A Necessary and Sufficient Condition for Solving a Rigid Body Problem

In this paper, the motion of a rigid body about a fixed point under the influence of a Newtonian force field is investigated. The Euler-Poisson equations are used to represent that motion. Three first integrals of these equations are well known. The exact solutions of these equations require, in general, a fourth algebraic first integral. The necessary and sufficient condition for some functions to be a fourth first integral of the governing equations is obtained