Chapter 9 - The Restricted Three-Body Problem

Abstract

The existence and stability of triangular libration points in the relativistic restricted three- body problem has been studied. It is found that L4;5 are unstable in the whole range 06 6 1=2 in contrast to the classical restricted three-body problem where they are stable for 0 << 0 , where is the mass parameter and 0 D 0:03852 ::: . Haghihara (1931) studied the relativistic one-body problem which is the reduced form of the two-body problem. He took Schwarzschild's field and showed that the orbit coincided with Eddington's orbit (1923). In 1938, Eddington and Clark studied the problem of n-bodies in general relativity theory but did not compare any property from Newtonion law of gravitation. Brumberg (1972, 1991) studied the problem in more detail and collected most of the important results on relativistic celestial mechanics. He not only obtained the equations of motion for the general problem of three bodies but also deduced the equations of motion for the restricted problem of three bodies. In this paper, we have studied the existence and linear stability of triangular libration points in the relativistic restricted three-body problem.