20th ed. - M.: 2010.- 416 p.

The book outlines the fundamentals of the mechanics of a material point, the system of material points and a solid body in a volume corresponding to the programs of technical universities. Many examples and tasks are given, the solutions of which are accompanied by appropriate guidelines. For students of full-time and correspondence technical universities.

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TABLE OF CONTENTS
Preface to the thirteenth edition 3
Introduction 5
SECTION ONE STATICS OF A SOLID STATE
Chapter I. Basic concepts initial provisions of articles 9
41. Absolutely rigid body; strength. Tasks of statics 9
12. Initial provisions of statics » 11
$ 3. Connections and their reactions 15
Chapter II. Composition of forces. System of converging forces 18
§four. Geometrically! Method of combining forces. Resultant of converging forces, decomposition of forces 18
f 5. Force projections on the axis and on the plane, Analytical method for setting and adding forces 20
16. Equilibrium of the system of converging forces_. . . 23
17. Solving problems of statics. 25
Chapter III. Moment of force about the center. Power couple 31
i 8. Moment of force about the center (or point) 31
| 9. A couple of forces. couple moment 33
f 10*. Equivalence and pair addition theorems 35
Chapter IV. Bringing the system of forces to the center. Equilibrium conditions... 37
f 11. Parallel force transfer theorem 37
112. Bringing the system of forces to a given center - . .38
§ 13. Conditions for the equilibrium of a system of forces. Theorem on the moment of the resultant 40
Chapter V. Flat system of forces 41
§ 14. Algebraic moments of force and couples 41
115. Reduction of a flat system of forces to the simplest form .... 44
§ 16. Equilibrium of a flat system of forces. The case of parallel forces. 46
§ 17. Problem solving 48
118. Balance of systems of bodies 63
§ 19*. statically determined and statically determined indefinable systems bodies (structures) 56"
f 20*. Definition of internal forces. 57
§ 21*. Distributed Forces 58
E22*. Calculation of flat trusses 61
Chapter VI. Friction 64
! 23. Laws of sliding friction 64
: 24. Rough bond reactions. Friction angle 66
: 25. Equilibrium in the presence of friction 66
(26*. Thread friction on a cylindrical surface 69
1 27*. Rolling friction 71
Chapter VII. Spatial system of forces 72
§28. Moment of force about the axis. Principal vector calculation
and the main moment of the system of forces 72
§ 29*. Reduction of the spatial system of forces to the simplest form 77
§thirty. Equilibrium of an arbitrary spatial system of forces. The case of parallel forces
Chapter VIII. Center of gravity 86
§31. Center of Parallel Forces 86
§ 32. Force field. Center of gravity of a rigid body 88
§ 33. Coordinates of the centers of gravity of homogeneous bodies 89
§ 34. Methods for determining the coordinates of the centers of gravity of bodies. 90
§ 35. Centers of gravity of some homogeneous bodies 93
SECTION TWO KINEMATICS OF A POINT AND A RIGID BODY
Chapter IX. Point kinematics 95
§ 36. Introduction to kinematics 95
§ 37. Methods for specifying the movement of a point. . 96
§38. Point velocity vector,. 99
§ 39
§40. Determining the speed and acceleration of a point with the coordinate method of specifying movement 102
§41. Solving problems of point kinematics 103
§ 42. Axes of a natural trihedron. Numerical speed value 107
§ 43. Tangent and normal acceleration of a point 108
§44. Some special cases of motion of a point in software
§45. Graphs of movement, speed and acceleration of point 112
§ 46. Problem solving< 114
§47*. Velocity and acceleration of a point in polar coordinates 116
Chapter X. Translational and rotational motions of a rigid body. . 117
§48. Translational movement 117
§ 49. Rotational motion of a rigid body around an axis. Angular Velocity and Angular Acceleration 119
§fifty. Uniform and uniform rotation 121
§51. Velocities and accelerations of points of a rotating body 122
Chapter XI. Plane-parallel motion of a rigid body 127
§52. Equations of plane-parallel motion (motion of a plane figure). Decomposition of motion into translational and rotational 127
§53*. Determination of trajectories of points of a plane figure 129
§54. Determining the velocities of points on a plane figure 130
§ 55. The theorem on the projections of the velocities of two points of the body 131
§ 56. Determination of the velocities of points of a plane figure using the instantaneous center of velocities. The concept of centroids 132
§57. Problem solving 136
§58*. Determination of accelerations of points of a plane figure 140
§59*. Instant center of acceleration "*"*
Chapter XII*. Motion of a rigid body around a fixed point and motion of a free rigid body 147
§ 60. Motion of a rigid body having one fixed point. 147
§61. Kinematic Euler equations 149
§62. Speeds and accelerations of body points 150
§ 63. General case of motion of a free rigid body 153
Chapter XIII. Complex point movement 155
§ 64. Relative, figurative and absolute motions 155
§ 65, Velocity addition theorem » 156
§66. The theorem on the addition of accelerations (Coriols' theorem) 160
§67. Problem solving 16*
Chapter XIV*. Complex motion of a rigid body 169
§68. The addition of translational movements 169
§69. Addition of rotations about two parallel axes 169
§70. Cylindrical gears 172
§ 71. Addition of rotations around intersecting axes 174
§72. Addition of translational and rotational movements. Screw movement 176
SECTION THREE DYNAMICS OF A POINT
Chapter XV: Introduction to dynamics. Laws of dynamics 180
§ 73. Basic concepts and definitions 180
§ 74. Laws of dynamics. Problems of the dynamics of a material point 181
§ 75. Systems of units 183
§76. Basic types of forces 184
Chapter XVI. Differential equations of motion of a point. Solving problems of point dynamics 186
§ 77. Differential equations, motions of a material point No. 6
