Mechanics is the science of moving bodies and the interactions between them during motion. At the same time, attention is paid to those interactions as a result of which the movement has changed or the deformation of bodies has 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, the actions of which are comparable to the magnitude of Planck's constant.
- What is technical mechanics? This is a science that reveals the principle of operation and the structure of mechanisms.
- What is theoretical mechanics? It is science and motion of bodies and 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, the movement of gases and liquids in natural conditions and technical systems, a polarizing or magnetizing medium in electric and magnetic fields, the stability and strength of technical and building structures, movement along the respiratory tract of air and blood through the vessels.
Newton's law lies at the foundation, with the help of it they describe the motion of bodies with small speeds in comparison with the speed of light.
There are the following sections in mechanics:
- 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 (body, the size of which can be ignored);
- absolutely rigid body (a body in which the distance between any points is unchanged);
- continuous medium (a body whose molecular structure is neglected).
If the rotation of the body with respect to the center of mass under the conditions of the problem under consideration can be neglected, or if it moves progressively, the body is equated to a material point. If you do not take into account the deformation of the body, then it must be considered absolutely non-deformable. 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 the 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, there follow equations that describe the behavior of a continuous medium. Continuum mechanics contains a number of independent sections - aerodynamics and hydrodynamics, theory of elasticity and plasticity, gas dynamics and magnetohydrodynamics, dynamics of the atmosphere and water surface, physicochemical mechanics of materials, mechanics of composites, biomechanics, space hydro-aeromechanics.
Now you know what mechanics are!
Definition
The founders of classical mechanics are G. Galilei (1564-1642) and I. Newton (1643-1727). The methods of classical mechanics are used to study the motion of any material bodies (except for microparticles) with speeds that are small compared to the speed of light in a vacuum. The movement of microparticles is considered in quantum mechanics, and the movement of bodies with speeds close to the speed of light - in relativistic mechanics (special theory of relativity).Mechanics is the part of physics that studies the movement and interaction of material bodies. In this case, mechanical motion is considered as a change over time in the relative position of bodies or their parts in space.
Properties of space and time, adopted in classical physics Let us give definitions to the above definitions.
One-dimensional space - a parametric characteristic in which the position of a point is described by one parameter.
Euclidean space and time means that they are not curved by themselves and are described in the framework of Euclidean geometry.
Uniformity of space means that its properties do not depend on the distance to the observer. The uniformity of time means that it does not stretch or contract, but flows uniformly. Isotropy of space means that its properties are independent of direction. Since time is one-dimensional, there is no need to talk about its isotropy. Time in classical mechanics is considered as the "arrow of time" directed from the past to the future. It is irreversible: you cannot go back to the past and "correct" something there.
Space and time are continuous (from Latin continuum - continuous, continuous), i.e. they can be split into smaller and smaller pieces for an arbitrarily long time. In other words, there are no “holes” in space and time, inside which they would be absent. Mechanics are divided into Kinematics and Dynamics
In this case, the speed of a material point is considered as the speed of its movement in space or, from a mathematical point of view, as a vector quantity equal to the time derivative of its radius of the vector:Kinematics studies the movement of bodies as a simple movement in space, introducing into consideration the so-called kinematic characteristics of movement: displacement, speed and acceleration.
The acceleration of a material point is considered as the rate of change of its speed or, from a mathematical point of view, as a vector quantity equal to the time derivative of its speed or the second time derivative of its radius vector:
Dynamics
In this case, body weight is considered as a measure of its inertia, i.e. resistance to the force acting on a given body, striving to change its state (set in motion or, conversely, stop, or change the speed of movement). Mass can also be considered as a measure of the gravitational properties of a body, i.e. its ability to interact with other bodies that also have mass and are at a certain distance from the given body. The momentum of a body is considered as a quantitative measure of its movement, defined as the product of body mass by its speed:Dynamics studies the motion of bodies in connection with the forces acting on them, operating on the so-called dynamic characteristics of motion: mass, momentum, force, etc.
Force is considered as a measure of mechanical action on a given material body from other bodies.
Mechanics is a science that is a branch of physics, the purpose of which is to study the principles of motion and interaction of individual material bodies. But the movement in the science of mechanics will be a change in position both in time and in space. Mechanics is considered to be a science, the task of which is to solve any problems of motion, balance and interaction of bodies. And the movement of the planet Earth around the Sun also obeys the laws of mechanics. On the other hand, the concept of mechanics also includes the creation of projects based on calculations for engines, machines, and their parts. In this case, one can speak not only about mechanics, but also about the mechanics of a continuous medium. Mechanics is also designed to solve the problems of motion of solid, gaseous, liquid bodies that have the ability to deform. Those. we are talking about material bodies that fill all space with a continuous continuous flow with a varying distance between points in the process of movement.
Mechanics is subdivided into: mechanics of continuous media, theoretical and special (about mechanisms and machines, soil mechanics, resistance, etc.) - according to the subject of study; classical, quantum and relativistic - in relation to the concepts of time, matter and space. The subject of the study of mechanics is mechanical systems. Every mechanical system exists with certain degrees of freedom. The state of a mechanical system is described by a system of generalized coordinates and impulses. Accordingly, the task of mechanics is to find out and investigate the properties of systems and to determine the presence of evolution in time.
Mechanical systems are closed, open and closed - in terms of interaction with the surrounding space; static and dynamic - according to the availability of the ability to change over time. The main and significant mechanical systems are recognized: a body of absolute elasticity, a physical pendulum, a body with the ability to deform, a mathematical pendulum, a material point. The school section of mechanics studies kinematics, dynamics, statics and conservation laws. While theoretical mechanics consists of celestial, nonholonomic, nonlinear dynamics, stability theory, catastrophe theory, and gyroscopes.
