Terms and definitions of dynamics theoretical mechanics. Theoretical mechanics

General theorems of the dynamics of a system of bodies. Theorems on the motion of the center of mass, on the change in the momentum, on the change in the main moment of the momentum, on the change in kinetic energy. Principles of d'Alembert, and possible displacements. General equation of dynamics. Lagrange's equations.

General theorems of rigid body dynamics and systems of bodies

General theorems of dynamics- this is a theorem on the movement of the center of mass of a mechanical system, a theorem on a change in the momentum, a theorem on a change in the main moment of the momentum (kinetic moment) and a theorem on a change in the kinetic energy of a mechanical system.

Theorem on the motion of the center of mass of a mechanical system

The theorem on the motion of the center of mass.
The product of the mass of the system and the acceleration of its center of mass is equal to the vector sum of all external forces acting on the system:
.

Here M is the mass of the system:
;
a C - acceleration of the center of mass of the system:
;
v C - speed of the center of mass of the system:
;
r C - radius vector (coordinates) of the center of mass of the system:
;
- coordinates (with respect to the fixed center) and masses of points that make up the system.

Theorem on the change in momentum (momentum)

The amount of motion (momentum) of the system is equal to the product of the mass of the entire system and the speed of its center of mass or the sum of the momentum (sum of impulses) of individual points or parts that make up the system:
.

Theorem on the change in momentum in differential form.
The time derivative of the amount of motion (momentum) of the system is equal to the vector sum of all external forces acting on the system:
.

Theorem on the change in momentum in integral form.
The change in the amount of motion (momentum) of the system for a certain period of time is equal to the sum of the impulses of external forces for the same period of time:
.

The law of conservation of momentum (momentum).
If the sum of all external forces acting on the system is zero, then the momentum vector of the system will be constant. That is, all its projections on the coordinate axes will retain constant values.

If the sum of the projections of external forces on any axis is equal to zero, then the projection of the momentum of the system on this axis will be constant.

Theorem on the change in the main moment of momentum (theorem of moments)

The main moment of the amount of motion of the system relative to a given center O is the value equal to the vector sum of the moments of the quantities of motion of all points of the system relative to this center:
.
Here square brackets denote the vector product.

Fixed systems

The following theorem refers to the case when the mechanical system has a fixed point or axis, which is fixed with respect to the inertial reference frame. For example, a body fixed with a spherical bearing. Or a system of bodies moving around a fixed center. It can also be a fixed axis around which a body or system of bodies rotates. In this case, the moments should be understood as the moments of impulse and forces relative to the fixed axis.

Theorem on the change in the main moment of momentum (theorem of moments)
The time derivative of the main moment of the momentum of the system with respect to some fixed center O is equal to the sum of the moments of all external forces of the system with respect to the same center.

The law of conservation of the main moment of momentum (moment of momentum).
If the sum of the moments of all external forces applied to the system relative to a given fixed center O is equal to zero, then the main moment of the system's momentum relative to this center will be constant. That is, all its projections on the coordinate axes will retain constant values.

If the sum of the moments of external forces about some fixed axis is equal to zero, then the moment of momentum of the system about this axis will be constant.

Arbitrary systems

The following theorem has a universal character. It is applicable to both fixed systems and freely moving ones. In the case of fixed systems, it is necessary to take into account the reactions of the bonds at the fixed points. It differs from the previous theorem in that the center of mass C of the system should be taken instead of the fixed point O.

Theorem of moments about the center of mass
The time derivative of the main angular momentum of the system about the center of mass C is equal to the sum of the moments of all external forces of the system about the same center.

Law of conservation of angular momentum.
If the sum of the moments of all external forces applied to the system about the center of mass C is equal to zero, then the main moment of the system's momentum about this center will be constant. That is, all its projections on the coordinate axes will retain constant values.

moment of inertia of the body

If the body rotates around the z axis with an angular velocity ω z , then its angular momentum (kinetic moment) relative to the z-axis is determined by the formula:
L z = J z ω z ,
where J z is the moment of inertia of the body about the z axis.

Moment of inertia of the body about the z-axis is determined by the formula:
,
where h k is the distance from a point of mass m k to the z axis.
For a thin ring of mass M and radius R or a cylinder whose mass is distributed along its rim,
J z = M R 2 .
For a solid homogeneous ring or cylinder,
.

