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 Thus, a new picture of the world and its basic laws was created to replace the outdated Aristotelian one.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.

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 movement as a direct change by a body of its place relative to other bodies, since in his physics the material world was inextricably linked with space and existed together 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 made it possible by the end of the 18th 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. A body can be represented as a system of 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 motion of a rigid body 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 .

Definition 1

Mechanics is an extensive branch of physics that studies the laws of changing the positions of physical bodies in space and time, as well as postulates based on Newton's laws.

Figure 1. Basic law of dynamics. Author24 - online exchange of student works

Often this scientific direction of physics is called “Newtonian mechanics”. Classical mechanics today is divided into the following sections:

• statics - examines and describes the balance of bodies;
• kinematics - studies the geometric features of movement without considering its causes;
• dynamics – studies the movement of material substances.

Mechanical movement is one of the simplest and at the same time the most common form of existence of living matter. Therefore, classical mechanics occupies an extremely significant place in natural science and is considered the main subsection of physics.

Basic laws of classical mechanics

Classical mechanics in its postulates studies the movement of working bodies at speeds that are much less than the speed of light. According to the special hypothesis of relativity, absolute space and time do not exist for elements moving at enormous speed. As a result, the nature of the interaction of substances becomes more complex, in particular, their mass begins to depend on the speed of movement. All this became the object of consideration of the formulas of relativistic mechanics, for which the light speed constant plays a fundamental role.

Classical mechanics is based on the following basic laws.

1. Galileo's principle of relativity. According to this principle, there are many reference systems in which any free body is at rest or moves with a constant speed in direction. These concepts in science are called inertial, and they move relative to each other rectilinearly and uniformly.
2. Newton's three laws. The first establishes the obligatory presence of the property of inertia in physical bodies and postulates the presence of such concepts of reference in which the movement of free matter occurs at a constant speed. The second postulate introduces the concept of force as the main measure of the interaction of active elements and, based on theoretical facts, postulates the relationship between the acceleration of a body, its size and inertia. Newton's third law - for every force acting on the first body there is a counteracting factor, equal in magnitude and opposite in direction.
3. The law of conservation of internal energy is a consequence of Newton's laws for stable, closed systems in which exclusively conservative forces act. The total mechanical force of a closed system of material bodies, between which only thermal energy acts, remains constant.

Parallelogram rules in mechanics

Certain consequences follow from Newton's three fundamental theories of body motion, one of which is the addition of the total number of elements according to the parallelogram rule. According to this idea, the acceleration of any physical substance depends on quantities that mainly characterize the action of other bodies, which determine the features of the process itself. The mechanical action on the object under study from the external environment, which radically changes the speed of movement of several elements at once, is called force. It can be multifaceted in nature.

In classical mechanics, which deals with speeds significantly lower than the speed of light, mass is considered one of the main characteristics of the body itself, regardless of whether it is moving or at rest. The mass of a physical body is independent of the interaction of the substance with other parts of the system.

Note 1

Thus, mass gradually came to be understood as the amount of living matter.

Establishing the concepts of mass and force, as well as the method of measuring them, allowed Newton to describe and formulate the second law of classical mechanics. So, mass is one of the key characteristics of matter, determining its gravitational and inertial properties.

The first and second principles of mechanics refer respectively to the systematic motion of a single body or material point. In this case, only the effect of other elements in a certain concept is taken into account. However, any physical action is an interaction.

The third law of mechanics already fixes this statement and states: an action always corresponds to an oppositely directed and equal reaction. In Newton's formulation, this postulate of mechanics is valid only for the case of a direct relationship of forces or when the action of one material body is suddenly transferred to another. In the case of movement over a long period of time, the third law applies when the time of transfer of action can be neglected.

In general, all the laws of classical mechanics are valid for the functioning of inertial reference systems. In the case of non-inertial concepts the situation is completely different. With accelerated movement of coordinates relative to the inertial system itself, Newton's first law cannot be used - free bodies in it will change their speed of movement over time and depend on the speed of movement and energy of other substances.

Limits of applicability of the laws of classical mechanics

Figure 3. Limits of applicability of the laws of classical mechanics. Author24 - online exchange of student works

As a result of the rather rapid development of physics at the beginning of the 20th century, a certain scope of application of classical mechanics was formed: its laws and postulates are fulfilled for the movements of physical bodies whose speed is significantly less than the speed of light. It was determined that with increasing speed, the mass of any substance will automatically increase.