Solid mechanics is, first of all, hydrostatics, aeromechanics, hydrodynamics, rheology, as well as the theory of elasticity and plasticity, gas dynamics, and fracture mechanics and composites. Most courses in the theory of mechanics are limited to the theory of solids. Deformable bodies are studied in the theory of elasticity and the theory of plasticity. And liquids and gases are studied in the mechanics of liquids and gases. Differential and integral calculus is the basis of classical mechanics. Calculus was developed by Newton and Leibniz. All 3 Newton's laws refer to different variational principles. Thus, classical mechanics is based on Newton's laws. But today there are 3 variants of the development of events in which the classical mechanics does not correspond to reality. For example, the properties of the microworld, here, to explain the laws, a transition from classical to quantum mechanics is required. Another example, these are speeds close to the speed of light - this requires a special theory of relativity. And the third option is systems with a large number of particles, when a transition to static physics is required.
Abstract on the topic:
HISTORY OF MECHANICS DEVELOPMENT
Completed: student 10 "A" class
Efremov A.V.
Checked by: Gavrilova O. P.
1. INTRODUCTION.
2. DEFINITION OF MECHANICS; ITS PLACE AMONG OTHER SCIENCES;
DEPARTMENTS OF MECHANICS.
4. HISTORY OF MECHANICS DEVELOPMENT:
The era preceding the establishment of the foundations of mechanics.
The period of creation of the foundations of mechanics.
Development of methods of mechanics in the 18th century.
Mechanics of the 19th and early 20th centuries
Mechanics in Russia and the USSR.
6. CONCLUSION.
7. APPENDIX.
1. INTRODUCTION.
For each person there are two worlds: internal and external; the senses are the intermediaries between these two worlds. The external world has the ability to influence the senses, cause them a special kind of changes, or, as they say, excite 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 inner world and, in turn, causes a subjective sensation in it, for the appearance of which the presence of consciousness is necessary. The subjective sensation perceived by the inner world is objectified, i.e. is transferred to outer space as something belonging to a certain place and a certain time.
In other words, through such objectification, we transfer our sensations into the external world, and space and time serve as the background on which these objective sensations are located. In those places of the space where they are placed, 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 between qualitative and quantitative differences, and the quantitative dissimilarity can refer either to tension (intensity), or to extension (extensiveness), or, finally, to the duration of the annoying objective reason.
Since the inferences accompanying any objectification are exclusively based on the perceived sensation, the complete sameness of these sensations will certainly entail the identity of objective causes, and this identity, apart from, and even against our will, persists even in cases where other sense organs indisputably testify us about the diversity of reasons. Here lies one of the main sources of undoubtedly erroneous inferences leading to the so-called deceptions of sight, hearing, etc. Another source is the lack of skill with new sensations. Perception in space and time of sensory impressions, which we compare with each other and to which we attach importance to objective reality that exists apart from our consciousness is called an external phenomenon. Changes in the color of bodies depending on lighting, the same water level in the vessels, the swing of the pendulum are external phenomena.
One of the powerful levers that move humanity along the path of its development is curiosity, which has the last, unattainable goal - the knowledge of the essence of our being, the true relationship of our inner world to the outer world. The result of curiosity was the acquaintance with a very large number of the most diverse phenomena that make up 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; DEPARTMENTS OF MECHANICS.
Mechanics (from the Greek mhcanich - skill related to machines; the science of machines) is the science of the simplest form of movement of matter - mechanical movement, representing the change in the spatial arrangement of bodies over time, and of the interactions between them associated with the movement of bodies. Mechanics explores the general laws connecting mechanical movements and interactions, accepting laws for the interactions themselves, obtained empirically and substantiated in physics. The methods of mechanics 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, 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 to be concentrated;
2) An absolutely rigid body, a set 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, a change in the relative position of elementary volumes is allowed. In contrast to a rigid body, to define the motion of a continuous medium, an infinite number of parameters are required. Continuous media include solid, liquid and gaseous bodies, reflected in the following abstract representations: ideally elastic body, plastic body, ideal fluid, viscous fluid, ideal gas and others. These abstract ideas about a material body reflect the actual properties of real bodies, essential in the given conditions.Accordingly, mechanics is divided into:
material point mechanics;
material point system mechanics;
mechanics of an absolutely rigid body;
mechanics of continuous medium.
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 a part of mechanics that deals with the study of the most general laws of motion, the formulation of its general provisions and theorems, and the application of methods 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.
In each of these sections, first of all, statics is highlighted, which unites issues related to the study of conditions for the balance of forces. Distinguish between statics of a rigid body and statics of a continuous medium: statics of an elastic body, hydrostatics and aerostatics (see Appendix). The motion 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 of time the distribution of displacements and velocities in space. The subject of dynamics is the mechanical motion of material bodies in connection with their interactions. The essential applications of mechanics are technical. The tasks posed by technology to mechanics are very diverse; these are questions 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) are technical sciences that use absolutely rigid body methods. Resistance 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 sections of mechanics have developed and continue to develop in close connection with the demands of practice; in the course of solving the problems of technology, mechanics as a branch of physics has developed in close connection with its other sections - with optics, thermodynamics and others. The foundations of so-called classical mechanics were generalized at the beginning of the 20th century. in connection with the discovery of physical fields and the laws of motion of microparticles. The content of the mechanics of fast-moving particles and systems (with velocities of the order of the speed of light) are presented in the theory of relativity, and the mechanics of micromotions - 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). Within the limits within which 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 motion are speed, which has a vector character, since it determines not only the rate of change of the path with time, but also the direction of motion, and acceleration is a vector, which is a measure of measuring the velocity vector in time. The vectors of angular velocity and angular acceleration serve as measures of the rotational motion of a rigid body. In the statics of an elastic body, the displacement vector and the corresponding strain tensor, including the concepts of relative elongations and shears, are of primary importance. The main measure of the interaction of bodies, which characterizes the change in time of the mechanical movement of the body, is force. The aggregates of the magnitude (intensity) of the force, expressed in certain units, the direction of the force (line of action) and the point of application determine quite unambiguously 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 says: every body maintains a state of rest or uniform and rectilinear motion until the applied forces force it to change this state. The first law can be used to determine inertial reference frames.