The Steiner-Huygens theorem.
Let Cz be the axis passing through the center of mass of the body, Oz be the axis parallel to it. Then the moments of inertia of the body about these axes are related by the relation:
J Oz = J Cz + M a 2 ,
where M is the body weight; a - distance between axles.

More generally:
,
where is the inertia tensor of the body.
Here is a vector drawn from the center of mass of the body to a point with mass m k .

Kinetic energy change theorem

Let a body of mass M perform translational and rotational motion with an angular velocity ω around some axis z. Then the kinetic energy of the body is determined by the formula:
,
where v C is the speed of movement of the center of mass of the body;
J Cz - moment of inertia of the body about the axis passing through the center of mass of the body parallel to the axis of rotation. The direction of the axis of rotation can change over time. This formula gives the instantaneous value of the kinetic energy.

Theorem on the change in the kinetic energy of the system in differential form.
The differential (increment) of the kinetic energy of the system during some of its displacement is equal to the sum of the differentials of work on this displacement of all external and internal forces applied to the system:
.

Theorem on the change in the kinetic energy of the system in integral form.
The change in the kinetic energy of the system during some of its displacement is equal to the sum of the work on this displacement of all external and internal forces applied to the system:
.

The work done by the force, is equal to the scalar product of the force vectors and the infinitesimal displacement of the point of its application :
,
that is, the product of the modules of the vectors F and ds and the cosine of the angle between them.

The work done by the moment of force, is equal to the scalar product of the vectors of the moment and the infinitesimal angle of rotation :
.

d'Alembert principle

The essence of d'Alembert's principle is to reduce the problems of dynamics to the problems of statics. To do this, it is assumed (or it is known in advance) that the bodies of the system have certain (angular) accelerations. Next, the forces of inertia and (or) moments of inertia forces are introduced, which are equal in magnitude and reciprocal in direction to the forces and moments of forces, which, according to the laws of mechanics, would create given accelerations or angular accelerations

Consider an example. The body makes a translational motion and external forces act on it. Further, we assume that these forces create an acceleration of the center of mass of the system . According to the theorem on the movement of the center of mass, the center of mass of a body would have the same acceleration if a force acted on the body. Next, we introduce the force of inertia:
.
After that, the task of dynamics is:
.
;
.

For rotational movement proceed in a similar way. Let the body rotate around the z axis and external moments of forces M e zk act on it. We assume that these moments create an angular acceleration ε z . Next, we introduce the moment of inertia forces M И = - J z ε z . After that, the task of dynamics is:
.
Turns into a static task:
;
.

The principle of possible movements

The principle of possible displacements is used to solve problems of statics. In some problems, it gives a shorter solution than writing equilibrium equations. This is especially true for systems with connections (for example, systems of bodies connected by threads and blocks), consisting of many bodies

The principle of possible movements.
For the equilibrium of a mechanical system with ideal constraints, it is necessary and sufficient that the sum of the elementary works of all active forces acting on it for any possible displacement of the system be equal to zero.

Possible system relocation- this is a small displacement, at which the connections imposed on the system are not broken.

Perfect Connections- these are bonds that do not do work when the system is moved. More precisely, the sum of work performed by the links themselves when moving the system is zero.

General equation of dynamics (d'Alembert - Lagrange principle)

The d'Alembert-Lagrange principle is a combination of the d'Alembert principle with the principle of possible displacements. That is, when solving the problem of dynamics, we introduce the forces of inertia and reduce the problem to the problem of statics, which we solve using the principle of possible displacements.

d'Alembert-Lagrange principle.
When a mechanical system moves with ideal constraints at each moment of time, the sum of elementary works of all applied active forces and all inertia forces on any possible displacement of the system is equal to zero:
.
This equation is called general equation of dynamics.

Lagrange equations

Generalized coordinates q 1 , q 2 , ..., q n is a set of n values ​​that uniquely determine the position of the system.

The number of generalized coordinates n coincides with the number of degrees of freedom of the system.

Generalized speeds are the derivatives of the generalized coordinates with respect to time t.

Generalized forces Q 1 , Q 2 , ..., Q n .
Consider a possible displacement of the system, in which the coordinate q k will receive a displacement δq k . The rest of the coordinates remain unchanged. Let δA k be the work done by external forces during such a displacement. Then
δA k = Q k δq k , or
.

