Classical mechanics (Newtonian mechanics)

The birth of physics as a science is associated with the discoveries of G. Galileo and I. Newton. Particularly significant is the contribution of I. Newton, who wrote down the laws of mechanics in the language of mathematics. His theory, which is often called classical mechanics, I. Newton outlined in his work “Mathematical Principles of Natural Philosophy” (1687).

The basis of classical mechanics is made up of three laws and two provisions regarding space and time.

Before considering I. Newton's laws, let us recall what a reference system and an inertial reference system are, since I. Newton's laws are not satisfied in all reference systems, but only in inertial reference systems.

A reference system is a system of coordinates, for example rectangular Cartesian coordinates, supplemented by a clock located at each point of a geometrically solid medium. A geometrically solid medium is called infinite set points, the distances between which are fixed. In I. Newton's mechanics, it is assumed that time flows regardless of the position of the clock, i.e. The clocks are synchronized and therefore time flows the same in all reference frames.

In classical mechanics, space is considered Euclidean, and time is represented by the Euclidean straight line. In other words, I. Newton considered space absolute, i.e. it is the same everywhere. This means that non-deformable rods with divisions marked on them can be used to measure lengths. Among the reference systems, we can distinguish those systems that, due to taking into account a number of special dynamic properties, differ from the rest.

The reference system in relation to which the body moves uniformly and rectilinearly is called inertial or Galilean.

The fact of the existence of inertial reference systems cannot be verified experimentally, since in real conditions It is impossible to isolate a part of matter, isolate it from the rest of the world so that the movement of this part of matter is not affected by other material objects. To determine in each specific case whether the reference frame can be taken as inertial, it is checked whether the velocity of the body is conserved. The degree of this approximation determines the degree of idealization of the problem.

For example, in astronomy, when studying motion celestial bodies The Cartesian ordinate system is often taken as an inertial reference system, the origin of which is at the center of mass of some “fixed” star, and the coordinate axes are directed to other “fixed” stars. In fact, stars move at high speeds relative to other celestial objects, so the concept of a “fixed” star is relative. But due to long distances between the stars the position we have given is sufficient for practical purposes.

For example, the best inertial reference frame for solar system will be one whose beginning coincides with the center of mass of the Solar system, which is practically located in the center of the Sun, since more than 99% of our mass is concentrated in the Sun planetary system. The coordinate axes of the reference system are directed to distant stars, which are considered stationary. Such a system is called heliocentric.

I. Newton formulated the statement about the existence of inertial reference systems in the form of the law of inertia, which is called Newton’s first law. This law states: Every body is in a state of rest or uniform rectilinear movement until influence from other bodies forces him to change this state.

Newton's first law is by no means obvious. Before G. Galileo, it was believed that this effect does not determine the change in speed (acceleration), but the speed itself. This opinion was based on such well-known everyday life facts, such as the need to continuously push a cart moving along a horizontal, level road so that its movement does not slow down. We now know that by pushing a cart, we balance the force exerted on it by friction. But without knowing this, it is easy to come to the conclusion that the impact is necessary to maintain the movement unchanged.

Newton's second law states: rate of change of particle momentum equal to the force acting on the particle:

Where T- weight; t- time; A-acceleration; v- velocity vector; p = mv- impulse; F- strength.

By force called vector quantity, characterizing the impact on a given body from other bodies. The modulus of this value determines the intensity of the impact, and the direction coincides with the direction of the acceleration imparted to the body by this impact.

Weight is a measure of the inertia of a body. Under inertia understand the intractability of the body to the action of force, i.e. the property of a body to resist a change in speed under the influence of a force. In order to express the mass of a certain body as a number, it is necessary to compare it with the mass reference body, taken as one.

Formula (3.1) is called the equation of particle motion. Expression (3.2) is the second formulation of Newton’s second law: the product of a particle's mass and its acceleration is equal to the force that acts on the particle.

Formula (3.2) is also valid for extended bodies if they move translationally. If several forces act on a body, then under the force F in formulas (3.1) and (3.2) their resultant is implied, i.e. sum of forces.