The second law, which establishes a quantitative relationship between a force applied to a point and a change in the momentum caused by this force, says: 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 lower the acceleration of the body, the greater its inertia. Mass is the measure of inertia. According to Newton's second law, the force is proportional to the product of the mass of a material point by its acceleration; with an appropriate choice of the unit of force, the latter can be expressed by the product of the mass of a point m by the acceleration a:
This vector equality represents the basic equation of the dynamics of a material point.
Newton's third law says: 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 one straight line in opposite directions. While the first two Newton's laws refer 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 made up of those accelerations that the point would have under the action of each force separately. The law of independence of the action of forces leads to the rule of the parallelogram of forces.
In addition to the previously named concepts, other measures of motion and action are used in mechanics.
The most important are the measures of motion: vector - the momentum p \u003d mv, equal to the product of mass by the velocity vector, and scalar - kinetic energy E k \u003d 1/2 mv 2, equal to half of the product of mass and square of velocity. In the case of rotational motion of a rigid body, its inertial properties are set by the tensor of inertia, which determines the moments of inertia and centrifugal moments about three axes passing through this point at each point of the body. 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 angular velocity. The measures of action of forces are: vector - elementary impulse of force F dt (product of force by element of time of its action), and scalar - elementary work F * dr (scalar product of vectors of force and elementary displacement of a position point); in rotary motion, the measure of the 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 motion of a continuous medium, caused by external forces applied to it, leads to the appearance in the medium of a stress state, 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 the three normal stresses at a given point, taken with the opposite sign, determines the pressure (see Appendix).
The study of the equilibrium and motion of a continuous medium is based on the laws of the relationship between the stress tensor and the strain tensor or strain rates. Such are 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, which more accurately characterize the phenomena occurring in real bodies. There are theories that take into account the previous history of body movement and stress, theories of creep, relaxation, and others (see Appendix).
The relationships between the measures of motion of a material point or a system of material points and the measures of action of forces are contained in the general theorems of dynamics: the quantities of motion, 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, that is, a system subject to predetermined constraints - mechanical connections (see Appendix), it is important to apply the general principles of mechanics - the principle of possible displacements and the D'Alembert principle. As 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 elementary work of all active forces acting on the system for any possible displacement of the system is equal to zero (for non-liberating connections) or it was equal to zero or less than zero (for releasing links). D'Alembert's principle for a free material point says: at each moment of time, the forces applied to the point can be balanced by adding to them the force of inertia.
When formulating problems, mechanics proceeds from the basic equations expressing the found laws of nature. Mathematical methods are used to solve these equations, and many of them were born and developed precisely in connection with the problems of mechanics. When setting a problem, it was always necessary to focus 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, in its complexity, does not lend itself to mathematical analysis, experimental research is widely used.
Experimental methods of mechanics are based on the developed technique of physical experiment. To record movements, both optical methods and methods of electrical registration are used, based on the preliminary transformation of mechanical movement into an electrical signal.
To measure forces, various dynamometers and scales are used, supplied with automatic devices and tracking systems. For measuring mechanical vibrations, a variety of radio engineering schemes are widely used. The experiment in continuum mechanics has achieved particular success. An optical method is used to measure voltage (see Appendix), which consists in observing a loaded transparent model in polarized light.
In recent years, strain gauging with the help of mechanical and optical strain gauges (see Appendix), as well as resistance strain gauges, has been greatly developed for measuring deformation.
To measure velocities and pressures in moving liquids and gases, thermoelectric, capacitive, induction and other methods are successfully used.
4. HISTORY OF MECHANICS DEVELOPMENT.
The history of mechanics, like that of 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 of solving them.
The era preceding the establishment of the foundations of mechanics. The era of the creation of the first tools of production and artificial structures should be recognized as the beginning of the accumulation of that 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 a few provisions were known relating to the simplest cases of equilibrium of bodies.
Statics arose earlier than all branches of mechanics. This section developed in close connection with the construction art of the ancient world.
The basic concept of statics - the concept of force - was initially closely associated with the muscular effort caused by the pressure of an object on the arm. Around the beginning of the IV century. BC e. the simplest laws of addition and balancing of forces applied to one point along the same straight line were already known. The problem of the lever attracted particular interest. The theory of leverage was created by the great scientist of antiquity Archimedes (III century BC) and is set forth in the work "On Levers". He established the rules for the addition and expansion of parallel forces, gave a definition of the concept of the center of gravity of a system of two weights suspended from a rod, and clarified the equilibrium conditions for such a system. Archimedes also discovered the basic laws of hydrostatics.
He applied his theoretical knowledge in the field of mechanics to various practical issues of construction and military technology. The concept of a moment of force, which plays a major role in all modern mechanics, is already hidden in the Archimedes' law. The great Italian scientist Leonardo da Vinci (1452 - 1519) introduced the concept of the shoulder of power under the guise of “potential leverage”.
The Italian mechanic Guido Ubaldi (1545 - 1607) applies the concept of moment in his block theory, where the concept of a chain hoist was introduced. Polyspast (Greek poluspaston, from polu - a lot and spaw - pull) - a system of movable and stationary blocks, bent around by a rope, are used to obtain a gain in strength and, less often, to obtain a gain in speed. Usually, it is customary to refer to statics as 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, using the famous method of exhaustion, indicated the position of the center of gravity of many regular geometric shapes, flat and spatial.
