QUALITATIVE-ANALYSIS OF PERIODIC OSCILLATIONS IN CLASSICAL AUTONOMOUS HAMILTONIAN-SYSTEMS

The paper deals with periodic oscillations of an autonomous Hamiltonian system which are qualitatively the same as the corresponding normal-mode oscillations of the linearized system. The conditions that guarantee the existence of a continuous branch of such solutions coinciding with a Lyapunov one-parameter family in the neighbourhood of the equilibrium point and reaching the boundary of a given region of the configuration space are obtained. Bilateral estimates of the oscillation periods are derived. Under a certain condition of concavity or convexity of the potential function, the Lyapunov family of periodic solutions has a unique continuation in the parameter to the boundary of the region; the corresponding period is a monotonic function of the parameter. As an example, a system of coupled oscillators is treated.