 3.2.2 18th century
    • 3.2.3 19th century
  • 3.3 Modern times
Notes
Literature

Introduction

Classical mechanics- a type of mechanics (a branch of physics that studies the laws of changes in the positions of bodies in space over time and the causes that cause them), based on Newton’s laws and Galileo’s principle of relativity. Therefore, it is often called “ Newtonian mechanics».

Classical mechanics is divided into:

• statics (which considers the equilibrium of bodies)
• kinematics (which studies the geometric property of motion without considering its causes)
• dynamics (which considers the movement of bodies).

There are several equivalent ways to formally describe classical mechanics mathematically:

• Newton's laws
• Lagrangian formalism
• Hamiltonian formalism
• Hamilton-Jacobi formalism

Classical mechanics gives very accurate results within the framework of everyday experience. However, its use is limited to bodies whose speeds are much lower than the speed of light, and whose sizes significantly exceed the sizes of atoms and molecules. A generalization of classical mechanics to bodies moving at an arbitrary speed is relativistic mechanics, and to bodies whose dimensions are comparable to atomic ones is quantum mechanics. Quantum theory fields considers quantum relativistic effects.

However, classical mechanics retains its significance because:

1. it is much easier to understand and use than other theories
2. over a wide range it describes reality quite well.

Classical mechanics can be used to describe the motion of objects such as tops and baseballs, many astronomical objects (such as planets and galaxies), and sometimes even many microscopic objects such as molecules.

Classical mechanics is a self-consistent theory, that is, within its framework there are no statements that contradict each other. However, its combination with other classical theories, for example classical electrodynamics and thermodynamics, leads to the emergence of insoluble contradictions. In particular, classical electrodynamics predicts that the speed of light is constant for all observers, which is incompatible with classical mechanics. At the beginning of the 20th century, this led to the need to create a special theory of relativity. When considered in conjunction with thermodynamics, classical mechanics leads to the Gibbs paradox, in which entropy cannot be accurately determined, and to the ultraviolet catastrophe, in which a black body must radiate an infinite amount of energy. Attempts to solve these problems led to the development of quantum mechanics.

1. Basic concepts

Classical mechanics operates on several basic concepts and models. Among them are:

2. Basic laws

2.1. Galileo's principle of relativity

The main principle on which classical mechanics is based is the principle of relativity, formulated on the basis of empirical observations by G. Galileo. According to this principle, there are infinitely many reference systems in which a free body is at rest or moving with a speed constant in magnitude and direction. These reference systems are called inertial and move relative to each other uniformly and rectilinearly. In all inertial systems reference, the properties of space and time are the same, and all processes in mechanical systems obey the same laws. This principle can also be formulated as the absence of absolute reference systems, that is, reference systems that are in any way distinguished relative to others.

2.2. Newton's laws

The basis of classical mechanics is Newton's three laws.

The first law establishes the presence of the property of inertia in material bodies and postulates the presence of such reference systems in which the movement free body occurs at a constant speed (such reference systems are called inertial).

Newton's second law introduces the concept of force as a measure of the interaction of a body and, based on empirical facts, postulates a relationship between the magnitude of the force, the acceleration of the body and its inertia (characterized by mass). In mathematical formulation, Newton's second law is most often written as follows:

where is the resulting vector of forces acting on the body; - body acceleration vector; m- body mass.

Newton's second law can also be written in terms of the change in momentum of a body:

In this form, the law is valid for bodies with variable mass, as well as in relativistic mechanics.

Newton's second law is not enough to describe the motion of a particle. Additionally, a description of force is required, obtained from consideration of the essence of the physical interaction in which the body participates.

Newton's third law clarifies some properties of the concept of force introduced in the second law. He postulates the presence for each force acting on the first body from the second, equal in magnitude and opposite in direction to the force acting on the second body from the first. The presence of Newton's third law ensures the fulfillment of the law of conservation of momentum for a system of bodies.

2.3. Law of conservation of momentum

The law of conservation of momentum is a consequence of Newton's laws for closed systems, that is, systems that are not acted upon external forces. From a more fundamental point of view, the law of conservation of momentum is a consequence of the homogeneity of space.

2.4. Law of energy conservation

The law of conservation of energy is a consequence of Newton's laws for closed conservative systems, that is, systems in which only conservative forces act. From a more fundamental point of view, the law of conservation of energy is a consequence of the homogeneity of time.

3. History

3.1. Ancient time

Classical mechanics originated in ancient times mainly in connection with problems that arose during construction. The first branch of mechanics to develop was statics, the foundations of which were laid in the works of Archimedes in the 3rd century BC. e. He formulated the lever rule, the theorem on the addition of parallel forces, introduced the concept of the center of gravity, and laid the foundations of hydrostatics (Archimedes' force).