General theorems on the centers of gravity of bodies of revolution were given by the Greek mathematician Papp (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. - in the Renaissance - the heyday of the arts and sciences 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 celestial bodies move around it in their circular orbits: the Moon, Mercury, Venus , Sun, Mars, Jupiter, Saturn.
The kinematic and dynamic studies of the Renaissance were focused mainly on clarifying the concepts of uneven and curvilinear motion of a point. Until that time, the generally accepted dynamic views of Aristotle, set forth in his "Problems of Mechanics", were not generally accepted.
So, he believed that to maintain a uniform and rectilinear movement of the body, a constantly acting force must be applied to it. This statement seemed to him to agree with everyday experience. Of course, Aristotle knew nothing about the fact that frictional force arises in this case. He also believed that the speed of free fall of bodies depends on their weight: "If half weight passes this much in some time, then the double weight passes the same amount in half time." Considering that everything consists of four elements - earth, water, air and fire, he writes: “Everything that can rush to the middle or center of the world is heavy; easily everything that rushes from the middle or center of the world ”. From this he concluded: since heavy bodies fall to the center of the Earth, this center is the focus of the world, and the Earth is motionless. Not yet possessing the concept of acceleration, which was later introduced by Galileo, the researchers of this era considered accelerated motion as consisting of separate uniform motions, each having their own speed in each interval. Galileo, at the age of 18, observing during the divine service the small damping oscillations of the chandelier and counting the time by the beats of the pulse, found that the period of the pendulum's oscillation does not depend on its span.
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 the earth's surface. Dropping bodies from the tower, he found that the time of the body's fall does not depend on its weight and is determined by the height of the fall. He was the first to prove that in a free fall of a body, 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 foundations of mechanics. Practice (mainly merchant shipping and military affairs)
puts before the mechanics of the XVI - XVII centuries. a number of important problems that occupied the minds of the best scientists of that time. “… Together with the emergence of cities, large buildings and the development of handicrafts, mechanics also developed. Soon it also becomes necessary for shipping and military affairs ”(F. Engels, Dialectics of Nature, 1952, p. 145). It was necessary to accurately investigate the flight of shells, the strength of large ships, the oscillations of the pendulum, the impact of the body. Finally, the victory of Copernicus' teachings raises the problem of the motion of celestial bodies. The heliocentric worldview by the beginning of the 16th century created the preconditions for the establishment of the laws of planetary motion by the German astronomer I. Kepler (1571 - 1630).
He formulated the first two laws of planetary motion:
1. All planets move along ellipses, in one of the focuses of which is the Sun.
2. The radius vector drawn from the Sun to the planet describes equal areas in equal time intervals.
The founder of mechanics is the great Italian scientist G. Galilei (1564 - 1642). He experimentally established the quantitative law of falling bodies in a void, according to which the distances traversed by a falling body at equal intervals of time relate to each other as consecutive 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 communicated to them in order to raise them to the height from which they fell. Passing to the limit, he showed that on the horizontal plane a heavy body will be at rest or will move uniformly and rectilinearly. Thus, he formulated the law of inertia. By adding the horizontal and vertical movements of the body (this is the first addition of finite independent movements in the history of mechanics), he proved that a body thrown at an angle to the horizon describes a parabola, and showed how to calculate the length of flight and the maximum height of the trajectory. For all his conclusions, he always stressed that we are talking about movement in the absence of resistance. In dialogues about two world systems, very figuratively, in the form of an artistic description, he showed that all the movements that can occur in the ship's cabin do not depend on whether the ship is at rest or moves in a straight line and evenly.
By this he established the principle of relativity of classical mechanics (the so-called principle of relativity of Galileo - Newton). In the particular case of the force of weight, Galileo closely linked the constancy of weight with the constancy of the acceleration of falling, but only Newton, introducing the concept of mass, gave an exact formulation of the relationship between force and acceleration (second law). Exploring the conditions of equilibrium of simple machines and floating bodies, Galileo, in essence, applies the principle of possible displacements (albeit in a rudimentary form). To him, science owes the first study of the strength of beams and the resistance of a fluid to bodies moving in it.
The French geometer and philosopher R. Descartes (1596 - 1650) put forward the fruitful idea of \u200b\u200bconservation of momentum. He applies mathematics to the analysis of motion and, introducing variables into it, establishes a correspondence between geometric images and algebraic equations.
But he did not notice the essential fact that the momentum is a directional quantity, and he added the momentum 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.
Galileo's follower 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 in the curvilinear motion of a point (centripetal acceleration). Huygens also solved a number of the most important problems of dynamics - the motion of a body in a circle, 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, center of oscillation of a physical pendulum. But his main merit is 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 \u200b\u200bconservation of momentum, he created a complete theory of the impact of elastic balls.
The merit of 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 first published in 1687, Newton summed up the achievements of his predecessors and indicated the ways of 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, resistance forces of the medium, viscous forces and many others), studies the laws of the dependence of these forces on the position and motion of bodies. The basic equation of dynamics, which is the 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 for moving in elliptical orbits. Back in his student years, Newton pondered over the issues of gravitation. In his papers they found the following entry: “From Kepler's rule that the periods of the 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 should 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 gives no proof, but I can assume that his reasoning was as follows. If we roughly 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 \u003d R 3 2 / R 3 1, (1.1) where T j and R j are the periods of revolution and radii of the orbits of two planets (j \u003d 1, 2) With uniform motion of planets in circular orbits with velocities V j their periods of circulation are determined by the equalities T j \u003d 2 p R j / V j
Therefore, T 2 / T 1 \u003d 2 p R 2 V 1 / V 2 2 p R 1 \u003d V 1 R 2 / V 2 R 1
Now relation (1.1) is reduced to the form V 2 1 / V 2 2 \u003d R 2 / R 1. (1.2)
By the years under consideration, Huygens had already established that the centrifugal force is proportional to the square of the velocity and inversely proportional to the radius of the circle, that is, F j \u003d kV 2 j / R j, where k is the proportionality coefficient.