If, with a possible displacement of the system, all coordinates change, then the work done by external forces during such a displacement has the form:
δA = Q 1 δq 1 + Q 2 δq 2 + ... + Q n δq n.
Then the generalized forces are partial derivatives of the displacement work:
.

For potential forces with potential Π,
.

Lagrange equations are the equations of motion of a mechanical system in generalized coordinates:

Here T is the kinetic energy. It is a function of generalized coordinates, velocities, and possibly time. Therefore, its partial derivative is also a function of generalized coordinates, velocities, and time. Next, you need to take into account that the coordinates and velocities are functions of time. Therefore, to find the total time derivative, you need to apply the rule of differentiation of a complex function:
.

References:
S. M. Targ, Short Course in Theoretical Mechanics, Higher School, 2010.

Consider the motion of a certain system of material volumes relative to a fixed coordinate system. When the system is not free, then it can be considered as free, if we discard the constraints imposed on the system and replace their action with the corresponding reactions.

Let us divide all the forces applied to the system into external and internal ones; both may include reactions of discarded

connections. Denote by and the main vector and the main moment of external forces relative to point A.

1. Theorem on the change in momentum. If is the momentum of the system, then (see )

i.e., the theorem is valid: the time derivative of the momentum of the system is equal to the main vector of all external forces.

Replacing the vector through its expression where is the mass of the system, is the velocity of the center of mass, equation (4.1) can be given a different form:

This equality means that the center of mass of the system moves as a material point whose mass is equal to the mass of the system and to which a force is applied that is geometrically equal to the main vector of all external forces of the system. The last statement is called the theorem on the motion of the center of mass (center of inertia) of the system.

If then from (4.1) it follows that the momentum vector is constant in magnitude and direction. Projecting it on the coordinate axis, we obtain three scalar first integrals of the differential equations of the system's double chain:

These integrals are called momentum integrals. When the speed of the center of mass is constant, i.e., it moves uniformly and rectilinearly.

If the projection of the main vector of external forces on any one axis, for example, on the axis, is equal to zero, then we have one first integral, or if two projections of the main vector are equal to zero, then there are two integrals of the momentum.

2. Theorem on the change of the kinetic moment. Let A be some arbitrary point in space (moving or stationary), which does not necessarily coincide with any particular material point of the system during the entire time of movement. We denote its velocity in a fixed system of coordinates by The theorem on the change in the angular momentum of a material system relative to point A has the form

If point A is fixed, then equality (4.3) takes a simpler form:

This equality expresses the theorem on the change of the angular momentum of the system relative to a fixed point: the time derivative of the angular momentum of the system, calculated relative to some fixed point, is equal to the main moment of all external forces relative to this point.

If then, according to (4.4), the angular momentum vector is constant in magnitude and direction. Projecting it on the coordinate axis, we obtain the scalar first integrals of the differential equations of the motion of the system:

These integrals are called integrals of angular momentum or area integrals.

If point A coincides with the center of mass of the system, Then the first term on the right side of equality (4.3) vanishes and the theorem on the change in angular momentum has the same form (4.4) as in the case of a fixed point A. Note (see 4 § 3) that in the case under consideration the absolute angular momentum of the system on the left side of equality (4.4) can be replaced by the equal angular momentum of the system in its motion relative to the center of mass.

Let be some constant axis or an axis of constant direction passing through the center of mass of the system, and let be the angular momentum of the system relative to this axis. From (4.4) it follows that

where is the moment of external forces about the axis. If during the whole time of motion then we have the first integral

In the works of S. A. Chaplygin, several generalizations of the theorem on the change in angular momentum were obtained, which were then applied in solving a number of problems on the rolling of balls. Further generalizations of the theorem on the change of the kpnetological moment and their applications in problems of the dynamics of a rigid body are contained in the works. The main results of these works are related to the theorem on the change in the angular momentum relative to the moving one, constantly passing through some moving point A. Let be a unit vector directed along this axis. Multiplying scalarly by both sides of equality (4.3) and adding the term to both its parts, we obtain

When the kinematic condition is met

equation (4.5) follows from (4.7). And if condition (4.8) is satisfied during the whole time of motion, then the first integral (4.6) exists.