From (3.2) it follows that when F= 0 (i.e. the body is not affected by other bodies) acceleration A is equal to zero, so the body moves rectilinearly and uniformly. Thus, Newton's first law is, as it were, included in the second law as his special case. But Newton’s first law is formed independently of the second, since it contains a statement about the existence of inertial reference systems in nature.

Equation (3.2) has such a simple form only with a consistent choice of units for measuring force, mass and acceleration. At independent choice units of measurement Newton's second law is written as follows:

Where To - proportionality factor.

The influence of bodies on each other is always in the nature of interaction. In the event that the body A affects the body IN with force FBA then the body IN affects the body And with by force F AB .

Newton's third law states that the forces with which two bodies interact are equal in magnitude and opposite in direction, those.

Therefore, forces always arise in pairs. Note that the forces in formula (3.4) are applied to different bodies, and therefore they cannot balance each other.

Newton's third law, like the first two, is satisfied only in inertial frames of reference. In non-inertial reference systems it is not valid. In addition, deviations from Newton's third law will be observed in bodies that move at speeds close to the speed of light.

It should be noted that all three of Newton's laws appeared as a result of generalization of data large number experiments and observations and are therefore empirical laws.

In Newtonian mechanics, not all reference systems are equal, since inertial and non-inertial reference systems differ from each other. This inequality indicates the lack of maturity of classical mechanics. On the other hand, all inertial frames of reference are equal and in each of them Newton's laws are the same.

G. Galileo in 1636 established that in the inertial frame of reference there are no mechanical experiments it is impossible to determine whether it is at rest or moving uniformly and in a straight line.

Let us consider two inertial frames of reference N And N", and the system jV" moves relative to the system N along the axis X With constant speed v(Fig. 3.1).

Rice. 3.1.

We will start counting time from the moment when the origin of coordinates O and o" coincided. In this case, the coordinates X And X" arbitrarily taken point M will be related by the expression x = x" + vt. With our choice of coordinate axes y - y z~ Z- In Newtonian mechanics it is assumed that in all reference systems time flows the same way, i.e. t = t". Consequently, we received a set of four equations:

Equations (3.5) are called Galilean transformations. They make it possible to move from the coordinates and time of one inertial reference system to the coordinates and time of another inertial reference system. Let us differentiate with respect to time / the first equation (3.5), keeping in mind that t = t therefore the derivative with respect to t will coincide with the derivative with respect to G. We get:

The derivative is the projection of the particle's velocity And in the system N

per axis X of this system, and the derivative is the projection of the particle velocity O"in the system N"on the axis X"of this system. Therefore we get

Where v = v x =v X "- projection of the vector onto the axis X coincides with the projection of the same vector onto the axis*".

Now we differentiate the second and third equations (3.5) and get:

Equations (3.6) and (3.7) can be replaced by one vector equation

Equation (3.8) can be considered either as a formula for converting the particle velocity from the system N" into the system N, or as the law of addition of speeds: the speed of a particle relative to the system Y is equal to the sum of the speed of the particle relative to the system N" and system speed N" relative to the system N. Let us differentiate equation (3.8) with respect to time and obtain:

therefore, particle accelerations relative to systems N and UU are the same. Strength F, N, equal to force F", which acts on a particle in the system N", those.

Relationship (3.10) will be satisfied, since the force depends on the distances between a given particle and the particles interacting with it (as well as on relative speeds particles), and these distances (and velocities) in classical mechanics are assumed to be the same in all inertial frames of reference. The mass also has the same numeric value in all inertial reference systems.

From the above reasoning it follows that if the relation is satisfied ta = F, then the equality will be satisfied ta = F". Reference systems N And N" were taken arbitrarily, so the result means that the laws of classical mechanics are the same for all inertial frames of reference. This statement is called Galileo's principle of relativity. We can say it differently: Newton's laws of mechanics are invariant under Galileo's transformations.

Quantities that have the same numerical value in all reference systems are called invariant (from lat. invariantis- unchanging). Examples of such quantities are electric charge, mass, etc.

Equations whose form does not change during such a transition are also called invariant with respect to the transformation of coordinates and time when moving from one inertial reference system to another. The quantities that enter into these equations may change when moving from one reference system to another, but the formulas that express the relationship between these quantities remain unchanged. Examples of such equations are the laws of classical mechanics.