If we now introduce into equality (1.2) the ratio V 2 j \u003d F j R j / k, then I will get F 1 / F 2 \u003d R 2 2 / R 2 1, (1.3) which establishes an inverse proportionality of the centrifugal forces of the planets to the squares of their distances before the Sun, Newton also carried out studies of the resistance of fluids 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 velocity. Newton discovered the basic law of internal friction in liquids and gases.
By the end of the 17th century. the fundamentals of mechanics have been elaborated. If the ancient centuries are considered the prehistory of mechanics, then the XVII century. can be considered as the period of creation of its foundations. Development of methods of mechanics in the XVIII century. In the XVIII 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 of general methods for solving problems of 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 the methods of analysis of infinitesimal to the solution of the equations of motion of a point. Euler's treatise "Mechanics, that is, the science of motion, set forth by the analytical method", published in St. Petersburg in 1736, contains general uniform methods for the analytical solution of problems of the dynamics of a point.
L. Euler - the founder of rigid body mechanics.
He owns the generally accepted method for the 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 of the rotational motion of a rigid body around a fixed center established by Euler. Euler established two integrals: the integral of the angular momentum
A 2 w 2 x + B 2 w 2 y + C 2 w 2 z \u003d m
and integral of living forces (integral of energy)
A w 2 x + B w 2 y + C w 2 z \u003d 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, which he discovered, which is a necessary addition to the law of momentum, formulated in general form in Newton's "Principles". Euler's “Mechanics” gives a close to modern formulation of the law of “living forces” for the case of rectilinear motion and notes the presence of such motions of a material point in which the change in living force when a point passes from one position to another does not depend on the shape of the trajectory. This was the beginning of 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 a ship and the theory of stability of elastic rods; Euler laid the foundation for the theory of calculating turbines by deriving the turbine equation; in applied mechanics, Euler's name is associated with 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 study of his predecessors - from Newton to Lagrange - by his own research on the stability of the solar system, solving the three-body problem , the motion of the moon and many other questions 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 set forth by Newton in the third book of the "Elements".
The problem of the figures of equilibrium and stability of a rotating liquid mass played a significant role in the development of mechanics.
The great Russian scientist MV Lomonosov (1711 - 1765) highly appreciated the importance of mechanics for natural science, physics and philosophy. He owns the materialistic interpretation of the processes of interaction between two bodies: “when one body accelerates the movement of the other and imparts part of its motion to it, then only in such a way that it itself loses the same part of the motion”. 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's 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, the same amount will be subtracted from something else. So, how much matter joins some body, the same amount will be taken away from another; how many hours I spend in sleep, as much I take away from vigil, etc. Since this law of nature is universal, it even extends to the rules of movement, and a body that impels another to move with its impetus loses its movement as much as it communicates another, moved by him. "
Lomonosov for the first time predicted the existence of absolute zero temperature, expressed the idea of \u200b\u200ba 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 displacements, which expresses the general conditions for the equilibrium of the system. This principle was first applied by the Dutch scientist S. Stevin (1548 - 1620) when considering the block equilibrium. 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 \u200b\u200ban “ideal” machine, devoid of internal losses for harmful resistance in the transmission mechanism. It looks as follows: if in the position of isolated equilibrium of a conservative system with stationary bonds the potential energy has a minimum, then this equilibrium position is stable.
The creation of the principles of the dynamics of a non-free system was facilitated by the problem of the motion 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 bonds acting on a moving material point can be balanced at any time by adding to them the force of inertia.
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 an analytical expression of D'Alembert's principle - “the general formula of dynamics”. How did Lagrange get it?
After Lagrange outlined the various principles of statics, he proceeds to establish "a general statics formula 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 +… \u003d 0. (2.1)
This equation represents a mathematical notation of the principle of possible displacements. In modern notation, this principle has the form
е n j \u003d 1 F j d r j \u003d 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: nowadays it is an arbitrarily conceivable movement of the point of application of the force, compatible with constraints, and in Lagrange it 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 it is called potential energy), having defined it by equality.
d П \u003d P dp + Q dq + R dr + ..., (2.3) in Cartesian coordinates, the function П (after integration) has the form
P \u003d A + Bx + Cy + Dz + ... + Fx 2 + Gxy + Hy 2 + Kxz + Lyz +
Mz 2 +… (2.4)
For further proof, Lagrange invents the famous method of undefined factors. Its essence is as follows. Consider the equilibrium of n material points, each of which is acted upon by the force F j. There are m links j r \u003d 0 between the coordinates of the points, depending only on their coordinates. Considering that d j r \u003d 0, equation (2.2) can be immediately reduced to the following modern form:
å n j \u003d 1 F j d r j + å m r \u003d 1 l r d j r \u003d 0, (2.5) where l r are undefined factors. Hence, the following equilibrium equations are obtained, which are called the Lagrange equations of the first kind:
X j + å m r \u003d 1 l r j r / x j \u003d 0, Y j + å m r \u003d 1 l r j r / y j \u003d 0,
Z j + å m r \u003d 1 l r j r / z j \u003d 0 (2.6) These equations need to add m constraint equations j r \u003d 0 (X j, Y j, Z j are the projections of the force F j)
Let us show how Lagrange uses this method to derive the equilibrium equations for an absolutely flexible and inextensible thread. First of all, referred to the unit of thread length (its dimension is equal to F / L).
The constraint equation for an inextensible thread has the form ds \u003d const, and, therefore, d ds \u003d 0. In equation (2.5), the sums transform into integrals over the length of the thread l ò l 0 F d rds + ò l 0 ld ds \u003d 0. (2.7 ) Taking into account the equality (ds) 2 \u003d (dx) 2 + (dy) 2 + (dz) 2, we find
d ds \u003d dx / ds d dx + dy / ds d dy + dz / ds d dz.
