History of the development of mechanics - abstract. Classical mechanics Message on mechanics

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Mechanics is the science of moving bodies and the interactions between them during movement. In this case, attention is paid to those interactions as a result of which the movement changed or deformation of bodies occurred. In this article we will tell you about what mechanics is.

Mechanics can be quantum, applied (technical) and theoretical.

What is quantum mechanics? This is a branch of physics that describes physical phenomena and processes whose actions are comparable to the value of Planck’s constant.
What is technical mechanics? This is a science that reveals the operating principle and structure of mechanisms.
What is theoretical mechanics? This is the science and movement of bodies and the general laws of motion.

Mechanics studies the movement of all kinds of machines and mechanisms, aircraft and celestial bodies, oceanic and atmospheric currents, plasma behavior, deformation of bodies, movement of gases and liquids in natural conditions and technical systems, polarizing or magnetizing medium in electrical and magnetic fields, stability and strength of technical and construction structures, movement of air and blood through the vessels through the respiratory tract.

Newton's law is fundamental; it is used to describe the motion of bodies with velocities that are small compared to the speed of light.

In mechanics there are the following sections:

kinematics (about the geometric properties of moving bodies without taking into account their mass and acting forces);
statics (about finding bodies in equilibrium using external influences);
dynamics (about moving bodies under the influence of force).

In mechanics, there are concepts that reflect the properties of bodies:

material point (a body whose dimensions can be ignored);
absolutely rigid body (a body in which the distance between any points is constant);
continuum (a body whose molecular structure is neglected).

If the rotation of the body relative to the center of mass under the conditions of the problem under consideration can be neglected or it moves translationally, the body is equated to a material point. If we do not take into account the deformation of the body, then it should be considered absolutely indeformable. Gases, liquids and deformable bodies can be considered as solid media in which particles continuously fill the entire volume of the medium. In this case, when studying the movement of a medium, the apparatus of higher mathematics is used, which is used for continuous functions. From the fundamental laws of nature - the laws of conservation of momentum, energy and mass - follow equations that describe the behavior of a continuous medium. In mechanics continuum contains a number of independent sections - aero- and hydrodynamics, theory of elasticity and plasticity, gas dynamics and magnetic hydrodynamics, dynamics of the atmosphere and water surface, physical and chemical mechanics of materials, mechanics of composites, biomechanics, space hydro-aeromechanics.

Now you know what mechanics are!

Abstract on the topic:

HISTORY OF MECHANICS DEVELOPMENT

Completed by: student of class 10 “A”

Efremov A.V.

Checked by: Gavrilova O.P.

1. INTRODUCTION.

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

MECHANICAL DIVISIONS.

4. HISTORY OF MECHANICS DEVELOPMENT:

The era that preceded 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.

6. CONCLUSION.

7. APPENDIX.

1. INTRODUCTION.

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 caused by the external world in the sense organ is transmitted to the internal world and, for its part, causes in it a subjective sensation, the appearance of which requires the presence of consciousness. The subjective sensation perceived by the inner world 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, which we compare with each other and to which we attach objective significance. reality that exists 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 mhcanich - skill related to machines; the science of machines) is the science of the simplest form of the movement of matter - mechanical movement, which represents a change over time in the spatial arrangement of 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, changes in the relative position of elementary volumes are allowed. 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 is divided into:

mechanics of a material point;

mechanics of a system of material points;

mechanics of an absolutely rigid body;

continuum mechanics.

The latter, in turn, is subdivided into the theory of elasticity, hydromechanics, 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 questions of the movement of machines and mechanisms, mechanics Vehicle 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 technical sciences emerged from mechanics. 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), which have common foundations with the theory of elasticity and hydrodynamics, 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 has developed in close connection with its other branches - with 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. The combination of the magnitude (intensity) of force, expressed in certain units, the direction of force (line of action) and the point of application quite uniquely determine force as a vector.

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: a given force F causes the less acceleration a of the 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 and the acceleration a:

This vector equality represents the basic equation of the dynamics of a material point.