§ 78. Solution of the first problem of dynamics (determination of forces from a given movement) 187
§ 79. Solution of the main problem of dynamics in the rectilinear motion of a point 189
§ 80. Examples of problem solving 191
§81*. Fall of a body in a resisting medium (in air) 196
§82. Solution of the main problem of dynamics, with curvilinear motion of a point 197
Chapter XVII. General theorems of point dynamics 201
§83. The amount of movement of the point. Force Impulse 201
§ S4. Theorem on the change in the momentum of a point 202
§ 85. The theorem on the change in the angular momentum of a point (theorem of moments) "204
§86*. Movement under the action of a central force. Law of areas.. 266
§ 8-7. Force work. Power 208
§88. Work Calculation Examples 210
§89. Theorem on the change in the kinetic energy of a point. ". . . 213J
Chapter XVIII. Non-free and relative motion of a point 219
§90. Non-free movement of a point. 219
§91. Relative movement of a point 223
§ 92. Influence of the Earth's rotation on the balance and motion of bodies... 227
Section 93*. Deviation of the incident point from the vertical due to the rotation of the Earth "230
Chapter XIX. Rectilinear fluctuations of a point. . . 232
§ 94. Free vibrations without taking into account the forces of resistance 232
§ 95. Free oscillations with viscous resistance (damped oscillations) 238
§96. Forced vibrations. Resonance 241
Chapter XX*. Motion of a body in the field of gravity 250
§ 97. Movement of a thrown body in the Earth's gravitational field "250
§98. Artificial satellites of the Earth. Elliptical trajectories. 254
§ 99. The concept of weightlessness. "Local reference systems 257
SECTION FOUR DYNAMICS OF A SYSTEM AND A RIGID BODY
G i a v a XXI. Introduction to system dynamics. moments of inertia. 263
§ 100. Mechanical system. Forces external and internal 263
§ 101. Mass of the system. Center of gravity 264
§ 102. Moment of inertia of a body about an axis. Radius of inertia. . 265
$ 103. Moments of inertia of a body about parallel axes. Huygens' theorem 268
§ 104*. centrifugal moments of inertia. Concepts about the main axes of inertia of the body 269
$105*. Moment of inertia of a body about an arbitrary axis. 271
Chapter XXII. The theorem on the motion of the center of mass of the system 273
$ 106. Differential equations of system motion 273
§ 107. The theorem on the motion of the center of mass 274
$ 108. Law of conservation of motion of the center of mass 276
§ 109. Problem solving 277
Chapter XXIII. Theorem on the change in the quantity of a movable system. . 280
$ BUT. Number of movement system 280
§111. Theorem on change of momentum 281
§ 112. Law of conservation of momentum 282
$113*. Application of the theorem to the motion of a liquid (gas) 284
§ 114*. Body of variable mass. Rocket movement 287
Gdawa XXIV. The theorem on the change in the moment of momentum of the system 290
§ 115. The main moment of the quantities of motion of the system 290
$ 116. Theorem on the change of the main moment of the momentum of the system (theorem of moments) 292
$117. The law of conservation of the main moment of momentum. . 294
$ 118. Problem solving 295
$119*. Application of the moment theorem to the motion of a liquid (gas) 298
§ 120. Equilibrium conditions for a mechanical system 300
Chapter XXV. Theorem on the change in the kinetic energy of the system. . 301.
§ 121. Kinetic energy of the system 301
$122. Some cases of calculating work 305
$ 123. Theorem on the change in the kinetic energy of the system 307
$ 124. Problem solving 310
$125*. Mixed tasks "314
$ 126. Potential force field and force function 317
$127, Potential energy. conservation law mechanical energy 320
Chapter XXVI. "Application of General Theorems to the Dynamics of a Rigid Body 323
$12&. Rotational motion of a rigid body around a fixed axis ". 323"
$ 129. Physical pendulum. Experimental determination of moments of inertia. 326
$130. Plane-parallel motion of a rigid body 328
$131*. Elementary theory of the gyroscope 334
$132*. Motion of a rigid body around a fixed point and motion of a free rigid body 340
Chapter XXVII. d'Alembert principle 344
$ 133. d'Alembert's principle for a point and a mechanical system. . 344
$ 134. Principal vector and principal moment of inertia forces 346
$ 135. Problem solving 348
$136*, Didemic reactions acting on the axis of a rotating body. Balancing of rotating bodies 352
Chapter XXVIII. The principle of possible displacements and the general equation of dynamics 357
§ 137. Classification of connections 357
§ 138. Possible displacements of the system. Number of degrees of freedom. . 358
§ 139. The principle of possible movements 360
§ 140. Solving problems 362
§ 141. General equation of dynamics 367
Chapter XXIX. Equilibrium conditions and equations of motion of the system in generalized coordinates 369
§ 142. Generalized coordinates and generalized velocities. . . 369
§ 143. Generalized forces 371
§ 144. Equilibrium conditions for a system in generalized coordinates 375
§ 145. Lagrange's equations 376
§ 146. Solving problems 379
Chapter XXX*. Small oscillations of the system around the position of stable equilibrium 387
§ 147. The concept of equilibrium stability 387
§ 148. Small free vibrations of a system with one degree of freedom 389
§ 149. Small damped and forced vibrations systems with one degree of freedom 392
§ 150. Small summary oscillations of a system with two degrees of freedom 394
Chapter XXXI. Elementary Impact Theory 396
§ 151. Basic equation of the theory of impact 396
§ 152. General theorems of the theory of impact 397
§ 153. Impact recovery factor 399
§ 154. Impact of the body on a fixed barrier 400
§ 155. Direct central impact of two bodies (impact of balls) 401
§ 156. Loss of kinetic energy during an inelastic impact of two bodies. Carnot's theorem 403
§ 157*. A blow to a rotating body. Impact Center 405
Index 409

