ЗйлеровьI рещения задчи о поступательно вращатльном движени трех твердьени трех твердьх тел

Аннотация

Настоящая статъя является непоцредсτвенньIм продолже мработьI начатой ранее автором (Вιдякин, 1976). Внолне еатественно что боиьшой теоретический интерес представляет χлеδуIощая пρоблема:

СуществуIот ли точньIе ьаСтньIе решения задачи о поступательно вращательном движениитрех абсолIотч тердьIх тел аналогиьнIх ЭйлеровьIм (прямолнейньIм) рениям задаи о движении трех материальньIх точечньIх тел.

Решенлю зтой проблемьI и поСвящается настоящая работа. При решении задчи предполагается, что злементарньIе частицьI тел взаимно притягиваутся по зела не имеут общих частей.

Abstract

The present paper is a continuation of the article (Vidyakin, 1976) in which we proved the existence of Lagrange (triangle) solutions in the general problem of the translatory-rotary motion of three absolutely rigid bodies.

In particular, we have found the conditions for the existence of Lagrange solutions in the case when all the bodies possess a symmetry with respect to three mutually perpendicular planes both in respect to the distribution of matter and in respect to the outward form. In this case the bodies effect simple translations along with the centres of the masses without rotation (in Lyapunov's system of coordinates).

If the rigid bodies possess a symmetry in respect to the axis and to the plane, perpendicular to this axis, then the Lagrange solutions of the ‘three floats’ (Duboshin, 1973), ‘three spokes’ (Kondurar, 1974), ‘three shafts’ (Vidyakin, 1976) types are admitted, as well as the solutions in the cases of combinations of the ‘float’, ‘spoke’ and ‘shaft’-bodies (Vidyakin, 1976).

Those solutions exist of certain conditions, imposed on the structure, orientation and rotation of the bodies, are observed.

In the general case (there) exist particular solutions which we have termed as ‘Near-Lagrangian’.

The present paper is to prove the existence of Euler (rectilinear) solutions in the problem of the translatory-rotary motion of three rigid bodies, assuming that the elementary particles of the rigid bodies are mutually attracted according to the Newtonian law.

In particular, we have found the conditions for the existence of Euler solutions in the case when all the bodies possess a symmetry in respect to three mutually perpendicular planes both in respect to the distribution of matter and in respect to the outward form. In this case the bodies are so disposed in the uniformly rotating coordinate system that two symmetry planes concur while the centres of the masses are disposed on one straight line.

In particular, if the bodies possess a symmetry in respect to the axis and to the plane perpendicular to this axis, then the Euler solutions of the ‘three floats’ (Duboshin, 1973), ‘three spokes’, ‘three shafts’ types as well as solutions in the cases of combinations of ‘float’-, ‘spoke’- and ‘shaft’-bodies and spheres, either homogeneous or possessing a spherically symmetric distribution of densities, are admitted.

The paper gives exact solutions for the cases when the attraction force function of the bodies has an approximate expression.