3.2. New time

3.2.1. 17th century

Dynamics as a branch of classical mechanics began to develop only in the 17th century. Its foundations were laid by Galileo Galilei, who was the first to correctly solve the problem of the motion of a body under the influence of a given force. Based on empirical observations, he discovered the law of inertia and the principle of relativity. In addition, Galileo contributed to the emergence of the theory of vibrations and the science of strength of materials.

Christiaan Huygens conducted research in the field of the theory of oscillations, in particular, he studied the movement of a point along a circle, as well as the oscillations of a physical pendulum. In his works, the laws of elastic impact of bodies were also formulated for the first time.

The laying of the foundations of classical mechanics ended with the work of Isaac Newton, who formulated the laws of mechanics in the most general form and discovered the law of universal gravitation. In 1684, he established the law of viscous friction in liquids and gases.

Also in the 17th century, in 1660, the law of elastic deformation was formulated, bearing the name of its discoverer Robert Hooke.

3.2.2. XVIII century

In the 18th century, analytical mechanics was born and intensively developed. Its methods for the problem of the motion of a material point were developed by Leonhard Euler, who laid the foundations of rigid body dynamics. These methods are based on the principle of virtual movements and on the D'Alembert principle. The development of analytical methods was completed by Lagrange, who managed to formulate the equations of the dynamics of a mechanical system in the most general form: using generalized coordinates and momenta. In addition, Lagrange took part in laying the foundations of the modern theory of oscillations.

An alternative method for the analytical formulation of classical mechanics is based on the principle of least action, which was first proposed by Maupertuis in relation to a single material point and generalized to the case of a system of material points by Lagrange.

Also in the 18th century, in the works of Euler, Daniel Bernoulli, Lagrange and D'Alembert, the foundations of a theoretical description of the hydrodynamics of an ideal fluid were developed.

3.2.3. 19th century

In the 19th century, the development of analytical mechanics took place in the works of Ostrogradsky, Hamilton, Jacobi, Hertz, and others. In the theory of oscillations, Routh, Zhukovsky and Lyapunov developed a theory of stability of mechanical systems. Coriolis developed the theory of relative motion, proving the theorem on the decomposition of acceleration into components. In the second half of the 19th century, kinematics was separated into a separate section of mechanics.

Advances in the field of continuum mechanics were especially significant in the 19th century. Navier and Cauchy formulated the equations of the theory of elasticity in a general form. In the works of Navier and Stokes, differential equations of hydrodynamics were obtained taking into account the viscosity of the liquid. Along with this, knowledge in the field of hydrodynamics of an ideal fluid is deepening: works by Helmholtz on vortices, Kirchhoff, Zhukovsky and Reynolds on turbulence, and Prandtl on boundary effects appear. Saint-Venant developed mathematical model, describing the plastic properties of metals.

3.3. Modern times

In the 20th century, the interest of researchers switched to nonlinear effects in the field of classical mechanics. Lyapunov and Henri Poincaré laid the foundations of the theory of nonlinear oscillations. Meshchersky and Tsiolkovsky analyzed the dynamics of bodies of variable mass. Aerodynamics stands out from continuum mechanics, the foundations of which were developed by Zhukovsky. In the middle of the 20th century, a new direction in classical mechanics was actively developing - chaos theory. The issues of stability of complex dynamic systems also remain important.

Notes

1. 1 2 3 4 Landau, Lifshits, p. 9
2. 1 2 Landau, Lifshits, p. 26-28
3. 1 2 Landau, Lifshits, p. 24-26
4. Landau, Lifshits, p. 14-16

Literature

• B. M. Yavorsky, A. A. Detlaf Physics for high school students and those entering universities. - M.: Academy, 2008. - 720 p. - ( Higher education). - 34,000 copies. - ISBN 5-7695-1040-4
• Sivukhin D.V. General course physics. - 5th edition, stereotypical. - M.: Fizmatlit, 2006. - T. I. Mechanics. - 560 s. - ISBN 5-9221-0715-1
• A. N. Matveev Mechanics and theory of relativity - www.alleng.ru/d/phys/phys108.htm. - 3rd ed. - M.: ONIX 21st century: Peace and Education, 2003. - 432 p. - 5000 copies. - ISBN 5-329-00742-9
• C. Kittel, W. Knight, M. Ruderman Mechanics. Berkeley course of physics.. - M.: Lan, 2005. - 480 p. - (Textbooks for universities). - 2000 copies. - ISBN 5-8114-0644-4
• Landau, L. D., Lifshits, E. M. Mechanics. - 5th edition, stereotypical. - M.: Fizmatlit, 2004. - 224 p. - (“Theoretical Physics”, Volume I). - ISBN 5-9221-0055-6
• G. Goldstein Classical mechanics. - 1975. - 413 p.
• S. M. Targ. Mechanics - www.femto.com.ua/articles/part_1/2257.html- article from the Physical Encyclopedia

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