Within any training course The study of physics begins with mechanics. Not from theoretical, not from applied and not computational, but from good old classical mechanics. This mechanics is also called Newtonian mechanics. According to legend, the scientist was walking in the garden, saw an apple fall, and it was this phenomenon that prompted him to discover the law of universal gravitation. Of course, the law has always existed, and Newton only gave it a form understandable to people, but his merit is priceless. In this article, we will not describe the laws of Newtonian mechanics in as much detail as possible, but we will outline the basics, basic knowledge, definitions and formulas that can always play into your hands.

Mechanics is a branch of physics, a science that studies the movement of material bodies and the interactions between them.

The word itself is of Greek origin and translates as "the art of building machines". But before building machines, we still have a long way to go, so let's follow in the footsteps of our ancestors, and we will study the movement of stones thrown at an angle to the horizon, and apples falling on heads from a height h.

Why does the study of physics begin with mechanics? Because it is completely natural, not to start it from thermodynamic equilibrium?!

Mechanics is one of the oldest sciences, and historically the study of physics began precisely with the foundations of mechanics. Placed within the framework of time and space, people, in fact, could not start from something else, no matter how much they wanted to. Moving bodies are the first thing we pay attention to.

What is movement?

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

It is after this definition that we quite naturally come to the concept of a frame of reference. Changing the position of bodies in space relative to each other. Key words here: relative to each other . After all, a passenger in a car moves relative to a person standing on the side of the road at a certain speed, and rests relative to his neighbor in a seat nearby, and moves at some other speed relative to a passenger in a car that overtakes them.