• By particle we mean a material point, i.e. a body whose dimensions can be neglected compared to the distance to other bodies.

Fundamentals of classical mechanics

Mechanics- a branch of physics that studies the laws of mechanical motion of bodies.

Body– a tangible material object.

Mechanical movement- change provisions body or its parts in space over time.

Aristotle represented this type of motion as a direct change by a body of its place relative to other bodies, since in his physics material world was inextricably linked with space, existing with it. He considered time to be a measure of the movement of a body. Subsequent changes in views on the nature of movement led to the gradual separation of space and time from physical bodies. Finally, absolutization Newton's concept of space and time generally took them beyond the limits of possible experience.

However, this approach allowed end of the XVIII century to build a complete system mechanics, now called classical. Classicism is that it:

1) describes most mechanical phenomena in the macrocosm using a small number of initial definitions and axioms;

2) strictly mathematically justified;

3) is often used in more specific areas of science.

Experience shows that classical mechanics applies to the description of the motion of bodies with speeds v<< с ≈ 3·10 8 м/с. Ее основные разделы:

1) statics studies the conditions of equilibrium of bodies;

2) kinematics - the movement of bodies without taking into account its causes;

3) dynamics - the influence of the interaction of bodies on their movement.

Basic mechanics concepts:

1) A mechanical system is a mentally identified set of bodies that are essential in a given task.

2) A material point is a body whose shape and dimensions can be neglected within the framework of this problem. The body can be represented as a system material points.

3) An absolutely rigid body is a body whose distance between any two points does not change under the conditions of a given problem.

4) The relativity of motion lies in the fact that a change in the position of a body in space can only be established in relation to some other bodies.

5) Reference body (RB) – an absolutely rigid body relative to which motion is considered in this problem.

6) Frame of reference (FR) = (TO + SC + clock). The origin of the coordinate system (OS) is combined with some TO point. Clocks measure periods of time.

Cartesian SK:

Figure 5

Position material point M is described radius vector of the point, – its projections on the coordinate axes.

If you set the initial time t 0 = 0, then the movement of point M will be described vector function or three scalar functions x(t),y(t), z(t).

Linear characteristics of the movement of a material point:

1) trajectory – line of motion of a material point (geometric curve),

2) path ( S) – the distance traveled along it in a period of time,

3) moving,

4) speed,

5) acceleration.

Any movement solid can be reduced to two main types - progressive And rotational around a fixed axis.

Forward movement- one in which the straight line connecting any two points of the body remains parallel to its original position. Then all points move equally, and the movement of the whole body can be described movement of one point.

Rotation around a fixed axis - a movement in which there is a straight line rigidly connected to the body, all points of which remain motionless in a given reference frame. The trajectories of the remaining points are circles with centers on this line. In this case it is convenient angular characteristics movements that are the same for all points of the body.

Angular characteristics of the movement of a material point:

1) angle of rotation (angular path), measured in radians [rad], where r– radius of the point’s trajectory,

2) angular displacement, the module of which is the angle of rotation over a short period of time dt,

3) angular velocity,

4) angular acceleration.

Figure 6

Relationship between angular and linear characteristics:

Dynamics uses concept of strength, measured in newtons (H), as a measure of the influence of one body on another. This impact is the cause of movement.

The principle of superposition of forces– the resulting effect of the influence of several bodies on a body is equal to the sum of the effects of the influence of each of these bodies separately. The quantity is called the resultant force and characterizes the equivalent effect on the body n tel.

Newton's laws generalize experimental facts of mechanics.

Newton's 1st law. There are reference systems relative to which a material point maintains a state of rest or uniform rectilinear motion in the absence of force acting on it, i.e. if , then .

Such motion is called motion by inertia or inertial motion, and therefore frames of reference in which Newton's 1st law is satisfied are called inertial(ISO).

Newton's 2nd law. , where is the momentum of the material point, m– its mass, i.e. if , then and, consequently, the movement will no longer be inertial.

Newton's 3rd law. When two material points interact, forces arise and are applied to both points, and .

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 movement.

