The Clebsch System

The Clebsch system is one of the few classical examples of rigid bodies whose equations of motion are known to be integrable in the sense of Liouville. The explicit solution of its equations of motion, however, is particularly hard, and it has defeated many attempts in the past. In this paper we present a novel and rather detailed study of these equations of motion. Our approach is based on an improved version of the method originally used, in 1889, by Sophia Kowalewski to solve the equations of motion of the top bearing her name. We improve her method in two important points, and we clarify that it concerns a class of dynamical systems which is wider than the class of Hamiltonian systems which are integrable in the sense of Liouville. We use the improved version of the method by Kowalewski to prove two results. First, without using the Hamiltonian structure of the Clebsch system, we show that the integration of the equations of motion may be achieved by computing four Abelian integrals. Next, taking into account its Hamiltonian structure, we show that two quadratures are sufficient to compute a complete integral of its Hamilton-Jacobi equation. In this way, the process of solution of the equations of motion of the Clebsch system is clarified both from the standpoint of Abel and from the standpoint of Jacobi.