That is why, in order to normally measure the parameters of moving objects and not get confused, we need reference system - rigidly interconnected reference body, coordinate system and clock. For example, the earth moves around the sun in a heliocentric frame of reference. In everyday life, we carry out almost all our measurements in a geocentric reference system associated with the Earth. The earth is a reference body relative to which cars, planes, people, animals move.

Mechanics, as a science, has its own task. The task of mechanics is to know the position of the body in space at any time. In other words, mechanics builds a mathematical description of motion and finds connections between physical quantities characterizing it.

In order to move further, we need the notion of “ material point ". They say that physics is an exact science, but physicists know how many approximations and assumptions have to be made in order to agree on this very accuracy. No one has ever seen a material point or sniffed an ideal gas, but they do exist! They are just much easier to live with.

A material point is a body whose size and shape can be neglected in the context of this problem.

Sections of classical mechanics

Mechanics consists of several sections

Kinematics
Dynamics
Statics

Kinematics from a physical point of view, studies exactly how the body moves. In other words, this section deals with the quantitative characteristics of movement. Find speed, path - typical tasks of kinematics

Dynamics solves the question of why it moves the way it does. That is, it considers the forces acting on the body.

Statics studies the equilibrium of bodies under the action of forces, that is, it answers the question: why does it not fall at all?

Limits of applicability of classical mechanics

Classical mechanics no longer claims to be a science that explains everything (at the beginning of the last century, everything was completely different), and has a clear scope of applicability. In general, the laws of classical mechanics are valid for the world familiar to us in terms of size (macroworld). They cease to work in the case of the world of particles, when classical mechanics is replaced by quantum mechanics. Also, classical mechanics is inapplicable to cases where the movement of bodies occurs at a speed close to the speed of light. In such cases, relativistic effects become pronounced. Roughly speaking, within the framework of quantum and relativistic mechanics - classical mechanics, this is a special case when the dimensions of the body are large and the speed is small.

Generally speaking, quantum and relativistic effects never go away; they also take place during the usual motion of macroscopic bodies at a speed much lower than the speed of light. Another thing is that the action of these effects is so small that it does not go beyond the most accurate measurements. Classical mechanics will thus never lose its fundamental importance.

We will continue to study the physical foundations of mechanics in future articles. For a better understanding of the mechanics, you can always refer to our authors, which individually shed light on the dark spot of the most difficult task.

The course covers: the kinematics of a point and a rigid body (and from different points of view it is proposed to consider the problem of the orientation of a rigid body), classical problems of the dynamics of mechanical systems and the dynamics of a rigid body, elements of celestial mechanics, the motion of systems of variable composition, the theory of impact, differential equations analytical dynamics.

The course presents all the traditional sections of theoretical mechanics, but special attention is paid to the most meaningful and valuable for theory and applications sections of dynamics and methods of analytical mechanics; statics is studied as a section of dynamics, and in the section of kinematics, the concepts necessary for the section of dynamics and the mathematical apparatus are introduced in detail.

Informational resources

Gantmakher F.R. Lectures on Analytical Mechanics. - 3rd ed. – M.: Fizmatlit, 2001.
Zhuravlev V.F. Fundamentals of theoretical mechanics. - 2nd ed. - M.: Fizmatlit, 2001; 3rd ed. – M.: Fizmatlit, 2008.
Markeev A.P. Theoretical mechanics. - Moscow - Izhevsk: Research Center "Regular and Chaotic Dynamics", 2007.

Requirements

The course is designed for students who own the apparatus of analytical geometry and linear algebra in the scope of the first-year program of a technical university.

Course program

1. Kinematics of a point
1.1. Problems of kinematics. Cartesian coordinate system. Decomposition of a vector in an orthonormal basis. Radius vector and point coordinates. Point speed and acceleration. Trajectory of movement.
1.2. Natural triangular. Expansion of velocity and acceleration in the axes of a natural trihedron (Huygens' theorem).
1.3. Curvilinear point coordinates, examples: polar, cylindrical and spherical coordinate systems. Velocity components and projections of acceleration on the axes of a curvilinear coordinate system.

2. Methods for specifying the orientation of a rigid body
2.1. Solid. Fixed and body-bound coordinate systems.
2.2. Orthogonal rotation matrices and their properties. Euler's finite turn theorem.
2.3. Active and passive points of view on orthogonal transformation. Addition of turns.
2.4. Finite rotation angles: Euler angles and "airplane" angles. Expression of an orthogonal matrix in terms of finite rotation angles.