Newton's third law states: an action always corresponds to 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 - the elementary impulse of force F dt (the product of force and the element of time of its action), and scalar - the elementary work F*dr (the scalar product of the force vectors and the 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 phenomenon, due to its complexity, cannot be 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.

For strain measurement, great development has been made in 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.

The history of mechanics, as well as other 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.

He applied his 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 poluspaston, from polu - a lot and spaw - I pull) - a system of movable and fixed blocks, encircled by 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.

Thus, he believed that in order to maintain uniform and linear motion of a body, a constant force must be applied to it. 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.

Having doubted 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 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 warfare)

poses the mechanics of the 16th – 17th centuries. a number of important problems occupying 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.

Galileo established the laws of motion of heavy bodies according to inclined plane, showing that, whether heavy bodies fall vertically or along an inclined plane, they always acquire such speeds as 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 are talking 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.

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 the most important problems of dynamics - the movement of a body in a circle, oscillations physical pendulum, 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 big number problems related mainly to celestial mechanics. In it, he was most interested in the reasons that made him move along elliptical orbits. While still a student, Newton began to think about the issues of gravitation. 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 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 two planets (j = 1, 2) When the planets move uniformly in circular orbits with speeds V j their circulation periods are determined by the equalities T j = 2 p R j / V j

Therefore, 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 establishes the inverse proportionality of the centrifugal forces of the planets to the squares of their distances before 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 view 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 the stability of the solar system, solving the three-body problem , the movement 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 of the kinetic theory of 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 at the end of the 18th century. 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.

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 “the general formula of statics for the equilibrium of any system of forces.” Starting with two forces, Lagrange establishes by induction the following general formula for the equilibrium of any system of forces:

P dp + 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 is, of course, not in the form of notation, but in the definition of variation: in our days, this is an arbitrarily conceivable movement of the point of application of force, compatible with connections, but for Lagrange, this is a small movement along the line of action of the force and in the direction of its action. Lagrange introduces into consideration the function P (now called potential energy), defining it by equality.

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

P = A + Bx + Sy + 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. Let us consider the equilibrium of n material points, each of which is acted upon by a force F j . There are m connections j r = 0 between the coordinates of the points, 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 are undetermined 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 you need to add m constraint equations j r = 0 (X j , Y j , Z j – projections of force F j)

The coupling equation for an inextensible thread has the form ds = const, and, therefore, d ds = 0. In equation (2.5), the sums go 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, we find

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 for s = 0 and s = l, and, therefore, the first term becomes zero. We enter the remaining part into equation (2.7), expand the scalar product F * dr and group the terms:

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 the variations d x, d y and d z are arbitrary and independent, 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 can 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 of the general equilibrium equation of the thread will express the sum of the moments of all forces l , which we can imagine acting on all elements of the thread. In fact, due to its inextensibility, each element resists the action external forces, and this resistance is usually considered as an active force, which is called tension. Thus, l represents the tension of the thread"

Turning to dynamics, Lagrange, taking bodies as points of mass m, writes that “the quantities 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 body m parallel to the x, y, z axes.”

The given accelerating forces P, Q, R, ..., according to Lagrange, act along the lines p, q, r, ..., are proportional to the masses, are directed to the corresponding centers and tend to reduce the distances to these centers. Therefore, variations of action lines will be - d p, - d q, - d r, ..., and virtual work 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 the modern form of the 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, by 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 technical problems 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. Ostrogradsky considered this kind of impact phenomena as the 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 it is possible to determine 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.

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.”

Development common methods integration of 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 vibrations of a 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, to 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.

The foundations of the theory of motion of a solid body in an ideal fluid were given by the German physicist G. Kirchhoff in 1869. With the advent of the medium in the middle of the 19th century. rifled guns, which was intended to give the projectile the rotation necessary for stability in flight, the problem of external ballistics turned out to be closely related to the dynamics of a heavy solid body. This formulation of the problem and its solution belong to the outstanding Russian scientist - artilleryman N.V. Mayevsky (1823 - 1892).