ò l 0 l d ds \u003d 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 \u003d (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 get d x \u003d d y \u003d d z \u003d 0 for s \u003d 0 and s \u003d l, and, therefore, the first term vanishes. We introduce the rest 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 \u003d 0
Since the variations d x, d y and d z are arbitrary and independent, then all square brackets must be equal to zero, which gives three equations of equilibrium of an absolutely flexible inextensible thread:
d / ds (l dx / ds) - X \u003d 0, d / ds (l dy / ds) - Y \u003d 0,
d / ds (l dz / ds) - Z \u003d 0. (2.8)
Lagrange explains the physical meaning of the factor l as follows: “Since the value ld ds can represent the moment of some force l (in modern terminology -“ virtual (possible) work ”) tending to reduce the length of the element ds, then the term ò ld 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 the elements of the thread. Indeed, 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 the tension of the thread "
Turning to dynamics, Lagrange, taking bodies as points of mass m, writes that “the quantities md 2 x / dt 2, md 2 y / dt 2, md 2 z / dt 2 (2.9) express the forces applied directly to move the 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,…, proportional to the masses, directed to the corresponding centers and tend to reduce the distance to these centers. Therefore, the variations of the lines of action will be - d p, - d q, - d r, ..., and the virtual work of the applied forces and forces (2.9) will be respectively equal
е m (d 2 x / dt 2 d x + d 2 y / dt 2 d y + d 2 z / dt 2 d z), - е (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
е m (d 2 x / dt 2 d x + d 2 y / dt 2 d y + d 2 z / dt 2 d z) + е (P d p
Q d q + R d r +…) \u003d 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 made 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 decomposes the forces P, Q, R, ... along the axes of rectangular coordinates and reduces this equation to the following form:
е (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 \u003d 0. (2.12)
Equation (2.12) completely coincides with the modern form of the general equation of dynamics up to signs:
е j (F j - m j d 2 r j / dt 2) d r j \u003d 0; (2.13) if we expand the scalar product, then we get the equation (2.12) (except for the signs in brackets)
Thus, continuing the work 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. Proceeding from the “general formula of dynamics”, Lagrange indicated two basic 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's equation 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, with various parameters (linear, angular, or a combination of these). For such a brilliant mathematician as Lagrange was, the problem of generalization naturally arose - to go to arbitrary, not concretized 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 \u003d 0 (L \u003d T - P)
The overwhelming majority of the problems solved in Analytical Mechanics reflect the technical problems of that time. From this point of view, it is necessary to especially highlight the group of the most important problems of dynamics, united by Lagrange under the general name “On small vibrations of any system of bodies”. This section provides the basis for modern vibration theory. Considering small movements, Lagrange showed that any such movement can be represented as the result of the superposition of simple harmonic oscillations.
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) the formulation of the variational principles of dynamics and the principle of conservation of mechanical energy;
3) development of methods for integrating the equations of dynamics.
In parallel with this, new fundamental problems of mechanics were advanced 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 connections that depend on time, introduced a new concept of unstoppable connections, that is, connections expressed analytically using inequalities, and generalized the principle of possible displacements and the general equation of dynamics to the case of such connections. Ostrogradskiy also has priority in considering differential relationships that impose restrictions on the speed of points in the system; analytically, such connections are expressed using non-integrable differential equalities or inequalities.
A natural addition, expanding the scope of the D'Alembert principle, was the application of the principle proposed by Ostrogradsky to systems subject to the action of instantaneous and impulsive forces arising from impacts on the system. Ostrogradsky considered such 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 XIX 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 caused the appearance of applied, or technical, mechanics in the first quarter of the century. In the first treatises on applied mechanics, the concepts of the work of forces were finally formed.
D'Alembert's principle, containing the most general formulation of the laws of motion of a non-free system, does not exhaust all the possibilities of posing dynamics problems. In the middle of the XVIII century. arose, and in the XIX century. new general principles of dynamics were developed - variational principles.
The first variational principle was the principle of least action, put forward in 1744 without any proof, as some general law of nature, by the French scientist P. Maupertuis (1698 - 1756). The principle of least action states, “that the path that it (the light) follows is the path for which the number of actions will be the least”.
The development of general methods for the integration of differential equations of dynamics refers mainly to the middle of the 19th century. The first step in reducing the differential equations of dynamics to a system of equations of the first order 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 equations of the first order for the case of constraints that do not depend on time was solved in 1834 by the English mathematician and physicist W. Hamilton (1805 - 1865). Its final completion belongs to Ostrogradskiy, who extended these equations to cases of nonstationary constraints.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, and also closely related to this theory, the problem of fluctuations 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 special 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 the stationary center, which does not coincide with the center of gravity of the body, is placed on the axis of symmetry, was considered by Lagrange. The solution of 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 instruments.
Examples of such devices are 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 works 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 the motion of a rigid body in an ideal fluid were given by the German physicist G. Kirchhoff in 1869. With the appearance in the middle of the 19th century. rifled guns, which was intended to give the projectile the rotation necessary for stability in flight, the task of external ballistics turned out to be closely related to the dynamics of a heavy rigid body. This formulation of the problem and its solution belongs to the outstanding Russian scientist - artilleryman N.V. Maevsky (1823 - 1892).
One of the most important problems in mechanics is the problem of the stability of the equilibrium and motion of material systems. The first general theorem on the stability of the equilibrium of a system under the action 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, applied 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. Routh, published in 1877, the study of stability by the method of small oscillations was reduced to considering the distribution of the roots of a certain "characteristic" equation and indicated the necessary and sufficient conditions under which these roots have negative real parts.
From a point of view different from that of Routh, the problem of stability of motion was considered in the work of NE 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 visual geometric form, so characteristic of the entire scientific work of the great mechanic.