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 displacement, 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:

Thus, 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 the moment of time t 1 corresponds to the radius vector , and , then during the interval the body will receive displacement . In this case average speedt is the quantity

which, in relation to the trajectory, represents a secant passing through points I and 2. Speed at time t is called a vector

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:

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 of time t 1 is the speed of the point, and at t 2 - , then the speed increment will be (Fig. 1.2). The average acceleration in this case

and instantaneous

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. The speed changes in magnitude and direction, the speed increment is divided into two quantities; - directed along (increase in speed in magnitude) and - directed perpendicularly (increment in speed in direction), i.e. = + (Fig. I.З). From (1.9) we obtain:

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:

5. Examples

I. Equally variable linear motion. This is motion 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:

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

If when moving m= const , That

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, where k basis classical mechanics Newton's laws lie. Kinematics studies...

Mechanics is the study of balance and movement of bodies (or their parts) in space and time. Mechanical movement is the simplest and at the same time (for humans) the most common form of existence of matter. Therefore, mechanics is exclusively important place in natural science and is the main subsection of physics. It historically arose and was formed as a science earlier than other subfields of natural science.

Mechanics includes statics, kinematics and dynamics. In statics the conditions of equilibrium of bodies are studied, in kinematics - the movements of bodies with geometric point vision, i.e. without taking into account the action of forces, and in dynamics - taking into account these forces. Statics and kinematics are often considered as an introduction to dynamics, although they also have independent significance.

Until now, by mechanics we meant classical mechanics, the construction of which was completed by the beginning of the 20th century. Within modern physics There are two more mechanics - quantum and relativistic. But we will look at classical mechanics in more detail.

Classical mechanics considers the movement of bodies at speeds much less than the speed of light. According to the special theory of relativity, for bodies moving at high speeds close to the speed of light, absolute time and absolute space do not exist. Hence, the nature of the interaction of bodies becomes more complex, in particular, the mass of a body turns out to depend on the speed of its movement. All this was the subject of consideration relativistic mechanics, for which the speed of light constant plays a fundamental role.

Classical mechanics is based on the following basic laws.

Galileo's principle of relativity

According to this principle, there are infinitely many reference frames in which free body rests or moves with a constant speed in magnitude and direction. These reference systems are called inertial and move relative to each other uniformly and rectilinearly. This principle can also be formulated as the absence of absolute reference systems, that is, reference systems that are in any way distinguished relative to others.

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

• 1. Stuff material body maintains a state of rest or uniform rectilinear motion until influence from other bodies forces it to change this state. The desire of a body to maintain a state of rest or uniform linear motion is called inertia. Therefore, the first law is also called the law of inertia.
• 2. The acceleration acquired by a body is directly proportional to the force acting on the body and inversely proportional to the mass of the body.
• 3. The forces with which interacting bodies act on each other are equal in magnitude and opposite in direction.

Newton's second law is known to us as

natural science classical mechanics law

F = m H a, or a = F/m,

where the acceleration a received by a body under the action of a force F is inversely proportional to the mass of the body m.

The first law can be obtained from the second, since in the absence of influence on the body from other forces, the acceleration is also zero. However, the first law is considered as independent law, since it asserts the existence of inertial frames of reference. IN mathematical formulation Newton's second law is most often written as follows:

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

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

Law of conservation of momentum

This law is a consequence of Newton's laws for closed systems, that is, systems that are not affected by external forces or actions external forces are compensated and the resulting force is zero. From a more fundamental point of view, there is a relationship between the law of conservation of momentum and homogeneity of space, expressed by Noether's theorem.

Law of Conservation of Energy

The law of conservation of energy is a consequence of Newton's laws for closed conservative systems, that is, systems in which only conservative forces act. The energy given by one body to another is always equal to the energy received by the other body. For quantification The process of energy exchange between interacting bodies in mechanics introduces the concept of the work of a force that causes movement. The force causing the movement of a body does work, and the energy of a moving body increases by the amount of work expended. As is known, a body of mass m moving with speed v has kinetic energy

Potential energy is mechanical energy systems of bodies that interact through force fields, such as gravitational forces. The work done by these forces when moving a body from one position to another does not depend on the trajectory of movement, but depends only on the initial and final position of the body in the force field. Gravitational forces are conservative forces, A potential energy of a body of mass m raised to a height h above the Earth's surface is equal to

E sweat = mgh,

where g is the acceleration of gravity.

