Non-integrability in Hamiltonian mechanics
This paper is a survey on theorems for proving non-integrability of Hamiltonian dynamical systems. In this review, the following two points of view are presented: 1. Properties of the variational equations around periodic solutions, in the real or complex time domain. 2. Destruction of the resonant tori of an integrable system under a perturbation. Let \(\dot x=J \nabla H\) be a Hamiltonian system, where \(x\) is a set of canonical coordinates and \(J\) is the standard symplectic structure. Further, let \(\dot z=A(t)z\) be the corresponding system of variational equations, where \(A=J \nabla^2H\) is a periodic matrix, evaluated on a periodic solution. The first reviewed theorem belongs to Poincaré (1892): If \(H\) possesses \(n\) integrals in involution, independent on a periodic solution, then it is non-isolated, i.e., all its characteristic exponents are zero. The last reviewed theorem (Ichtiaroglou and Meletlidou 1996) gives the conditions, under which the perturbed Hamiltonian \(H= H_0(I_j)+ \varepsilon H_1 (I_j,w_j)\) (where \((I_j,w_j)\) are action-angle variables of the integrable part \(H_0)\) does not possess a complete set of analytic integrals of motion.
scientific article; zbMATH DE number 1051498
scientific article