A rigorous formulation of the problem of stability of motion and an indication of the most general methods of its solution, as well as a specific consideration of some of the most important problems of the theory of stability, belong to A. M. Lyapunov, and were presented by him in his fundamental essay "General problem of stability of motion" (1892). He gave the definition of a stable equilibrium position, which looks as follows: if for a given r (radius of the sphere) one can choose such an arbitrarily small but not equal to zero value of h (initial energy) that in all subsequent time the particle does not go beyond 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 some functions, from the comparison of the signs of which with the signs of their derivatives with respect to time, one can conclude about the stability or instability of the considered state of motion (“the second Lyapunov method”). With the help of this method, Lyapunov, in his stability theorems in the first approximation, indicated the limits of applicability of the method of small oscillations of a material system around its stable equilibrium position (first described in Lagrange's “Analytical Mechanics”).
The subsequent development of the theory of small fluctuations in the XIX century. was mainly due to the influence of resistances leading to damping of oscillations and external disturbing forces that create forced oscillations. The theory of forced vibrations and the theory of resonance appeared in response to the demands of 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 of the theory of oscillations, was regulator construction. The founder of the modern dynamics of the regulation process is the Russian scientist and engineer I.A.Vyshnegradskiy (1831 - 1895). In 1877, in his work “On Direct Controllers”, Vyshnegradskiy was the first to formulate the well-known inequality that must be satisfied by a stably operating machine equipped with a controller.
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 the pitching of a ship in 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 solving the most important technical problems. In the XX century. problems of electrical engineering, radio engineering, the theory of automatic control of machines and production processes, technical acoustics, and others gave rise to a new field of 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. By the end of the XIX century. a special group of mechanical problems was distinguished - the motion of bodies of variable mass. The fundamental role in the creation of a new area of \u200b\u200btheoretical 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 point of variable mass”.
In the XIX and early XIX centuries. the foundations were laid for two important areas 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 of problems in this area was indicated by N. Ye. Zhukovsky.
By the end of the XIX century. mechanics has reached a high level of development. XX century brought a deep critical revision of a number of basic provisions of classical mechanics and was marked by the emergence of the mechanics of fast motions proceeding with speeds close to the speed of light. The mechanics of fast movements, 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 mechanics engineering in Russia and the USSR. Mechanics in pre-revolutionary Russia, thanks to the fruitful scientific activities 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 to cope with the tasks set before it by domestic technology, but also to contribute to the development of technology throughout the world. The works of the “father of Russian aviation” N. Ye. Zhukovsky laid the foundations of aerodynamics and aviation science in general. The works of N. Ye. Zhukovsky and S. A. Chaplygin were of fundamental importance in the development of modern hydro-aeromechanics. SA Chaplygin is the author of fundamental research in the field of gas dynamics, which indicated the development of high-speed aerodynamics for many decades ahead. A. N. Krylov's work on the theory of stability of a ship's roll in waves, research on the buoyancy of their hull, and the theory of compass deviation put him among the founders of modern science of shipbuilding.
One of the important factors that contributed to the development of mechanics in Russia was the high level of teaching it in higher education. Much has been done in this respect by M. V. Ostrogradskii and his followers. Problems of motion stability are of the greatest technical importance in problems of the theory of automatic control. I. N. Voznesensky (1887 - 1946) played an outstanding role in the development of the theory and technology of regulation of machines and production processes. Problems of rigid body dynamics developed mainly in connection with the theory of gyroscopic phenomena.
Soviet scientists have achieved significant results in the field of elasticity theory. They carried out research on the theory of plate bending and general solutions of problems in the theory of elasticity, on the plane problem of the theory of elasticity, on the variational methods of the theory of elasticity, on structural mechanics, on the theory of plasticity, on the theory of an ideal fluid, on the dynamics of a compressible fluid and gas dynamics, on the theory filtration of movements, which contributed to the rapid development of Soviet hydro-aerodynamics, 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, confirmed the USSR's leading role in this field. The formulation, theoretical consideration and organization of the experimental study of nonlinear oscillations are an important achievement of L.I. Mandel'shtam (1879 - 1944) and ND Papaleksi (1880 - 1947) and their school (A.A.
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 formulated by AA Andronov (1901 - 1952) in connection with the problem of continuous 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, the problems of the theory of oscillations, the dynamics of a rigid body and the theory of stability of motion. 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. The problems of vibrations are closely intertwined with the issues of radio engineering, automatic regulation and control of movements, 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 stability of motion. These tasks are posed by the dynamics of flight, the dynamics of the ship, the theory of gyroscopic systems and instruments, used mainly in air navigation and ship navigation. In the theory of motion stability, the first place is given to the study of Lyapunov's “special cases”, the stability of periodic and unsteady motions, and the main research tool is the so-called “second Lyapunov method”.
In the theory of elasticity, along with problems for a body obeying Hooke's law, the greatest attention is paid to the issues of plasticity and creep in the details of machines and structures, the 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 acquiring great importance. In close connection with the theory of plasticity, the mechanics of a free-flowing medium is being developed. The dynamic problems of the theory of elasticity are associated with seismology, the propagation of elastic and plastic waves along the rods, and dynamic phenomena arising from impact. The most important problems of hydro-aerodynamics are associated with the problems of high speeds in aviation, ballistics, turbine construction and engine construction.
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 motions.
The problems of high-speed aerodynamics are closely intertwined with the issues of heat transfer, combustion and explosions. The study of the motion of a compressible gas at high speeds presupposes the main problem of gas dynamics, and at low speeds it is associated with problems of dynamic meteorology. The problem of turbulence, which has not yet received a theoretical solution, is of fundamental importance for hydroaerodynamics. In practice, they continue to use numerous empirical and semi-empirical formulas.