One of the most important problems in mechanics is the problem of the stability of equilibrium and motion of material systems. The first general theorem on the stability of the equilibrium of a system under the influence of generalized forces belongs to Lagrange and is stated in “Analytical Mechanics”. According to this theorem, a sufficient condition for equilibrium is the presence of a minimum potential energy in the equilibrium position. The method of small oscillations, used by Lagrange to prove the theorem on the stability of equilibrium, turned out to be fruitful for studying the stability of steady motions. In “Treatise on the stability of a given state of motion.”

The English scientist E. Rous, published in 1877, reduced the study of stability by the method of small vibrations to the consideration of the distribution of roots of some “characteristic” equation and indicated the necessary and sufficient conditions under which these roots have negative real parts.

From a different point of view than that of Routh, the problem of motion stability was considered in the work of N. E. Zhukovsky (1847 - 1921) “On the strength of motion” (1882), in which orbital stability was studied. The criteria for this stability, established by Zhukovsky, are formulated in a clear geometric form, so characteristic of all the scientific work of the great mechanic.

A rigorous formulation of the problem of motion stability and an indication of the most general methods for solving it, as well as a specific consideration of individual most important problems of the theory of stability belong to A. M. Lyapunov, and were set out by him in the fundamental work “The General Problem of Motion Stability” (1892). They were given a definition of a stable equilibrium position, which looks like this: if for a given r (the radius of the sphere), one can choose one that is arbitrarily small, but not equal to zero value h (initial energy), that at all subsequent times the particle will not go beyond the sphere of radius r, then the equilibrium position at this point is called stable. Lyapunov connected the solution of the stability problem with the consideration of certain functions, from a comparison of the signs of which with the signs of their time derivatives one can conclude about the stability or instability of the state of motion under consideration (“the second Lyapunov method”). Using this method, Lyapunov, in his theorems on stability to the first approximation, indicated the limits of applicability of the method of small oscillations of a material system around the position of its stable equilibrium (first set forth in Lagrange’s “Analytical Mechanics”).

Subsequent development of the theory of small oscillations in the 19th century. was associated mainly with taking into account the influence of resistances leading to attenuation of oscillations, and external disturbing forces creating forced oscillations. The theory of forced oscillations and the doctrine of resonance appeared in response to requests from machine technology and, first of all, in connection with the construction of railway bridges and the creation of high-speed steam locomotives. Another important branch of technology, the development of which required the application of methods from the theory of oscillations, was control engineering. The founder of modern dynamics of the regulation process is the Russian scientist and engineer I. A. Vyshnegradsky (1831 – 1895). In 1877, in his work “On Direct Action Regulators,” Vyshnegradsky first formulated the well-known inequality that a stably operating machine equipped with a regulator must satisfy.

The further development of the theory of small oscillations was closely connected with the emergence of individual major technical problems. The most important works on the theory of ship rolling during waves belong to the outstanding Soviet scientist

A.N. Krylov, whose entire activity was devoted to the application of modern achievements of mathematics and mechanics to the solution of the most important technical problems. In the 20th century problems of electrical engineering, radio engineering, the theory of automatic control of machines and production processes, technical acoustics and others brought to life new area science – the theory of nonlinear oscillations. The foundations of this science were laid in the works of A. M. Lyapunov and the French mathematician A. Poincaré, and further development, as a result of which a new, rapidly growing discipline was formed, is due to the achievements of Soviet scientists. TO end of the 19th century V. A special group of mechanical problems emerged - the movement of bodies of variable mass. The fundamental role in the creation of a new field of theoretical mechanics - the dynamics of variable mass - belongs to the Russian scientist I. V. Meshchersky (1859 - 1935). In 1897, he published his fundamental work “Dynamics of a Variable Mass Point.”

In the 19th and early 19th centuries. the foundations were laid for two important branches of hydrodynamics: viscous fluid dynamics and gas dynamics. The hydrodynamic theory of friction was created by the Russian scientist N.P. Petrov (1836 – 1920). The first rigorous solution to problems in this area was indicated by N. E. Zhukovsky.

