The main purpose of this chapter is to ensure that the student understands the conceptual structure of classical mechanics. As a result of studying the material in this chapter, the student should:
know
- basic concepts of classical mechanics and how to control them;
- principles of least action and invariance, Newton's laws, concepts of force, determinism, mass, extension, duration, time, space;
be able to
- determine the place of any concept within classical mechanics;
- give any mechanical phenomenon a conceptual interpretation;
- explain mechanical phenomena through dynamics;
own
- conceptual understanding of current problem situations related to the interpretation of physical concepts;
- a critical attitude towards the views of various authors;
- the theory of conceptual transduction.
Keywords: principle of least action, Newton's laws, space, time, dynamics, kinematics.
Creation of classical mechanics
Few doubt that Newton accomplished a scientific feat with the creation of classical mechanics. It consisted in the fact that for the first time the differential law of motion of physical objects was presented. Thanks to Newton's work, physical knowledge was raised to a height to which it had never been before. He managed to create a theoretical masterpiece that determined the main direction of development of physics for at least more than two centuries. It is difficult to disagree with those scientists who associate the beginning of scientific physics with Newton. In the future, it is necessary not only to identify the main content of classical mechanics, but also, if possible, to understand its conceptual components, being ready to take a critical view of Newton’s conclusions. After him, physics went through a three-century journey. It is clear that even the brilliantly gifted Newton could not anticipate all of its innovations.
The set of concepts that Newton chose is of considerable interest. This is, firstly, a set of elementary concepts: mass, force, extension, duration of a certain process. Secondly, derived concepts: in particular, speed and acceleration. Thirdly, two laws. Newton's second law expresses the relationship between the force acting on an object, its mass, and the acceleration it acquires. According to Newton's third law, the forces that objects exert on each other are equal in magnitude, opposite in direction, and applied to different bodies.
But what about the principles in Newton's theory? Most modern researchers are confident that the role of the principle in Newton’s mechanics is played by the law, which he called the first. It is usually given in the following formulation: every body continues to be maintained in a state of rest or uniform and rectilinear motion until and unless it is forced by applied forces to change this state. The piquancy of the situation lies in the fact that, at first glance, this position seems to follow directly from Newton’s second law. If the sum of the forces applied to an object is equal to zero, then for a body with constant mass () the acceleration () is also equal to zero, which exactly corresponds to the content of Newton’s first law. Nevertheless, physicists are quite justified in not considering the first law
Newton is just a special case of his second law. They believe that Newton had good reason to consider the first law to be the main concept of classical mechanics, in other words, he gave it the status of a principle. In modern physics, the first law is usually formulated in this way: there are such reference systems, called inertial, relative to which a free material point retains the magnitude and direction of its speed indefinitely. It is believed that Newton expressed precisely this circumstance, albeit awkwardly, with his first law. Newton's second law is satisfied only in those frames of reference for which the first law is valid.
Thus, Newton's first law is, in fact, necessary to introduce the idea of invariance of Newton's second and third laws. Consequently, it plays the role of the invariance principle. According to the author, instead of formulating Newton's first law, it would be possible to introduce the principle of invariance: there are reference systems in which Newton's second and third laws are invariant.
So, everything seems to be in place. In accordance with Newton's ideas, the supporter of the mechanics he created has at his disposal elementary and derivative concepts, as well as laws and the principle of invariance. But even after this statement, numerous controversial points are revealed that convince us of the need to continue the study of the conceptual content of Newtonian mechanics. Avoiding it, it is impossible to understand the true content of classical mechanics.
conclusions
- 1. Newton's scientific feat was that he wrote down the differential law of motion of physical objects under the influence of forces.
- 2. Newton's first law is the principle of invariance.
- Strictly speaking, Newton's first law is a principle. That is why we are talking not about three, but about two Newton’s laws. ( Note auto.)
INTRODUCTION
Physics is a science of nature that studies the most general properties of the material world, the most general forms of motion of matter that underlie all natural phenomena. Physics establishes the laws that these phenomena obey.
Physics also studies the properties and structure of material bodies and indicates ways of practical use of physical laws in technology.
In accordance with the variety of forms of matter and its movement, physics is divided into a number of sections: mechanics, thermodynamics, electrodynamics, physics of vibrations and waves, optics, physics of the atom, nucleus and elementary particles.
At the intersection of physics and other natural sciences, new sciences arose: astrophysics, biophysics, geophysics, physical chemistry, etc.
