Solving problems in dynamics. The principle of d'Alembert

As a separate science, theoretical mechanics is a doctrine that unites the general laws of mechanical motion and the interaction of material bodies. Development of this science was originally received as a division of physics, taking as a basis axiomatics, it was separated into a separate branch of natural science.

Solving the problems of dynamics within the framework of the subject of theoretical mechanics is greatly facilitated by the use of the d'Alembert principle. It consists in the fact that the balancing of all active forces that act on the points of the mechanical system, and the reactions of existing links, occurs due to the account of the so-called inertia forces. Mathematically, this is expressed as the summation of all the above elements, the result of which is zero.

D'Alembert himself, Jean Leron (1717-1783), is known to the world as a great enlightener who achieved high achievements in the most diverse fields of natural science. Mathematics, mechanics, and philosophy have undergone an analysis of his inquisitive mind. As a result, the works of D'Alembert touched upon material systems (the d'Alembert principle), describing their differential equations, namely the rules of compilation. Jean Leron substantiated the theory of perturbation of the planets, he paid much attention to the study of the theory of series and differential equations, mathematical analysis. A Frenchman by nationality, D'Alembert became an honorary foreign member of the St. Petersburg Academy of Sciences.

Merit scientist Frenchman, who developed the principle of solving complex problems of dynamics, which also bears his name, is that due to its application for the consideration of dynamic processes it is allowed to use simpler methods of static mechanics. Due to the simplicity and availability of this principle (the principle of d'Alembert) has found wide application in engineering practice.

We apply the d'Alembert principle for a material point

To establish a unified approach, an algorithm for studying a single mechanical system, the D'Alembert principle helps. In this case, there is no dependence on the conditions imposed on its motion. Dynamic differential equations of motion are reduced to the form of equilibrium equations. For example, taking for consideration a non-free certain material point M, moving along the curve AB as a result of the action of active forces with the resultant F, we can use the notation N for the reaction force (the effect of the curve AB on M). We introduce the forces F, N, and Ф into the basic equation describing the dynamics of the point, we obtain a convergent system that expresses the equilibrium condition of a particular system. In this case, the quantity $ describes the action of inertia forces and has a negative value. This is the use of the d'Alembert principle in calculations with reference to a material point.

It should be taken into account that with this approach, we obtain a rather conventional force coupling equation used to balance the inertial force system. But despite this, the D'Alembert principle provides a convenient and simple solution for dynamic problems.

Application of the d'Alembert principle for a mechanical system

Having achieved a positive result in solving the problem of dynamics for a material point, one can safely proceed to a more complex version of this problem, where the d'Alembert principle is used for a mechanical system.

The equation for the system differs little from the equation for the point. The essential difference lies in the fact that calculation for a mechanical non-free system at any time implies finding the resultant forces, the sums of the reactions of the bonds and the forces of inertia of the material points.

The use of the above methods and principles does not contradict the basic law of physics. On the contrary, even with a certain amount of overlap that facilitates the decision process. This method did not come from scratch, all the main conclusions are based on the basic laws of Newton, the principles of Herman-Euler, which were developed in the principles of d'Alembert.