By the end of the 19th century. mechanics has reached a high level of development. XX century brought a deep critical revision of a number of basic principles of classical mechanics and was marked by the emergence of the mechanics of fast movements occurring at speeds close to the speed of light. The mechanics of rapid motions, as well as the mechanics of microparticles, were further generalizations of classical mechanics.

Newtonian mechanics retained a vast field of activity in the basic questions of mechanical engineering in Russia and the USSR. Mechanics in pre-revolutionary Russia, thanks to the fruitful scientific work of M. V. Ostrogradsky, N. E. Zhukovsky, S. A. Chaplygin, A. M. Lyapunov, A. N. Krylov and others, achieved great success and was able not only cope with the tasks put before her domestic technology, but also contribute to the development of technology throughout the world. The works of the “father of Russian aviation” N. E. Zhukovsky laid the foundations of aerodynamics and aviation science in general. The works of N. E. Zhukovsky and S. A. Chaplygin were of primary importance in the development of modern hydroaeromechanics. S. A. Chaplygin carried out fundamental research in the field of gas dynamics, which indicated for many decades ahead the ways of developing high-speed aerodynamics. A. N. Krylov’s works on the theory of stability of ship rolling in waves, studies on the buoyancy of their hulls, and the theory of compass deviation placed him among the founders of the modern science of shipbuilding.

One of the important factors that contributed to the development of mechanics in Russia was the high level of its teaching in higher education. In this regard, much has been done by M. V. Ostrogradsky and his followers. The issues of motion stability are of greatest technical importance in problems of the theory of automatic control. An outstanding role in the development of the theory and technology of regulating machines and production processes belongs to I. N. Voznesensky (1887 - 1946). Problems of rigid body dynamics developed mainly in connection with the theory of gyroscopic phenomena.

Soviet scientists achieved significant results in the field of elasticity theory. They conducted research on the theory of plate bending and general solutions to problems in the theory of elasticity, on the plane problem of the theory of elasticity, on variational methods of the theory of elasticity, on structural mechanics, on the theory of plasticity, on the theory of ideal fluid, on the dynamics of compressible fluids and gas dynamics, on the theory filtration of movements, which contributed to the rapid development of Soviet hydroaerodynamics, dynamic problems in the theory of elasticity were developed. The results of paramount importance obtained by scientists of the Soviet Union on the theory of nonlinear oscillations established the USSR's leading role in this field. The formulation, theoretical consideration and organization of the experimental study of nonlinear oscillations constitute an important merit of L. I. Mandelstam (1879 - 1944) and N. D. Papaleksi (1880 - 1947) and their school (A. A. Andronov and others).

The foundations of the mathematical apparatus of the theory of nonlinear oscillations are contained in the works of A. M. Lyapunov and A. Poincaré. Poincaré’s “limit cycles” were posed by A. A. Andronov (1901 – 1952) in connection with the problem of undamped oscillations, which he called self-oscillations. Along with methods based on the qualitative theory of differential equations, the analytical direction of the theory of differential equations developed.

5. PROBLEMS OF MODERN MECHANICS.

The main problems of modern mechanics of systems with a finite number of degrees of freedom include, first of all, problems of the theory of oscillations, rigid body dynamics and the theory of motion stability. In the linear theory of oscillations, it is important to create effective methods for studying systems with periodically changing parameters, in particular, the phenomenon of parametric resonance.

To study the motion of nonlinear oscillatory systems, both analytical methods and methods based on the qualitative theory of differential equations are being developed. Problems of vibrations are closely intertwined with issues of radio engineering, automatic regulation and motion control, as well as with the tasks of measuring, preventing and eliminating vibrations in transport devices, machines and building structures. In the field of rigid body dynamics, most attention is paid to problems in the theory of oscillations and the theory of motion stability. These problems are posed by flight dynamics, ship dynamics, and the theory of gyroscopic systems and instruments used mainly in air navigation and ship navigation. In the theory of motion stability, the study of Lyapunov’s “special cases”, the stability of periodic and unsteady motions, comes first, and the main research tool is the so-called “second Lyapunov method”.