Total mechanical energy is equal to the sum of kinetic and potential energy.

LECTURE 1

INTRODUCTION TO CLASSICAL MECHANICS

Classical mechanics studies the mechanical motion of macroscopic objects that move at speeds much less than the speed of light (=3 10 8 m/s). Macroscopic objects are understood as objects whose dimensions are m (on the right is the size of a typical molecule).

Physical theories that study systems of bodies whose movement occurs at speeds much lower than the speed of light are classified as non-relativistic theories. If the speeds of the particles of the system are comparable to the speed of light, then such systems belong to relativistic systems, and they must be described on the basis of relativistic theories. The basis of all relativistic theories is special theory relativity (SRT). If the sizes of the studied physical objects are small, then such systems belong to quantum systems, and their theories belong to the quantum theories.

Thus, classical mechanics should be considered as a non-relativistic, non-quantum theory of particle motion.

1.1 Frames of reference and principles of invariance

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

Space in classical mechanics is considered three-dimensional (to determine the position of a particle in space, three coordinates must be specified), subject to Euclidean geometry (the Pythagorean theorem is valid in space) and absolute. Time is one-dimensional, unidirectional (changing from past to future) and absolute. The absoluteness of space and time means that their properties do not depend on the distribution and movement of matter. In classical mechanics it is accepted as fair next statement: Space and time are not related to each other and can be considered independently of each other.

The motion is relative and, therefore, to describe it it is necessary to choose reference body, i.e. the body relative to which motion is considered. Since movement occurs in space and time, to describe it one should choose one or another coordinate system and clock (arithmetize space and time). Due to the three-dimensionality of space, each of its points is associated with three numbers (coordinates). The choice of one or another coordinate system is usually dictated by the condition and symmetry of the problem at hand. IN theoretical reasoning we will usually use a rectangular Cartesian coordinate system (Figure 1.1).

In classical mechanics, to measure time intervals, due to the absoluteness of time, it is sufficient to have one clock placed at the origin of the coordinate system (this issue will be discussed in detail in the theory of relativity). The reference body and the clocks and scales (coordinate system) associated with this body form reference system.

Let us introduce the concept of a closed physical system. Closed physical system is a system of material objects in which all objects of the system interact with each other, but do not interact with objects that are not part of the system.

As experiments show, in relation to a number of reference systems they turn out to be valid the following principles invariance.

The principle of invariance with respect to spatial shifts(space is homogeneous): the flow of processes inside a closed physical system is not affected by its position relative to the reference body.

The principle of invariance under spatial rotations(space is isotropic): the flow of processes inside a closed physical system is not affected by its orientation relative to the reference body.

The principle of invariance with respect to time shifts(time is uniform): the course of processes within a closed physical system is not affected by the time when the processes begin.

The principle of invariance under mirror reflections(space is mirror-symmetric): processes occurring in closed mirror-symmetric physical systems are themselves mirror-symmetric.

Those reference systems in relation to which space is homogeneous, isotropic and mirror-symmetrical and time is homogeneous are called inertial systems countdown(ISO).

Newton's first law claims that ISOs exist.

There is not one, but an infinite number of ISOs. The reference system that moves relative to the ISO rectilinearly and uniformly will itself be the ISO.

The principle of relativity states that the course of processes in a closed physical system is not affected by its linear uniform motion relative to the reference system; the laws describing the processes are the same in different ISOs; the processes themselves will be the same if the initial conditions are the same.

1.2 Basic models and sections of classical mechanics

In classical mechanics, when describing real physical systems, a number of abstract concepts, which are answered by real physical objects. The main concepts include: closed physical system, material point (particle), absolutely rigid body, continuous medium and a number of others.

Material point (particle)– body, size and internal structure which can be neglected when describing its movement. Moreover, each particle is characterized by its own specific set of parameters - mass, electric charge. The material point model does not consider structural internal characteristics particles: moment of inertia, dipole moment, intrinsic moment (spin), etc. The position of a particle in space is characterized by three numbers (coordinates) or a radius vector (Fig. 1.1).

Absolutely rigid body

A system of material points, the distances between which do not change during their movement;

A body whose deformations can be neglected.

Real physical process viewed as a continuous sequence elementary events.