The inconsistency of principles in classical mechanics was mainly based on the fact that the future, in a certain sense, is completely in the present - this determines the probability of accurately predicting the behavior of a system at any period of time.

Note 2

The Newtonian method immediately became the main tool for understanding the essence of nature and all life on the planet. The laws of mechanics and methods of mathematical analysis soon showed their effectiveness and significance. The physical experiment, which was based on measuring technology, provided scientists with unprecedented accuracy.

Physical knowledge increasingly became a central industrial technology, stimulating the general development of other important natural sciences.

In physics, all previously isolated electricity, light, magnetism and heat became whole and combined into the electromagnetic hypothesis. And although the nature of gravity itself remained uncertain, its actions could be calculated. The concept of Laplace's mechanistic determinism was established and implemented, which is based on the ability to accurately determine the behavior of bodies at any time if the initial conditions are initially determined.

The structure of mechanics as a science seemed quite reliable and solid, and also almost complete. As a result, the impression was that the knowledge of physics and its laws was close to its end - the foundation of classical physics showed such a powerful force.

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. I. Newton outlined his theory, which is often called classical mechanics, 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 coordinate system, for example, rectangular Cartesian coordinates, supplemented by a clock located at each point of a geometrically solid medium. A geometrically solid medium is an infinite set of 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 and 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 the motion of 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 the large distances between the stars, the position given by us is sufficient for practical purposes.

For example, the best inertial reference system for the Solar System will be one whose origin coincides with the center of mass of the Solar System, which is practically located at the center of the Sun, since more than 99% of the mass of our planetary system is concentrated in the Sun. 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 motion until the influence of other bodies forces it 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 facts known from everyday life, 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 is called a vector quantity that characterizes the influence 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 of the reference body, taken as a unit.

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 its 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. With an independent choice of 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 from a large number of 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 an inertial frame of reference, no mechanical experiments can determine whether it is at rest or moving uniformly and rectilinearly.

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 at 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.

Relation (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 the relative velocities of the particles), and these distances (and velocities) in classical mechanics are assumed to be the same in all inertial frames of reference. Mass also has the same numerical value in all inertial frames of reference.

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.

This is a branch of physics that studies motion based on Newton's laws. Classical mechanics is divided into:
The basic concepts of classical mechanics are the concepts of force, mass and motion. Mass in classical mechanics is defined as a measure of inertia, or the ability of a body to maintain a state of rest or uniform linear motion in the absence of forces acting on it. On the other hand, forces acting on a body change the state of its motion, causing acceleration. The interaction of these two effects is the main theme of Newtonian mechanics.
Other important concepts in this branch of physics are energy, momentum, and 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.
The foundations of classical mechanics were laid by Galileo, as well as Copernicus and Kepler, in the study of the laws of motion of celestial bodies, and for a long time mechanics and physics were considered in the context of astronomical events.
In his works, Copernicus noted that the calculation of the patterns of movement of celestial bodies can be significantly simplified if we move away from the principles laid down by Aristotle and consider the Sun, and not the Earth, as the starting point for such calculations, i.e. make the transition from geocentric to heliocentric systems.
The ideas of the heliocentric system were further formalized by Kepler in his three laws of motion of celestial bodies. In particular, it followed from the second law that all planets of 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 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 of its basic laws was created to replace the outdated Aristotelian one.
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, instead of classical mechanics, both characteristics are characterized by quantum field theory. For systems with a very large number of components, or degrees of freedom, classical mechanics may also be adequate, but methods of statistical mechanics are used
Classical mechanics is preserved because, firstly, it is much easier to use than other theories, and, secondly, it has great possibilities for approximation and application for a very wide class of physical objects, starting with the usual ones, such as a top or a ball , many astronomical objects (planets, galaxies) and very microscopic ones).
Although classical mechanics is broadly compatible with other classical theories such as classical electrodynamics and thermodynamics, there are some inconsistencies between these theories that were discovered in the late 19th century. They can be solved by methods of more modern physics. In particular, classical electrodynamics predicts that the speed of light is constant, which is incompatible with classical mechanics and led to the creation of special relativity. The principles of classical mechanics are considered together with the statements of classical thermodynamics, which leads to the Gibbs paradox, according to which it is impossible to accurately determine the value of entropy, and to the ultraviolet catastrophe, in which a black body must radiate an infinite amount of energy. Quantum mechanics was created to overcome these inconsistencies.
Objects that are studied by mechanics are called mechanical systems. The task of mechanics is to study the properties of mechanical systems, in particular their evolution over time.
The basic mathematical apparatus of classical mechanics is differential and integral calculus, developed specifically for this by Newton and Leibniz. In its classical formulation, mechanics is based on Newton's three laws.
The following is a presentation of the basic concepts of classical mechanics. For simplicity, we will consider only the material point of the object, the dimensions of which can be neglected. The movement of a material point is characterized by several parameters: its position, mass, and forces applied to it.
In reality, the dimensions of every object that classical mechanics deals with are non-zero. Material points, such as an electron, obey the laws of quantum mechanics. Objects of non-zero size can experience more complex movements, since their internal state can change, for example, a ball can also rotate. However, for such bodies the results are obtained for material points, considering them as aggregates of a large number of interacting material points. Such complex bodies behave like material points if they are small on the scale of the problem under consideration.
Radius vector and its derivatives
The position of a material point object is determined relative to a fixed point in space called the origin. It can be specified by the coordinates of this point (for example, in a rectangular 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.
Trajectory
A trajectory is the totality of all positions of a material point moving in the general case; it is a curved line, the shape of which depends on the nature of the point’s movement and the chosen reference system.
Moving
Displacement is a vector connecting the initial and final positions of a material point.
Speed
Speed, or the ratio of movement to the time during which it occurs, is defined as the first derivative of movement to time:

In classical mechanics, speeds can be added and subtracted. For example, if one car is traveling west at a speed of 60 km/h, and catches up with another, which is moving in the same direction at a speed of 50 km/h, then relative to the second car, the first one is moving west at a speed of 60-50 = 10 km/h But in the future, fast cars move slower at a speed of 10 km/h to the east.
To determine the relative speed, in any case, the rules of vector algebra are applied to construct speed vectors.
Acceleration
Acceleration, or the rate of change of speed, is the derivative of speed to time or the second derivative of displacement to time:

The acceleration vector can change in magnitude and direction. In particular, if the speed decreases, sometimes acceleration and deceleration, but in general any change in speed.
Strength. Newton's second law
Newton's second law states that the acceleration of a material point is directly proportional to the force acting on it, and the acceleration vector is directed along the line of action of this force. In other words, this law relates the force that acts on a body with its mass and acceleration. Then Newton's second law looks like this:

Magnitude m v called impulse. Usually, mass m does not change with time, and Newton's law can be written in a simplified form

Where A acceleration, which was defined above. Body weight m Not always over time. For example, the mass of a rocket decreases as fuel is used. Under such circumstances, the last expression does not apply, and the full form of Newton's second law must be used.
Newton's second law is not enough to describe the motion of a particle. It requires determining the force that acts on it. For example, a typical expression for the friction force when a body moves in a gas or liquid is defined as follows:

Where? some constant called the friction coefficient.
After all the forces have been determined, based on Newton’s second law, we obtain a differential equation called the equation of motion. In our example with only one force acting on the particle, we get:

Integrating, we get:

Where is the initial speed. This means that the speed of our object decreases exponentially to zero. This expression in turn can be integrated again to obtain an expression for the radius vector r of the body as a function of time.
If several forces act on a particle, then they are added according to the rules of vector addition.
Energy
If strength F acts on a particle, which as a result moves to? r, then the work performed is equal to:

If the mass of the particle has become, then the yearning work done with all forces, from Newton’s second law

Where T kinetic energy. For a material point it is defined as

For complex objects consisting of many particles, the kinetic energy of the body is equal to the sum of the kinetic energies of all particles.
A special class of conservative forces can be expressed by the gradient of a scalar function known as potential energy V:

If all forces acting on a particle are conservative, and V 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.
Newton's laws have several important consequences for rigid bodies (see angular momentum)
There are also two important alternative formulations of classical mechanics: Lagrange mechanics and Hamiltonian mechanics. They are equivalent to Newtonian mechanics, but are sometimes useful for analyzing certain problems. They, like other modern formulations, do not use the concept of force, instead referring to other physical quantities such as energy.