Integrability of some Problems in Dynamical Systems

Abstract

This thesis is concerned with the study of the existence of second isolating integral and integrals of motion for the Henon-Heiles system. This study focuses on the Henon-Heiles Hamiltonian system and the three known integrable cases. In this thesis, we maintained the algebraic solutions and prove these results by numerical methods. This thesis consists of three chapters, appendix and list of references. Chapter 1: In this chapter, we introduce the important theorems in this area, definitions, Kolmogorov, Arnold and Moser theorem (KAM theorem), Henon-Heiles system and introduction for Differentiable Dynamical System. Chapter II: In this chapter, we introduce the separability of three integrable cases of the 1-Henon-Heiles system and its solutions. I) The first case (Case 1}, is separable in the coordinates u = x + y, v = x- y and its solution can be expressed through the elliptic function. 2) The second case (Case II), the corresponding Hamilton-Jacobi equation is separable in the translated parabolic coordinates and its solution can be expressed in quadratures. Also, we introduce the method of invariant variational principles which we used in the third case, by introducing some examples which explain this method. Here we are concerned with the study of the conservation laws for physical system.