On resonant classical Hamiltonians with two equal frequencies

Abstract

This paper contains a detailed study of the flow that the classical Hamiltonian $$H = \tfrac{1}{2}(x_1^2 + y_1^2 ) - \tfrac{1}{2}(x_2^2 + y_2^2 ) + \mathcal{O}_3 $$ induces near the origin of its phase spaceR 4. Here the perturbation term\(\mathcal{O}_3 \) represents a convergent power series. In particular, criteria for the existence and stability of periodic orbits are developed and expressed in terms of canonical invariants that are extracted from the perturbation term.