In the theory of elasticity, along with problems for a body subject to Hooke's law, the greatest attention is drawn to the issues of plasticity and creep in parts of machines and structures, calculation of the stability and strength of thin-walled structures. The direction that aims to establish the basic laws of the relationship between stresses and strains and strain rates for models of real bodies (rheological models) is also gaining great importance. The mechanics of granular media is developed in close connection with the theory of plasticity. Dynamic problems of the theory of elasticity are associated with seismology, the propagation of elastic and plastic waves along rods and dynamic phenomena arising during impact. The most important problems of hydroaerodynamics are associated with problems of high speeds in aviation, ballistics, turbine construction and engine building.

This includes, first of all, the theoretical determination of the aerodynamic characteristics of bodies at sub-, near- and supersonic speeds, both in steady and unsteady motion.

Problems of high-speed aerodynamics are closely intertwined with issues of heat transfer, combustion and explosions. The study of the movements of compressible gas at high speeds involves the main problem of gas dynamics, and at low speeds it is associated with problems of dynamic meteorology. Of primary importance for hydroaerodynamics is the problem of turbulence, which has not yet received a theoretical solution. In practice, numerous empirical and semi-empirical formulas continue to be used.

The hydrodynamics of heavy fluids faces problems of the spatial theory of waves and wave resistance of bodies, wave formation in rivers and canals, and a number of problems related to hydraulic engineering.

Of great importance for the latter, as well as for issues of oil production, are the problems of filtration movement of liquids and gases in porous media.

6. CONCLUSION.

Galileo-Newtonian mechanics has come a long way in development and did not immediately win the right to be called classical. Its successes, especially in the 17th-18th centuries, established experiment as the main method for testing theoretical constructs. Almost until the end of the 18th century, mechanics occupied a leading position in science, and its methods had a great influence on the development of all natural sciences.

Subsequently, Galileo–Newtonian mechanics continued to develop intensively, but its leading position gradually began to be lost. Electrodynamics, the theory of relativity, quantum physics, nuclear energy, genetics, electronics, and computer technology began to move to the forefront of science. Mechanics has given way to its place as a leader in science, but has not lost its importance. As before, all dynamic calculations of any mechanisms operating on land, under water, in the air and in space are based to one degree or another on the laws of classical mechanics. Based on far from obvious consequences of its basic laws, devices have been built that autonomously, without human intervention, determine the location of submarines, surface ships, and aircraft; systems have been built that autonomously orient spacecraft and direct them to the planets of the solar system, Halley's comet. Analytical mechanics, a component of classical mechanics, retains “incomprehensible efficiency” in modern physics. Therefore, no matter how physics and technology develop, classical mechanics will always take its rightful place in science.

7. APPENDIX.

Fluid mechanics is a branch of physics that deals with the study of the laws of motion and equilibrium of a fluid and its interaction with washed solids.

Aeromechanics is the science of the equilibrium and movement of gaseous media and solids in a gaseous medium, primarily in air.

Gas mechanics is a science that studies the movement of gases and liquids under conditions where the compressibility property is essential.

Aerostatics is a part of mechanics that studies the conditions of equilibrium of gases (especially air).

Kinematics is a branch of mechanics that studies the movements of bodies without taking into account the interactions that determine these movements. Basic concepts: instantaneous speed, instantaneous acceleration.

Ballistics is the science of projectile movement. External ballistics studies the movement of a projectile in the air. Internal ballistics studies the movement of a projectile under the action of powder gases, the mechanical freedom of which is limited by any force.

Hydraulics is the science of the conditions and laws of equilibrium and movement of fluids and the methods of applying these laws to the solution of practical problems. Can be defined as applied fluid mechanics.

An inertial coordinate system is a coordinate system in which the law of inertia is satisfied, i.e. in which the body, when compensating for external influences exerted on it, moves uniformly and rectilinearly.

Pressure is a physical quantity equal to the ratio of the normal component of the force with which a body acts on the surface of the support in contact with it to the area of contact, or otherwise - the normal surface force acting per unit area.

Viscosity (or internal friction) is the property of liquids and gases to provide resistance when one part of the liquid moves relative to another.

