On The Geometrical Approach to The Problem of Integrability of Hamiltonian Systems

Abstract

In this research we treat the problem of integrability of Hamiltonian systems . There have been several methods for treating this problem depending on different situations. These methods include the first integral method obtained via the Poission bracket and generalized in the context of Lie bracket . The latter generalization is based on Hamiltonian mechanics and symplectic structure. The method that we emphasized in this research is the Cartan method of moving frame. We have utilized this method of moving frame where the killing tensor is major entity that is involved in the treatment .In particular, we have used the intrinsic geometry provided by the Guass and main curvature to extract the separable system of coordinates by employing the method of moving frame. Then the corresponding Killing tensor , the potential function and the first integrals are recovered .We have applied this procedure of separation of variables to surfaces of rotation and surface of constants curvature