Physics is the theoretical basis of technology. The development of physics served as the foundation for the creation of such new branches of technology as space technology, nuclear technology, quantum electronics, etc. In turn, the development of technical sciences contributes to the creation of completely new methods of physical research, which determine the progress of physics and related sciences.
PHYSICAL FOUNDATIONS OF CLASSICAL MECHANICS
I. Mechanics. General concepts
Mechanics is a branch of physics that examines the simplest form of motion of matter - mechanical motion.
Mechanical motion is understood as a change in the position of the body being studied in space over time relative to a certain goal or system of bodies conventionally considered motionless. Such a system of bodies together with a clock, for which any periodic process can be chosen, is called reference system(S.O.). S.O. often chosen for reasons of convenience.
For a mathematical description of movement with S.O. They associate a coordinate system, often rectangular.
The simplest body in mechanics is a material point. This is a body whose dimensions can be neglected in the conditions of the present problem.
Any body whose dimensions cannot be neglected is considered as a system of material points.
Mechanics are divided into kinematics, which deals with the geometric description of motion without studying its causes, dynamics, which studies the laws of motion of bodies under the influence of forces, and statics, which studies the conditions of equilibrium of bodies.
2. Kinematics of a point
Kinematics studies the spatiotemporal movement of bodies. It operates with such concepts as movement, path 
, time t, speed 
, acceleration.
The line that a material point describes during its movement is called a trajectory. According to the shape of the movement trajectories, they are divided into rectilinear and curvilinear. Vector , connecting the initial I and final 2 points is called movement (Fig. I.I).
Each moment of time t has its own radius vector 
:
T 
In this way, the movement of a point can be described by a vector function.
which we define vector way of specifying movement, or three scalar functions
x= x(t); y= y(t); z= z(t) , (1.2)
which are called kinematic equations. They determine the movement task coordinate way.
The movement of a point will also be determined if for each moment of time the position of the point on the trajectory is established, i.e. addiction
It determines the movement task natural way.
Each of these formulas represents law movement of the point.
3. Speed
If time t 1 corresponds to the radius vector 
, A 
, then for the interval
the body will move 
. In this case average speed
t is the quantity
,
(1.4)
which, in relation to the trajectory, represents a secant passing through points I and 2. Speed at time t is called a vector
, (1.5)
From this definition it follows that the speed at each point of the trajectory is directed tangentially to it. From (1.5) it follows that the projections and magnitude of the velocity vector are determined by the expressions:
If the law of motion (1.3) is given, then the magnitude of the velocity vector will be determined as follows:
, (1.7)
Thus, knowing the law of motion (I.I), (1.2), (1.3), you can calculate the vector and modulus of the doctor of speed and, conversely, knowing the speed from formulas (1.6), (1.7), you can calculate the coordinates and path.
4. Acceleration
During arbitrary movement, the velocity vector continuously changes. The quantity characterizing the rate of change of the velocity vector is called acceleration.
If in. moment in time t 1 point speed 
, and when t 2 - 
, then the speed increment will be (Fig.1.2). Average acceleration n 
in this case
and instantaneous
,
(1.9)
For the projection and acceleration module we have: , (1.10)
If a natural method of movement is given, then acceleration can be determined this way. Speed changes in magnitude and direction, speed increment 
decomposed into two sizes; 
- directed along (speed increment in magnitude) and 
- directed perpendicularly (speed increment in direction), i.e. = + (Fig. I.З). From (1.9) we obtain:
(1.11); 
(1.12)
Tangential (tangential) acceleration characterizes the rate of change in magnitude (1.13)
normal (centripetal acceleration) characterizes the speed of change in direction. To calculate a n consider
OMN and MPQ under the condition of small movement of the point along the trajectory. From the similarity of these triangles we find PQ:MP=MN:OM:
The total acceleration in this case is determined as follows:
, (1.15)
5. Examples
I. Equally variable rectilinear motion. This is a movement with constant acceleration ( 
) . From (1.8) we find
or
, Where v 0 - speed at time t 0 . Believing t 0 =0, we find
,
and the distance traveled S from formula (I.7):
Where S 0 is a constant determined from the initial conditions.
2. Uniform movement in a circle. In this case, the speed changes only in direction, that is 
- centripetal acceleration.
