Лагранжевы и Близкие к ним решения задачи о поступательно-вращательном движении трёх твёрдых тел

Abstract

The present paper is a direct continuation of papers by Duboshin, 1973; Kondurar, 1974 and Vidyakin, 1972 in which the existence of one kind of Lagrange (triangle) and Euler (rectilinear) solutions of the general problem of the translatory-rotary motion of three absolutely rigid bodies was proved.

In the present paper the following problem is solved: Whether there exist exact special solutions in the general problem of the translatory-rotary motion of three rigid bodies, which have three mutually perpendicular symmetry planes, analogous to the Lagrange solutions in the problem of three point bodies.

In this paper it is proved that such solutions exist if the bodies possess a definite structure and orientation. In particular, the conditions for the existence of Lagrange solutions of the problem of three rigid bodies have been found.

We showed that bodies possessing three mutually perpendicular symmetry planes (both in form and in the distribution of the masses) are situated in a rotating system of coordinates in such a way that one symmetry plane passes through the centres of mass of the bodies and the other symmetry planes are oriented in a definite manner. The orientation of the bodies depends on their structures. In particular, if rigid bodies possess a geometric-dynamic symmetry in respect to an axis and the plane perpendicular to this axis, then, according to the terminology of G. N. Duboshin (Duboshin, 1961) the solutions of the ‘three floats’, ‘three spokes’, ‘three shafts’ types and the cases of the combinations of the bodies: ‘float’, ‘spoke’ and ‘shaft’ are admitted.

In the general case the centres of mass of the bodies form the apexes of an unequilateral triangle.