Creep is a process of small continuous plastic deformation occurring in metals under conditions of prolonged static loading.

Relaxation is the process of establishing static equilibrium in a physical or physicochemical system. During the relaxation process, macroscopic quantities characterizing the state of the system asymptotically approach their equilibrium values.

Mechanical connections are restrictions imposed on the movement or position of a system of material points in space and carried out using surfaces, threads, rods and others.

Mathematical relationships between coordinates or their derivatives, characterizing the implemented mechanical connections of movement restrictions, are called constraint equations. For the movement of the system to be possible, the number of constraint equations must be less than the number of coordinates that determine the position of the system.

The optical method for studying stress is a method for studying stress in polarized light, based on the fact that particles of an amorphous material become optically anisotropic when deformed. In this case, the main axes of the refractive index ellipsoid coincide with the main directions of deformation, and the main light vibrations, passing through a deformed plate of polarized light, receive a path difference.

Strain gauge - a device for measuring tensile or compressive forces applied to any system by the deformations caused by these forces

Celestial mechanics is a branch of astronomy devoted to the study of the movement of cosmic bodies. Now the term is used differently and the subject of celestial mechanics is usually considered only general methods of studying the motion and force field of solar system bodies.

The theory of elasticity is a branch of mechanics that studies displacements, elastic deformations and stresses that arise in a solid body under the influence of external forces, from heating and other influences. Its goal is to determine quantitative relationships characterizing the deformation or internal relative movements of particles of a solid body under the influence of external influences in a state of equilibrium or small internal relative motion.

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Story development all-wheel drive(4WD) in cars... . We wish you an interesting pastime. Story all-wheel drive Story all-wheel drive: Civic Shuttle... which for a person not familiar with mechanics and reading technical drawings, the image shown...

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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 a subjective sensation in it, the appearance of which requires the presence of consciousness.

The subjective sensation perceived by the inner world 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.

The perception in space and time of sensory impressions, which we compare with each other and to which we attach the meaning of an objective reality that exists apart from 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 acquaintance with a very large number of 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, which represents a change over time in the spatial arrangement of 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:

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;
mechanics of an absolutely rigid body;
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, the formulation of its general provisions and theorems, as well as the application of the methods of mechanics to the study of the motion of a material point, a system of a finite number of material points and an absolutely rigid body.

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 technical sciences emerged from mechanics. 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), which have common foundations with the theory of elasticity and hydrodynamics, 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 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 motion until 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 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 use mathematical methods, and many of them originated and 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 deformation, strain measuring using mechanical and optical strain gauges (see Appendix), as well as resistance strain gauges, has gained great development in recent years. 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

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 represented already fairly developed scientific systems, in the field of mechanics only certain 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.

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, a constant force must be applied to it. 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.

The period of creation of the fundamentals of mechanics. Practice (mainly merchant shipping and military affairs) confronts the mechanics of the 16th - 17th centuries. a number of important problems occupying 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 are talking about movement in the absence of resistance. In dialogues about two systems of the world, very figuratively, in the form 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 in a straight line and uniformly. 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. While still a student, Newton began to think about the issues of gravitation. 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.”

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)

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 establishes the inverse proportionality of the centrifugal forces of the 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.

Development of mechanical methods in the 18th century. In the 18th century. production needs - 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 general techniques solving problems of the mechanics of a material point, a system of points of a rigid body, developed in “Analytical Mechanics” (1788) by J. Lagrange (1736 – 1813).

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,

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.

The history of the creation of the dynamics of a non-free system is associated with the development of the principle of possible movements, expressing General terms 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 at the end of the 18th century. 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.

After Lagrange has laid down the various principles of statics, he proceeds to establish “the general formula of statics for the equilibrium 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 the 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 Fj. 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 Fj 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 Fj).

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 the physical meaning of the factor l: “Since the quantity 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 the modern form of the 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).

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.

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 the middle of the 19th century. the principle of conservation of energy was formulated: for any physical system it is possible to determine 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 by the beginning of the 19th century. 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 general principles of dynamics—variational principles—were developed. 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 of a 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, to 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.