I. Basic concepts
The movement of bodies in space is the result of their mechanical interaction with each other, as a result of which a change in the movement of bodies or their deformation occurs. As a measure of mechanical interaction in dynamics, a quantity is introduced - force 
. For a given body, force is an external factor, and the nature of the movement depends on the properties of the body itself - compliance with external influences exerted on it or the degree of inertia of the body. The measure of inertia of a body is its mass T, depending on the amount of body matter.
Thus, the basic concepts of mechanics are: moving matter, space and time as forms of existence of moving matter, mass as a measure of inertia of bodies, force as a measure of mechanical interaction between bodies. The relationships between these concepts are determined by laws! movements that were formulated by Newton as a generalization and clarification of experimental facts.
2. Laws of mechanics
1st law. Every body maintains a state of rest or uniform rectilinear motion as long as external influences do not change this state. The first law contains the law of inertia, as well as the definition of force as a cause that violates the inertial state of the body. To express it mathematically, Newton introduced the concept of momentum or momentum of a body:
(2.1)
then if 
2nd law. The change in momentum is proportional to the applied force and occurs in the direction of action of this force. Selecting units of measurement m and so that the proportionality coefficient is equal to unity, we get
or 
(2.2)
If when moving m= const , That
or 
(2.3)
In this case, the 2nd law is formulated as follows: force is equal to the product of the body’s mass and its acceleration. This law is the basic law of dynamics and allows us to find the law of motion of bodies based on given forces and initial conditions. 3rd law.
The forces with which two bodies act on each other are equal and directed in opposite directions, i.e. 
, (2.4)
Newton's laws acquire a specific meaning after the specific forces acting on the body are indicated. For example, often in mechanics the movement of bodies is caused by the action of such forces: gravitational force 
, where r is the distance between bodies, is the gravitational constant; gravity - the force of gravity near the surface of the Earth, P=
mg; friction force 
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The pinnacle of I. Newton’s scientific creativity is his immortal work “Mathematical Principles of Natural Philosophy,” first published in 1687. In it, he summarized the results obtained by his predecessors and his own research and created for the first time a single, harmonious system of terrestrial and celestial mechanics, which formed the basis of all classical physics.
Here Newton gave definitions of the initial concepts - the amount of matter equivalent to mass, density; momentum equivalent to impulse and various types of force. Formulating the concept of the amount of matter, he proceeded from the idea that atoms consist of some single primary matter; density was understood as the degree of filling a unit volume of a body with primary matter.
This work sets out Newton's doctrine of universal gravitation, on the basis of which he developed the theory of the motion of planets, satellites and comets that form the solar system. Based on this law, he explained the phenomenon of tides and the compression of Jupiter. Newton's concept was the basis for many technological advances over time. On its foundation, many methods of scientific research in various fields of natural science were formed.
The result of the development of classical mechanics was the creation of a unified mechanical picture of the world, within the framework of which all the qualitative diversity of the world was explained by differences in the movement of bodies, subject to the laws of Newtonian mechanics.
Newton's mechanics, in contrast to previous mechanical concepts, made it possible to solve the problem of any stage of movement, both previous and subsequent, and at any point in space with known facts causing this movement, as well as the inverse problem of determining the magnitude and direction of action of these factors at any point with known basic elements of motion. Thanks to this, Newtonian mechanics could be used as a method for the quantitative analysis of mechanical motion.
Law of Universal Gravitation.
The law of universal gravitation was discovered by I. Newton in 1682. According to his hypothesis, attractive forces act between all bodies of the Universe, directed along the line connecting the centers of mass. For a body in the form of a homogeneous ball, the center of mass coincides with the center of the ball.
In subsequent years, Newton tried to find a physical explanation for the laws of planetary motion discovered by I. Kepler at the beginning of the 17th century, and to give a quantitative expression for gravitational forces. So, knowing how the planets move, Newton wanted to determine what forces act on them. This path is called the inverse problem of mechanics.
If the main task of mechanics is to determine the coordinates of a body of known mass and its speed at any moment in time from known forces acting on the body, then when solving the inverse problem it is necessary to determine the forces acting on the body if it is known how it moves.
The solution to this problem led Newton to the discovery of the law of universal gravitation: “All bodies are attracted to each other with a force directly proportional to their masses and inversely proportional to the square of the distance between them.”
There are several important points to make regarding this law.
1, its action explicitly extends to all physical material bodies in the Universe without exception.
2 the gravitational force of the Earth at its surface equally affects all material bodies located anywhere on the globe. Right now the force of gravity is acting on us, and we really feel it as our weight. If we drop something, under the influence of the same force it will uniformly accelerate towards the ground.
