Stability of motion of the restricted circular and charged three-body problem

Abstract

Stabiliity is applied to characterize type of motion in which the moving body is confined to certain limited regions and in this sense we may say that the motion of the body in question is stable. This method has been used in the past chiefly in connection with the classical restricted problem of three bodies.

In this paper we consider a dynamical system defined by the Lagrangian

$$L = \tfrac{1}{2}(\dot x^2 + \dot y^2 + \dot z^2 ) + x\dot y - y\dot x + \tfrac{1}{2}(x^2 + y^2 ) + \frac{{q - \mu }}{{r_1 }} + \frac{{\mu - q}}{{r_2 }}$$

that is moving under the action of a potential function; namely,

$$\Omega = - \tfrac{1}{2}(x^2 + y^2 ) - \frac{{q - \mu }}{{r_1 }} - \frac{{\mu - q}}{{r_2 }}$$

.

If any small disturbing influences are applied to the restricted circular three charged body problem which is defined by the LagrangianL, it may deviate only slightly from the equilibrium condition of motion or it may depart from it further and further. In this paper the relations for the stability of this system will be given.