3. Spatial motion of a rigid body
3.1. Translational and rotational motion of a rigid body. Angular velocity and angular acceleration.
3.2. Distribution of velocities (Euler's formula) and accelerations (Rivals' formula) of points of a rigid body.
3.3. Kinematic invariants. Kinematic screw. Instant screw axle.

4. Plane-parallel motion
4.1. The concept of plane-parallel motion of the body. Angular velocity and angular acceleration in the case of plane-parallel motion. Instantaneous center of speed.

5. Complex motion of a point and a rigid body
5.1. Fixed and moving coordinate systems. Absolute, relative and figurative movement of a point.
5.2. The theorem on the addition of velocities in the case of a complex motion of a point, relative and figurative velocities of a point. The Coriolis theorem on the addition of accelerations for a complex motion of a point, relative, translational and Coriolis accelerations of a point.
5.3. Absolute, relative and portable angular velocity and angular acceleration of a body.

6. Motion of a rigid body with a fixed point (quaternion presentation)
6.1. The concept of complex and hypercomplex numbers. Algebra of quaternions. Quaternion product. Conjugate and inverse quaternion, norm and modulus.
6.2. Trigonometric representation of the unit quaternion. Quaternion method of specifying body rotation. Euler's finite turn theorem.
6.3. Relationship between quaternion components in different bases. Addition of turns. Rodrigues-Hamilton parameters.

7. Exam work

8. Basic concepts of dynamics.
8.1 Momentum, angular momentum (kinetic moment), kinetic energy.
8.2 Power of forces, work of forces, potential and total energy.
8.3 Center of mass (center of inertia) of the system. The moment of inertia of the system about the axis.
8.4 Moments of inertia about parallel axes; the Huygens–Steiner theorem.
8.5 Tensor and ellipsoid of inertia. Principal axes of inertia. Properties of axial moments of inertia.
8.6 Calculation of the angular momentum and kinetic energy of the body using the inertia tensor.

9. Basic theorems of dynamics in inertial and non-inertial frames of reference.
9.1 Theorem on the change in the momentum of the system in an inertial frame of reference. The theorem on the motion of the center of mass.
9.2 Theorem on the change in the angular momentum of the system in an inertial frame of reference.
9.3 Theorem on the change in the kinetic energy of the system in an inertial frame of reference.
9.4 Potential, gyroscopic and dissipative forces.
9.5 Basic theorems of dynamics in non-inertial frames of reference.

10. Movement of a rigid body with a fixed point by inertia.
10.1 Euler dynamic equations.
10.2 Euler case, first integrals of dynamical equations; permanent rotations.
10.3 Interpretations of Poinsot and Macculag.
10.4 Regular precession in the case of dynamic symmetry of the body.

11. Motion of a heavy rigid body with a fixed point.
11.1 General formulation of the problem of the motion of a heavy rigid body around.
fixed point. Euler dynamic equations and their first integrals.
11.2 Qualitative Analysis motion of a rigid body in the Lagrange case.
11.3 Forced regular precession of a dynamically symmetric rigid body.
11.4 The basic formula of gyroscopy.
11.5 The concept of the elementary theory of gyroscopes.

12. Dynamics of a point in the central field.
12.1 Binet's equation.
12.2 Orbit equation. Kepler's laws.
12.3 The scattering problem.
12.4 The problem of two bodies. Equations of motion. Area integral, energy integral, Laplace integral.

13. Dynamics of systems of variable composition.
13.1 Basic concepts and theorems on the change of basic dynamic quantities in systems of variable composition.
13.2 Movement of a material point of variable mass.
13.3 Equations of motion of a body of variable composition.

14. Theory of impulsive movements.
14.1 Basic concepts and axioms of the theory of impulsive movements.
14.2 Theorems about changing the basic dynamic quantities during impulsive motion.
14.3 Impulsive motion of a rigid body.
14.4 Collision of two rigid bodies.
14.5 Carnot's theorems.