The action of universal gravitational forces in nature explains many phenomena: the movement of planets in the solar system, artificial satellites of the Earth - all of them are explained on the basis of the law of universal gravitation and the laws of dynamics.
Newton was the first to express the idea that gravitational forces determine not only the movement of the planets of the solar system; they act between any bodies in the Universe. One of the manifestations of the force of universal gravitation is the force of gravity - this is the common name for the force of attraction of bodies towards the Earth near its surface.
The force of gravity is directed towards the center of the Earth. In the absence of other forces, the body falls freely to the Earth with the acceleration of gravity.
Three principles of mechanics.
Newton's laws of mechanics, three laws underlying the so-called. classical mechanics. Formulated by I. Newton (1687).
First Law: “Every body continues to be maintained in its state of rest or uniform and linear motion until and unless it is compelled by applied forces to change that state.”
Second law: “The change in momentum is proportional to the applied driving force and occurs in the direction of the straight line along which this force acts.”
Third law: “An action always has an equal and opposite reaction, otherwise, the interactions of two bodies on each other are equal and directed in opposite directions.” N. z. m. appeared as a result of a generalization of numerous observations, experiments and theoretical studies of G. Galileo, H. Huygens, Newton himself, and others.
According to modern concepts and terminology, in the first and second laws, a body should be understood as a material point, and motion should be understood as motion relative to an inertial reference system. The mathematical expression of the second law in classical mechanics has the form or mw = F, where m is the mass of a point, u is its speed, and w is acceleration, F is the acting force.
N. z. m. cease to be valid for the movement of objects of very small sizes (elementary particles) and for movements at speeds close to the speed of light
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The interaction of these two effects is the main theme of Newtonian mechanics.
Other important concepts in this branch of physics are energy, momentum, angular momentum, which can be transferred between objects during interaction. The energy of a mechanical system consists of its kinetic (energy of motion) and potential (depending on the position of the body relative to other bodies) energies. Fundamental conservation laws apply to these physical quantities.
1. History
The foundations of classical mechanics were laid by Galileo, as well as Copernicus and Kepler, in the study of the patterns of motion of celestial bodies, and for a long time mechanics and physics were considered in the context of describing astronomical events.
The ideas of the heliocentric system were further formalized by Kepler in his three laws of motion of celestial bodies. In particular, Kepler's second law states that all planets in the solar system move in elliptical orbits, with the Sun as one of their focuses.
The next important contribution to the foundation of classical mechanics was made by Galileo, who, exploring the fundamental laws of mechanical motion of bodies, in particular under the influence of the forces of gravity, formulated five universal laws of motion.
But still, the laurels of the main founder of classical mechanics belong to Isaac Newton, who in his work “Mathematical Principles of Natural Philosophy” carried out a synthesis of those concepts in the physics of mechanical motion that were formulated by his predecessors. Newton formulated three fundamental laws of motion, which were named after him, as well as the law of universal gravitation, which drew a line under Galileo's studies of the phenomenon of free falling bodies. Thus, a new picture of the world and its basic laws was created to replace the outdated Aristotelian one.
2. Limitations of classical mechanics
Classical mechanics provides accurate results for the systems we encounter in everyday life. But they become incorrect for systems whose speed approaches the speed of light, where it is replaced by relativistic mechanics, or for very small systems where the laws of quantum mechanics apply. For systems that combine both of these properties, relativistic quantum field theory is used instead of classical mechanics. For systems with a very large number of components, or degrees of freedom, classical mechanics can also be adequate, but methods of statistical mechanics are used
Classical mechanics is widely used because, firstly, it is much simpler and easier to use than the theories listed above, and, secondly, it has great potential for approximation and application for a very wide class of physical objects, starting with familiar, such as a top or a ball, in great astronomical objects (planets, galaxies) and very microscopic ones (organic molecules).
3. Mathematical apparatus
Basic mathematics classical mechanics- differential and integral calculus, developed specifically for this by Newton and Leibniz. In its classical formulation, mechanics is based on Newton's three laws.
4. Statement of the basics of the theory
The following is a presentation of the basic concepts of classical mechanics. For simplicity, we will use the concept of a material point as an object whose dimensions can be neglected. The movement of a material point is determined by a small number of parameters: position, mass and forces applied to it.