The hydrodynamics of a heavy fluid is faced with the problems of the spatial theory of waves and wave drag of bodies, wave formation in rivers and canals, and a number of problems associated with hydraulic engineering.
Problems of filtration movement of liquids and gases in porous media are of great importance for the latter, as well as for issues of oil production.
6. CONCLUSION.
Galileo - Newton mechanics has come a long way of development and did not immediately win the right to be called classical. Her successes, especially in the 17th and 18th centuries, established experiment as the main method for testing theoretical constructions. 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 science.
In the future, 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 come to the forefront of science. Mechanics gave way to a leader in science, but did not lose its significance. As before, all dynamic calculations of any mechanisms operating on the ground, under water, in the air and in space are based to one degree or another on the laws of classical mechanics. On the basis of far from obvious consequences from its basic laws, devices are built, autonomously, without human intervention, determining the location of submarines, surface ships, aircraft; systems have been built that autonomously orient spacecraft and direct them to the planets of the solar system, Halley's comet. Analytical mechanics - an integral part of classical mechanics - retains the "inconceivable efficiency" in modern physics. Therefore, no matter how physics and technology develop, classical mechanics will always take its rightful place in science.
7. APPENDIX.
Hydromechanics 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 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 when the property of compressibility is essential.
Aerostatics is a part of mechanics that studies the equilibrium conditions for 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 propellant gases, the mechanical freedom of which is limited by any effort.
Hydraulics is the science of the conditions and laws of equilibrium and motion of fluids and the ways of applying these laws to solving practical problems. Can be defined as applied fluid mechanics.
An inertial coordinate system is a coordinate system in which the law of inertia is fulfilled, 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 the body acts on the surface of the support in contact with it, to the contact area, or otherwise - the normal surface force acting per unit area.
Viscosity (or internal friction) is the property of liquids and gases to resist when one part of the liquid moves relative to another.
Creep is a process of small continuous plastic deformation that occurs in metals under conditions of prolonged static loading.
Relaxation is the process of establishing static equilibrium in a physical or physicochemical system. In the process of relaxation, the macroscopic quantities characterizing the state of the system asymptotically approach their equilibrium values.
Mechanical links 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 relations between the coordinates or their derivatives, which characterize the mechanical constraints of motion, are called constraint equations. For the system to move, 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 stresses is a method for studying stresses in polarized light, based on the fact that particles of an amorphous material become optically anisotropic upon deformation. In this case, the main axes of the refractive index ellipsoid coincide with the main directions of deformation, and the main light oscillations, passing through the deformed plate of polarized light, receive a path difference.
Strain gauge - a device for measuring tensile or compressive forces applied to any system due to deformations caused by these forces
Celestial mechanics is a section of astronomy devoted to the study of the motion 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 bodies of the solar system.
The theory of elasticity is a branch of mechanics that studies displacements, elastic deformations and stresses that arise in a solid under the influence of external forces, from heating and from other influences. The task is to determine quantitative relationships characterizing the deformation or internal relative displacements of particles of a solid body that is under the influence of external influences in a state of equilibrium or small internal relative motion.
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History development computing technology (14)
Abstract \u003e\u003e InformaticsEfficiency. In 1642 the French mechanic Blaise Pascal designed the first in ... generations - for its short history development have already changed four ... -to date Since the 90s in stories development computing technology, it's time for the fifth ...
History development computer facilities (1)
Abstract \u003e\u003e InformaticsHistory development computer facilities First counting ... hours. 1642 - French mechanic Blaise Pascal developed a more compact ... Electronic computers: XX century In stories computer technology, there is a kind of periodization ...
In any curriculum, physics starts with mechanics. Not theoretical, not applied or computational, but good old classical mechanics. This mechanics is also called Newtonian mechanics. According to legend, the scientist was walking in the garden, saw an apple falling, and it was this phenomenon that pushed him to the discovery of the law of universal gravitation. Of course, the law has always existed, and Newton only gave it a form that people can understand, 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 basics, 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 constructing machines, we are still like the Moon, so we will follow in the footsteps of our ancestors, and we will study the movement of stones thrown at an angle to the horizon, and apples falling on heads from a height of h.
Why does the study of physics begin with mechanics? Because it is completely natural, not to start it from thermodynamic equilibrium ?!
Mechanics is one of the oldest sciences, and historically the study of physics began with the foundations of mechanics. Placed within the framework of time and space, people, in fact, could not start from something else, with all their desire. Moving bodies are the first thing we turn our attention to.
What is movement?
Mechanical movement 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 at a certain speed, and rests relative to his neighbor on the seat next to him, and moves at a different speed relative to a passenger in a car that overtakes 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 of our measurements in a geocentric frame of reference associated with the Earth. The earth is a reference body relative to which cars, airplanes, people, 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 constructs 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 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. Nobody has ever seen a material point or smelled ideal gas, but they are! It's just much easier to live with them.
Material point is a body whose size and shape can be neglected in the context of this task.
Sections of classical mechanics
Mechanics consists of several sections
- Kinematics
- Dynamics
- Statics
Kinematicsfrom a physical point of view, it studies exactly how the body moves. In other words, this section deals with the quantitative characteristics of movement. Find speed, path - typical kinematic problems
Dynamics solves the question of why it moves that way. That is, it considers the forces acting on the body.
Statics studies the balance of bodies under the action of forces, that is, answers the question: why does it not fall at all?
The 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 true for the world we are accustomed to in size (macrocosm). They stop working in the case of the particle world, when quantum mechanics replaces the classical one. Also, classical mechanics is inapplicable 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 anywhere; they also take place during the ordinary motion of macroscopic bodies with a speed much less 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. Thus, classical mechanics will never lose their fundamental importance.
We will continue to study the physical foundations of mechanics in the next articles. For a better understanding of the mechanics, you can always refer to to our authorswho individually shed light on the dark spot of the most difficult task.
