Some remarks on the generalized many-body problem

Abstract

This paper deals with the generalized problem of motion of a system of a finite number of bodies (material points).

We suppose here that every point of the system acts on another one with a force (attractive or repulsive) directed along the straight line connecting these two points, and proportional to the product of their masses and a certain function of time, mutual distance and its derivatives of the first and second order (Duboshin, 1970).

The laws of forces are different for different pairs of points and, generally speaking, the validity of the third axiom of dynamics (law of action and reaction) is not assumed in advance.

With these general assumptions we find the conditions for the laws of the forces under which the problem admits the first integrals, analogous to the classic integrals of the many-body problem with the Newton's law of attraction.

It is shown furthermore, that in this generalized problem it is possible to obtain an equation, analogous to the classic equation of Lagrange-Jacobi and deduce the conditions of stability or instability of the system in Lagrange's sense.

The results obtained may be applied for the investigation of motion in some isolated stellar systems, where the laws of mechanics may be different from the laws in our solar system.