On the implicit equations of a mechanical systems motion

It is shown that the forces acting on the points of the mechanical system may depend on their accelerations. The differential equations of a mechanical system motion prove to be implicit. It is not allowed with respect to higher derivatives. There are fundamental mathematical problems related to the possibility and the only solution of these equations with respect to higher derivatives. Implicit equations of motion are typical for mechanical systems with dry friction sliding and rolling. In the dynamics of material point such problems do not arise. But in more complex mechanical systems, including the study of the motion of a solid whole mass is concentrated at one point, as well as in systems with one degree of freedom, such a situation is very characteristic. The paper discusses four fairly simple examples of mechanical systems movement, which is described by implicit differential equations of motion.


Introduction
The axioms of dynamics are based on the principle of determinism [1], which states that the initial state of a mechanical system uniquely determines the further behavior of the system under the action of given forces. The principle of determinism is a special case of the principle of the repetition of experience in physics. If you conduct the same experiment under the same conditions, you will get the same result.
The principle of determinism served as the basis for the formulation of Newton's second law, according to which the differential equation of motion of a material point under the action of force has the form ) , where mthe mass of the point, rits radius vector, r v   -the speed of the point, and the force F is a function of the position of the point, its velocity and time. If the initial state of the point is then the main problem of dynamics on determining the further motion of a point is the Cauchy problem and has a unique solution if the conditions of the existence and uniqueness theorem of the solution of differential equations are satisfied [1][2][3][4][5].

Statement of the problem
The differential equation of motion of the point (1) is transferred to the mechanical system. For a system of material points, the differential equations of motion have the form [1][2][3][4][5] where k m -the mass of the k -th point of the system, k rits radius vector. The force k F acting on the k -th point of the system is a function of the position and velocities of all points of the system and time Consequently, the equations of motion of a mechanical system form a system of differential equations resolved with respect to the highest derivatives.
However, situations are possible when the forces acting on the points of the system also depend on the accelerations of the points of the system The system of differential equations of motion of a mechanical system is implicit. It is not resolved with respect to higher derivatives. Fundamental mathematical problems arise related to the possibility and uniqueness of the solution of these equations with respect to higher derivatives.
The naturally implicit form of the equations of motion is preserved when passing to generalized coordinates using general dynamics theorems or Lagrange equations of the second kind.
Similar situations are characteristic of mechanical systems with dry friction [8][9][10][11]. In the dynamics of the point, such problems do not arise. But in more complex mechanical systems, including investigation of a rigid body motion, which whole mass is concentrated at one point, as well as in systems with one degree of freedom, this situation is very characteristic. This is explained by the fact that in accordance with the Coulomb law during sliding, the dry friction force where f -the coefficient of sliding friction, N -the normal reaction, v -the relative velocity of sliding. The normal reaction may depend on the accelerations of the points of the system, and then the equations of motion are implicit (3) -(4). In problems with dry friction, another complication arises due to the fact that, when  If bodies can not only slide relative to each other, but also roll, then in addition to the sliding friction force, a sliding friction moment arises, which is determined by similar relationships, and it is possible to alternate the phases of sliding, rolling with slipping and rolling without slipping.
Such systems were the subject of discussion in connection with the Painleve paradoxes [8][9][10][11], related to the fact that the equations of motion in some cases turn out to be insoluble with respect to higher derivatives or have several solutions.
The non-deterministic behavior of a mechanical system, as noted above, is also possible in systems without friction with equations of motion resolved with respect to higher derivatives.
The mathematical conditions for the existence and uniqueness of the solution of implicit differential equations of motion of mechanical systems in a general form are devoted to [12][13].
Let us dwell on some examples of implicit equations of motion of mechanical systems with dry sliding and rolling friction.
This article considers a slightly modified Painleve example and three more new examples of mechanical systems with friction, which lead to implicit differential equations of motion of the system. In the first two examples, the differential equations of motion are uniquely resolved with respect to the highest derivative. In the last example (mathematical pendulum), situations arise similar to the Painleve paradoxes. With a sufficiently large coefficient of friction, the equations of motion cannot be solved with respect to the highest derivative.

Uniform rod (ladder)
A uniform rod (ladder), (Figure 1), is in contact with a horizontal floor and a vertical wall. The mass of the rod is equal m , the length of the rod -l AB 2  . Point C is the center of mass of the rod, l CB AC   . The sliding friction coefficients at points A and B are equal f . The position of the rod is determined by the angle that it forms with the vertical.
At the initial time, the velocity of point A is directed downward, respectively, Denote by y x, the coordinates of the rod center of mass C , then We denote by The theorem on the center of mass motion has the form Solving these relations relatively, we obtain In accordance with the theorem on the change in the angular momentum relative to the center of mass From the last two relations it follows Given (7), we obtain the implicit differential equation of motion which is easily resolved with respect to the highest derivative The differential equation of motion (8) has a first integral. We denote by     the angular velocity, and using the replacement exclude time from equation (8). Then we get an inhomogeneous linear differential equation with constant coefficients for the dependence of the square of the angular velocity on the angle of rotation The integration constant c is determined from the initial conditions (6).

Wheel with a displaced center of mass
Differentiating these ratios twice, we obtain 2 2 cos sin , sin cos In accordance with the theorem on the center of mass motion and the theorem on the change in the angular momentum relative to the center of mass, we have Then, taking into account (10), from the first two of equations (11) Substituting these relations into the third of equations (11) The implicit differential equation of wheel motion has the form l rl l r (13) Note that the condition for not jumping the disk above the supporting surface (12) must be substituted    from equation (13), and then this condition has the form l g l rl l r According to function those can take any value modulo not exceeding 1.
From the theorem on the change in the angular momentum relative to the center of mass, we obtain Substituting (16), (17) into (18), we obtain From the law of Coulomb follows (5) kfN where When We substitute (20) into (16), and solve (16), (17) ) sin cos )( cos (  These contradictions to the basic principles of mechanics published by Painleve in a monograph [8] are called Painleve paradoxes [9][10][11].