If the connections of the system are ideal and allow rotation of the system as a rigid body around the axis and in the number of virtual displacements, then the main moment of reactions about the axis and is equal to zero, and then the value on the right side of equation (4.5) is the main moment of all external active forces about the axis and . The equality to zero of this moment and the satisfiability of relation (4.8) will be in the case under consideration sufficient conditions for the existence of the integral (4.6).

If the direction of the axis and is unchanged, then condition (4.8) can be written as

This equality means that the projections of the velocity of the center of mass and the velocity of point A on the axis and on the plane perpendicular to this are parallel. In the work of S. A. Chaplygin, instead of (4.9), a less general condition is required where X is an arbitrary constant.

Note that condition (4.8) does not depend on the choice of a point on . Indeed, let P be an arbitrary point on the axis. Then

and hence

In conclusion, we note the geometric interpretation of Resal's equations (4.1) and (4.4): the vectors of the absolute velocities of the ends of the vectors and are equal, respectively, to the main vector and the main moment of all external forces relative to the point A.

The use of OZMS in solving problems is associated with certain difficulties. Therefore, additional relationships are usually established between the characteristics of motion and forces, which are more convenient for practical use. These ratios are general theorems of dynamics. They, being consequences of the OZMS, establish dependencies between the speed of change of some specially introduced measures of movement and the characteristics of external forces.

Theorem on the change in momentum. Let's introduce the concept of the momentum vector (R. Descartes) of a material point (Fig. 3.4):

i i = t v G (3.9)

Rice. 3.4.

For the system, we introduce the concept principal momentum vector of the system as a geometric sum:

Q \u003d Y, m "V r

In accordance with the OZMS: Xu, - ^ \u003d i), or X

R(E) .

Taking into account that /w, = const we get: -Ym,!" = R(E),

or in final form

do / di \u003d A (E (3.11)

those. the first time derivative of the main momentum vector of the system is equal to the main vector of external forces.

The theorem on the motion of the center of mass. Center of gravity of the system called a geometric point, the position of which depends on t, etc. on the mass distribution /r/, in the system and is determined by the expression of the radius vector of the center of mass (Fig. 3.5):

where g s - radius vector of the center of mass.

Rice. 3.5.

Let's call = t with the mass of the system. After multiplying the expression

(3.12) on the denominator and differentiating both parts of the semi-

valuable equality we will have: g s t s = ^t.U. = 0, or 0 = t s U s.

Thus, the main momentum vector of the system is equal to the product of the mass of the system and the velocity of the center of mass. Using the momentum change theorem (3.11), we obtain:

t with dU s / dі \u003d A (E), or

Formula (3.13) expresses the theorem on the motion of the center of mass: the center of mass of the system moves as a material point with the mass of the system, which is affected by the main vector of external forces.

Theorem on the change in moment of momentum. Let us introduce the concept of moment of momentum of a material point as a vector product of its radius-vector and momentum:

to oh, = bl X that, (3.14)

where to OI - angular momentum of a material point relative to a fixed point O(Fig. 3.6).

Now we define the angular momentum of a mechanical system as a geometric sum:

K () \u003d X ko, \u003d ShchU,? O-15>

Differentiating (3.15), we get:

Ґ сік--- X t i w. + g yu X t i

Given that = U G U i X t i u i= 0, and formula (3.2), we obtain:

сіК a /с1ї - ї 0 .

Based on the second expression in (3.6), we finally have a theorem on the change in the angular momentum of the system:

The first time derivative of the angular momentum of the mechanical system relative to the fixed center O is equal to the main moment of the external forces acting on this system relative to the same center.

When deriving relation (3.16), it was assumed that O- fixed point. However, it can be shown that in a number of other cases the form of relation (3.16) does not change, in particular, if, in the case of plane motion, the moment point is chosen at the center of mass, the instantaneous center of velocities or accelerations. In addition, if the point O coincides with a moving material point, equality (3.16), written for this point, will turn into the identity 0 = 0.

Theorem on the change in kinetic energy. When a mechanical system moves, both the “external” and internal energy of the system change. If the characteristics of the internal forces, the main vector and the main moment, do not affect the change in the main vector and the main moment of the number of accelerations, then internal forces can be included in the estimates of the processes of the energy state of the system. Therefore, when considering changes in the energy of the system, one has to consider the movements of individual points, to which internal forces are also applied.

The kinetic energy of a material point is defined as the quantity

T^myTsg. (3.17)

The kinetic energy of a mechanical system is equal to the sum of the kinetic energies of the material points of the system:

notice, that T > 0.