15. Test

Learning Outcomes

As a result of mastering the discipline, the student must:

Know:
- basic concepts and theorems of mechanics and the methods of studying the motion of mechanical systems arising from them;
Be able to:
- correctly formulate problems in terms of theoretical mechanics;
- develop mechanical and mathematical models that adequately reflect the main properties of the phenomena under consideration;
- apply the acquired knowledge to solve relevant specific problems;
Own:
- skills in solving classical problems of theoretical mechanics and mathematics;
- the skills of studying the problems of mechanics and building mechanical and mathematical models that adequately describe a variety of mechanical phenomena;
- skills of practical use of methods and principles of theoretical mechanics in solving problems: force calculation, determination of the kinematic characteristics of bodies at various ways tasks of motion, determination of the law of motion of material bodies and mechanical systems under the action of forces;
- skills to independently master new information in the process of production and scientific activity using modern educational and information technologies;

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.

Content

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 Theoretical Mechanics, Higher School, 2010.

Point kinematics.

1. The subject of theoretical mechanics. Basic abstractions.

Theoretical mechanicsis a science in which the general laws are studied mechanical movement and mechanical interaction of material bodies

Mechanical movementcalled the movement of a body in relation to another body, occurring in space and time.

Mechanical interaction is called such an interaction of material bodies, which changes the nature of their mechanical movement.

Statics - This is a branch of theoretical mechanics, which studies methods for converting systems of forces into equivalent systems and establishes the conditions for the equilibrium of forces applied to a solid body.

Kinematics - is the branch of theoretical mechanics that deals with the movement of material bodies in space from a geometric point of view, regardless of the forces acting on them.

Dynamics - This is a branch of mechanics that studies the movement of material bodies in space, depending on the forces acting on them.

Objects of study in theoretical mechanics:

material point,

system of material points,

Absolutely rigid body.

Absolute space and absolute time are independent of each other. Absolute space - three-dimensional, homogeneous, motionless Euclidean space. Absolute time - flows from the past to the future continuously, it is homogeneous, the same at all points in space and does not depend on the movement of matter.

2. The subject of kinematics.

Kinematics - this is a branch of mechanics that studies the geometric properties of the motion of bodies without taking into account their inertia (i.e. mass) and the forces acting on them

To determine the position of a moving body (or point) with the body in relation to which the movement of this body is being studied, rigidly, some coordinate system is connected, which together with the body forms reference system.

The main task of kinematics is to, knowing the law of motion of a given body (point), to determine all the kinematic quantities that characterize its motion (velocity and acceleration).

3. Methods for specifying the movement of a point

· natural way

Should be known:

Point movement trajectory;

Start and direction of counting;

The law of motion of a point along a given trajectory in the form (1.1)

· Coordinate method

Equations (1.2) are the equations of motion of the point M.

The equation for the trajectory of point M can be obtained by eliminating the time parameter « t » from equations (1.2)

· Vector way

(1.3)

Relationship between coordinate and vector methods for specifying the movement of a point

(1.4)

Relationship between coordinate and natural ways point movement assignments

Determine the trajectory of the point, excluding time from equations (1.2);

-- find the law of motion of a point along a trajectory (use the expression for the arc differential)

After integration, we obtain the law of motion of a point along a given trajectory:

The connection between the coordinate and vector methods of specifying the movement of a point is determined by equation (1.4)

4. Determining the speed of a point with the vector method of specifying the movement.

Let at the momenttthe position of the point is determined by the radius vector , and at the moment of timet 1 – radius-vector , then for a period of time the point will move.

(1.5)

point average speed,

the direction of the vector is the same as the vector

The speed of a point at a given time

To get the speed of a point at a given moment of time, it is necessary to make a passage to the limit

(1.6)

(1.7)

The speed vector of a point at a given time is equal to the first derivative of the radius vector with respect to time and is directed tangentially to the trajectory at a given point.

(unit¾ m/s, km/h)

Mean acceleration vector has the same direction as the vectorΔ v , that is, directed towards the concavity of the trajectory.

Acceleration vector of a point at a given time is equal to the first derivative of the velocity vector or the second derivative of the point's radius vector with respect to time.

(unit - )

How is the vector located in relation to the trajectory of the point?

In rectilinear motion, the vector is directed along the straight line along which the point moves. If the trajectory of the point is a flat curve, then the acceleration vector , as well as the vector cp, lies in the plane of this curve and is directed towards its concavity. If the trajectory is not a plane curve, then the vector cp will be directed towards the concavity of the trajectory and will lie in the plane passing through the tangent to the trajectory at the pointM and a line parallel to the tangent at an adjacent pointM 1 . AT limit when the pointM 1 tends to M this plane occupies the position of the so-called contiguous plane. Therefore, in general case the acceleration vector lies in the contiguous plane and is directed towards the concavity of the curve.