Elementary event is a phenomenon with zero spatial extent and zero duration (for example, a bullet hitting a target). An event is characterized by four numbers – coordinates; three spatial coordinates (or radius - vector) and one time coordinate: . The movement of a particle is represented as a continuous sequence of the following elementary events: the passage of a particle through this point space at a given time.

The law of particle motion is considered given if the dependence of the radius vector of the particle (or its three coordinates) on time is known:

Depending on the type of objects being studied, classical mechanics is divided into mechanics of particles and systems of particles, mechanics of an absolutely rigid body, mechanics continuum(mechanics elastic bodies, hydromechanics, aeromechanics).

According to the nature of the problems being solved, classical mechanics is divided into kinematics, dynamics and statics. Kinematics studies the mechanical movement of particles without taking into account the reasons, causing change the nature of the movement of particles (forces). The law of motion of particles of the system is considered given. According to this law, the velocities, accelerations, and trajectories of movement of particles in the system are determined in kinematics. Dynamics considers the mechanical movement of particles taking into account the reasons causing a change in the nature of the movement of particles. The forces acting between the particles of the system and on the particles of the system from bodies not included in the system are considered known. The nature of forces in classical mechanics is not discussed. Statics can be considered as a special case of dynamics, where the conditions are studied mechanical balance particles of the system.

According to the method of describing systems, mechanics is divided into Newtonian and analytical mechanics.

1.3 Event coordinate transformations

Let's consider how the coordinates of events are transformed when moving from one ISO to another.

1. Spatial shift. IN in this case the transformations look like this:

Where is the spatial shift vector, which does not depend on the event number (index a).

2. Time shift:

Where is the time shift.

3. Spatial rotation:

Where is the vector of infinitesimal rotation (Fig. 1.2).

4. Time inversion (time reversal):

5. Spatial inversion (reflection at a point):

6. Galileo's transformations. We consider the transformation of the coordinates of events during the transition from one ISO to another, which moves relative to the first rectilinearly and uniformly with speed (Fig. 1.3):

Where is the second ratio postulated(!) and expresses the absoluteness of time.

Differentiating with respect to time the right and left side transformation of spatial coordinates taking into account the absolute nature of time, using the definition speed, as the derivative of the radius vector with respect to time, the condition that =const, we obtain classical law speed addition

Here we should especially pay attention to the fact that when deriving the last relation necessary take into account the postulate about the absolute nature of time.

Rice. 1.2 Fig. 1.3

Differentiating with respect to time again using the definition acceleration, as a derivative of speed with respect to time, we obtain that the acceleration is the same with respect to different ISOs (invariant with respect to Galilean transformations). This statement mathematically expresses the principle of relativity in classical mechanics.

WITH mathematical point In terms of transformations 1-6 form a group. Really, this group contains a single transformation – identity transformation, corresponding to the absence of transition from one system to another; for each of the transformations 1-6 there is inverse conversion, which returns the system to its initial state. The operation of multiplication (composition) is introduced as a sequential application of the corresponding transformations. It should be especially noted that the group of rotation transformations does not obey the commutative (commutation) law, i.e. is non-Abelian. Full group transformations 1-6 are called the Galilean group of transformations.

1.4 Vectors and scalars

Vector called physical quantity, which is transformed as the radius vector of the particle and is characterized by its numerical value and direction in space. With respect to the operation of spatial inversion, vectors are divided into true(polar) and pseudovectors(axial). During spatial inversion, the true vector changes its sign, the pseudovector does not change.

Scalars characterized only by their numerical value. With respect to the operation of spatial inversion, scalars are divided into true And pseudoscalars. During spatial inversion, the true scalar does not change, but the pseudoscalar changes its sign.

Examples. The radius vector, velocity, and acceleration of a particle are true vectors. Rotation angle vectors, angular velocity, angular acceleration– pseudovectors. The cross product of two true vectors is a pseudovector, vector product true vector to pseudovector – true vector. Dot product two true vectors - a true scalar, a true vector per pseudovector - a pseudoscalar.

It should be noted that in a vector or scalar equality, terms on the right and left must be of the same nature in relation to the operation of spatial inversion: true scalars or pseudoscalars, true vectors or pseudovectors.