We define the force power as the scalar product of the force vector by the velocity vector:

The theorem on the motion of the center of mass. Differential equations of motion of a mechanical system. The theorem on the motion of the center of mass of a mechanical system. Law of conservation of motion of the center of mass.

Theorem on the change in momentum. The amount of movement of a material point. Elemental impulse of force. Impulse of force over a finite period of time and its projections on the coordinate axes. Theorem on the change in momentum of a material point in differential and finite forms.

The amount of movement of the mechanical system; its expression in terms of the mass of the system and the velocity of its center of mass. The theorem on the change in the momentum of a mechanical system in differential and finite forms. Law of conservation of mechanical momentum

(The concept of a body and a point of variable mass. Meshchersky's equation. Tsiolkovsky's formula.)

Theorem on the change in moment of momentum. The moment of momentum of a material point relative to the center and relative to the axis. The theorem on the change in the angular momentum of a material point. Central force. Conservation of angular momentum of a material point in the case of a central force. (The concept of sector speed. The law of areas.)

The main moment of momentum or the kinetic moment of a mechanical system about the center and about the axis. The angular momentum of a rotating rigid body about the axis of rotation. Theorem on the change in the kinetic moment of a mechanical system. The law of conservation of the kinetic moment of a mechanical system. (Theorem on the change in the angular momentum of a mechanical system in relative motion with respect to the center of mass.)

Theorem on the change in kinetic energy. Kinetic energy of a material point. Elementary work of force; analytical expression for elementary work. The work of a force on the final displacement of the point of its application. The work of the force of gravity, the force of elasticity and the force of gravity. Theorem on the change in the kinetic energy of a material point in differential and finite forms.

Kinetic energy of a mechanical system. Formulas for calculating the kinetic energy of a rigid body during translational motion, during rotation around a fixed axis, and in the general case of motion (in particular, during plane-parallel motion). Theorem on the change in the kinetic energy of a mechanical system in differential and finite forms. Equality to zero of the sum of the work of internal forces in a solid. Work and power of forces applied to a rigid body rotating around a fixed axis.

The concept of a force field. Potential force field and force function. Expression of force projections in terms of force function. Surfaces of equal potential. The work of a force on the final displacement of a point in a potential force field. Potential energy. Examples of potential force fields: a uniform gravitational field and a gravitational field. The law of conservation of mechanical energy.

Rigid Body Dynamics. Differential equations of translational motion of a rigid body. Differential equation of rotation of a rigid body around a fixed axis. physical pendulum. Differential equations of plane motion of a rigid body.

d'Alembert principle. d'Alembert's principle for a material point; force of inertia. d'Alembert's principle for a mechanical system. Bringing the forces of inertia of the points of a rigid body to the center; principal vector and principal moment of inertia forces.

(Determination of dynamic reactions of bearings during rotation of a rigid body around a fixed axis. The case when the axis of rotation is the main central axis of inertia of the body.)

The principle of possible displacements and the general equation of dynamics. Relationships imposed on a mechanical system. Possible (or virtual) displacements of a material point and a mechanical system. The number of degrees of freedom of the system. Ideal connections. The principle of possible movements. General equation of dynamics.

Equations of system motion in generalized coordinates (Lagrange equations). Generalized system coordinates; generalized speeds. Expression of elementary work in generalized coordinates. Generalized forces and their calculation; the case of forces with potential. Equilibrium conditions for the system in generalized coordinates. Differential equations of system motion in generalized coordinates or Lagrange equations of the 2nd kind. Lagrange equations in case of potential forces; Lagrange function (kinetic potential).

The concept of equilibrium stability. Small free oscillations of a mechanical system with one degree of freedom around the position of stable equilibrium of the system and their properties.

Elements of the impact theory. Impact phenomenon. Impact force and impact impulse. Action of impact force on a material point. Theorem on the change in the momentum of a mechanical system upon impact. Direct central impact of the body on a fixed surface; elastic and inelastic impacts. Impact recovery coefficient and its experimental determination. Direct central blow of two bodies. Carnot's theorem.

BIBLIOGRAPHY

Basic

Butenin N. V., Lunts Ya-L., Merkin D. R. Course of theoretical mechanics. Vol. 1, 2. M., 1985 and previous editions.