As part of any educational course, the study of physics begins with mechanics. Not from theoretical, not from applied or computational, but from good old classical mechanics. This mechanics is also called Newtonian mechanics. According to legend, a scientist was walking in the garden and saw an apple falling, and it was this phenomenon that prompted him to discover the law of universal gravitation. Of course, the law has always existed, and Newton only gave it a form understandable to people, but his merit is priceless. In this article we will not describe the laws of Newtonian mechanics in as much detail as possible, but we will outline the fundamentals, basic knowledge, definitions and formulas that can always play into your hands.

Mechanics is a branch of physics, a science that studies the movement of material bodies and the interactions between them.

The word itself is of Greek origin and is translated as “the art of building machines.” But before we build machines, we are still like the Moon, so let’s follow in the footsteps of our ancestors and study the movement of stones thrown at an angle to the horizon, and apples falling on our heads from a height h.

Why does the study of physics begin with mechanics? Because this is completely natural, shouldn’t we start with thermodynamic equilibrium?!

Mechanics is one of the oldest sciences, and historically the study of physics began precisely with the foundations of mechanics. Placed within the framework of time and space, people, in fact, could not start with something else, no matter how much they wanted. Moving bodies are the first thing we pay attention to.

What is movement?

Mechanical motion is a change in the position of bodies in space relative to each other over time.

It is after this definition that we quite naturally come to the concept of a frame of reference. Changing the position of bodies in space relative to each other. Key words here: relative to each other . After all, a passenger in a car moves relative to a person standing on the side of the road with certain speed, and is at rest relative to its neighbor on the seat next to them, and moves at some other speed relative to the passenger in the car that is overtaking them.

That is why, in order to normally measure the parameters of moving objects and not get confused, we need reference system - rigidly interconnected reference body, coordinate system and clock. For example, the earth moves around the sun in a heliocentric frame of reference. In everyday life, we carry out almost all our measurements in a geocentric reference system associated with the Earth. The earth is a body of reference relative to which cars, planes, people, and animals move.

Mechanics, as a science, has its own task. The task of mechanics is to know the position of a body in space at any time. In other words, mechanics builds a mathematical description of motion and finds connections between the physical quantities that characterize it.

In order to move further, we need the concept “ material point " They say that physics is an exact science, but physicists know how many approximations and assumptions have to be made in order to agree on this very accuracy. No one has ever seen a material point or smelled an ideal gas, but they exist! They are simply much easier to live with.

A material point is a body whose size and shape can be neglected in the context of this problem.

Sections of classical mechanics

Mechanics consists of several sections

Kinematics
Dynamics
Statics

Kinematics from a physical point of view, it studies exactly how a body moves. In other words, this section deals with the quantitative characteristics of movement. Find speed, path - typical kinematics problems

Dynamics solves the question of why it moves the way it does. That is, it considers the forces acting on the body.

Statics studies the balance of bodies under the influence of forces, that is, answers the question: why doesn’t it fall at all?

Limits of applicability of classical mechanics

Classical mechanics no longer claims to be a science that explains everything (at the beginning of the last century everything was completely different), and has a clear framework of applicability. In general, the laws of classical mechanics are valid in the world we are accustomed to in size (macroworld). They stop working in the case of the particle world, when quantum mechanics replaces classical mechanics. Also, classical mechanics is not applicable to cases when the movement of bodies occurs at a speed close to the speed of light. In such cases, relativistic effects become pronounced. Roughly speaking, within the framework of quantum and relativistic mechanics - classical mechanics, this is a special case when the dimensions of the body are large and the speed is small.

Generally speaking, quantum and relativistic effects never go away; they also occur during the ordinary motion of macroscopic bodies at a speed much lower than the speed of light. Another thing is that the effect of these effects is so small that it does not go beyond the most accurate measurements. Classical mechanics will thus never lose its fundamental importance.

We will continue to study the physical foundations of mechanics in future articles. For a better understanding of the mechanics, you can always refer to to our authors, which will individually shed light on the dark spot of the most difficult task.