In reality, the dimensions of every object that classical mechanics deals with are non-zero. A material point, such as an electron, obeys the laws of quantum mechanics. Objects with non-zero dimensions have much more complex behavior, because their internal state can change - for example, a ball can also rotate while moving. Nevertheless, the results obtained for material points can be applied to such bodies if we consider them as a collection of many interacting material points. Such complex objects can behave like material points if their sizes are insignificant on the scale of a specific physical problem.
4.1. Position, radius vector and its derivatives
The position of an object (material point) is determined relative to a fixed point in space, which is called the origin. It can be specified by the coordinates of this point (for example, in the Cartesian coordinate system) or by a radius vector r, drawn from the origin to this point. In reality, a material point can move over time, so the radius vector is generally a function of time. In classical mechanics, in contrast to relativistic mechanics, it is believed that the flow of time is the same in all reference systems.
4.1.1. Trajectory
A trajectory is the totality of all positions of a moving material point - in the general case, it is a curved line, the appearance of which depends on the nature of the point’s movement and the chosen reference system.
4.1.2. Moving
.If all forces acting on a particle are conservative, and V is the total potential energy obtained by adding the potential energies of all forces, then
| . |
Those. total energy E = T + V persists over time. This is a manifestation of one of the fundamental physical laws of conservation. In classical mechanics it can be useful practically, because many types of forces in nature are conservative.
Mechanics- is a branch of physics that studies the simplest form of motion of matter - mechanical movement, which consists in changing the position of bodies or their parts over time. The fact that mechanical phenomena occur in space and time is reflected in any law of mechanics that explicitly or implicitly contains space-time relationships - distances and time intervals.
Mechanics sets itself two main tasks:
the study of various movements and generalization of the results obtained in the form of laws with the help of which the nature of movement in each specific case can be predicted. The solution to this problem led to the establishment by I. Newton and A. Einstein of the so-called dynamic laws;
finding general properties inherent in any mechanical system during its movement. As a result of solving this problem, the laws of conservation of such fundamental quantities as energy, momentum and angular momentum were discovered.
Dynamic laws and the laws of conservation of energy, momentum and angular momentum are the basic laws of mechanics and form the content of this chapter.
§1. Mechanical movement: basic concepts
Classical mechanics consists of three main sections - statics, kinematics and dynamics. Statics examines the laws of the addition of forces and the conditions of equilibrium of bodies. Kinematics provides a mathematical description of all kinds of mechanical motion, regardless of the reasons that cause it. Dynamics studies the influence of interaction between bodies on their mechanical motion.
In practice everything physical problems are solved approximately: real complex movement is considered as a set of simple movements, a real object replaced by an idealized model this object, etc. For example, when considering the movement of the Earth around the Sun, the size of the Earth can be neglected. In this case, the description of the movement is greatly simplified - the position of the Earth in space can be determined by one point. Among the models of mechanics, the defining ones are material point and absolutely rigid body.
Material point (or particle)- this is a body whose shape and dimensions can be neglected in the conditions of this problem. Any body can be mentally divided into a very large number of parts, no matter how small compared to the size of the whole body. Each of these parts can be considered as a material point, and the body itself - as a system of material points.
If the deformations of a body during its interaction with other bodies are negligible, then it is described by the model absolutely solid body.
Absolutely rigid body (or rigid body) - this is a body, the distances between any two points of which do not change during movement. In other words, it is a body whose shape and dimensions do not change during its movement. An absolutely rigid body can be considered as a system of material points rigidly connected to each other.
The position of a body in space can only be determined in relation to some other bodies. For example, it makes sense to talk about the position of a planet in relation to the Sun, or an airplane or ship in relation to the Earth, but it is impossible to indicate their positions in space without reference to any specific body. An absolutely rigid body, which serves to determine the position of the object of interest to us, is called a reference body. To describe the movement of an object, some coordinate system is associated with a reference body, for example, a rectangular Cartesian coordinate system. The coordinates of an object allow you to determine its position in space. The smallest number of independent coordinates that must be specified to completely determine the position of a body in space is called the number of degrees of freedom. So, for example, a material point moving freely in space has three degrees of freedom: the point can make three independent movements along the axes of a Cartesian rectangular coordinate system. An absolutely rigid body has six degrees of freedom: to determine its position in space, three degrees of freedom are needed to describe translational motion along the coordinate axes and three to describe rotation about the same axes. To measure time, the coordinate system is equipped with a clock.
The combination of a reference body, a coordinate system associated with it, and a set of clocks synchronized with each other form a reference system.