Dobronravov V. V., Nikitin N. N. Course of theoretical mechanics. M., 1983.

Starzhinsky V. M. Theoretical mechanics. M., 1980.

Targ S. M. A short course in theoretical mechanics. M., 1986 and previous editions.

Yablonsky A. A., Nikiforova V. M. Course of theoretical mechanics. Part 1. M., 1984 and previous editions.

Yablonsky A. A. Course of theoretical mechanics. Part 2. M., 1984 and previous editions.

Meshchersky I.V. Collection of problems in theoretical mechanics. M., 1986 and previous editions.

Collection of problems in theoretical mechanics / Ed. K. S. Kolesnikova. M., 1983.

Additional

Bat M. I., Dzhanelidze G. Yu., Kelzon A. S. Theoretical mechanics in examples and tasks. Ch. 1, 2. M., 1984 and previous editions.

Collection of problems in theoretical mechanics / 5raznichen / co N. A., Kan V. L., Mintsberg B. L. et al. M., 1987.

Novozhilov I. V., Zatsepin M. F. Standard calculations in theoretical mechanics based on a computer. M., 1986,

Collection of assignments for term papers in theoretical mechanics / Ed. A. A. Yablonsky. M., 1985 and previous editions (contains examples of problem solving).

With a large number of material points that make up the mechanical system, or if it includes absolutely rigid bodies () that perform non-translational motion, the use of a system of differential equations of motion in solving the main problem of the dynamics of a mechanical system turns out to be practically impossible. However, when solving many engineering problems, there is no need to determine the movement of each point of the mechanical system separately. Sometimes it is enough to draw conclusions about the most important aspects of the process of motion under study without completely solving the system of equations of motion. These conclusions from the differential equations of motion of a mechanical system constitute the content of the general theorems of dynamics. General theorems, firstly, free from the need in each individual case to carry out those mathematical transformations that are common for different problems and are once and for all carried out when deriving theorems from differential equations of motion. Secondly, general theorems give a connection between the general aggregated characteristics of the motion of a mechanical system, which have a clear physical meaning. These general characteristics, such as momentum, momentum, kinetic energy of a mechanical system are called measures of motion of a mechanical system.

The first measure of motion is the amount of motion of a mechanical system

The momentum of a material point is a vector measure of its movement, equal to the product of the mass of the point and its speed:

.

The momentum of a mechanical system is a vector measure of its motion, equal to the sum of the quantities of motion of its points:

, (13.1)

We transform the right side of formula (23.1):

where
is the mass of the whole system,
is the speed of the center of mass.

Consequently, the momentum of a mechanical system is equal to the momentum of its center of mass if the entire mass of the system is concentrated in it:

.

Impulse of force

The product of a force and the elementary time interval of its action
is called the elementary impulse of force.

Impulse of force over a period of time is called the integral of the elementary impulse of the force

.

Theorem on the change in the momentum of a mechanical system

Let for each point
mechanical system act resultant of external forces and resultant of internal forces .

Consider the basic equations of the dynamics of a mechanical system

Adding term by term equations (13.2) for n points of the system, we get

(13.3)

The first sum on the right side is equal to the main vector external forces of the system. The second sum is equal to zero by the property of the internal forces of the system. Consider the left side of equality (13.3):

Thus, we get:

, (13.4)

or in projections on the coordinate axes

(13.5)

Equalities (13.4) and (13.5) express the theorem on the change in the momentum of a mechanical system:

The time derivative of the momentum of a mechanical system is equal to the main vector of all external forces of the mechanical system.

This theorem can also be represented in integral form by integrating both parts of equality (13.4) over time within the limits of t 0 to t:

, (13.6)

where
, and the integral on the right side is the momentum of external forces behind

time t-t 0 .

Equality (13.6) represents the theorem in integral form:

The increment of momentum of a mechanical system over a finite time is equal to the momentum of external forces during this time.

The theorem is also called momentum theorem.

In projections onto the coordinate axes, the theorem can be written as:

Consequences (laws of conservation of momentum)

one). If the main vector of external forces for the considered period of time is equal to zero, then the momentum of the mechanical system is constant, i.e. if
,
.

2). If the projection of the main vector of external forces on any axis for the considered period of time is equal to zero, then the projection of the momentum of the mechanical system on this axis is constant,

those. if
then
.