New Periodic Solutions of Singular Hamiltonian Systems with Fixed Energies

TL;DR

This paper establishes the existence of new non-trivial periodic solutions with prescribed energy for singular second order Hamiltonian systems using variational methods, extending classical theorems in the field.

Contribution

It introduces novel variational approaches to find periodic solutions in Hamiltonian systems with singular potentials, broadening the scope of previous results.

Findings

Proved existence of non-trivial periodic solutions with fixed energy.

Extended classical theorems to systems with unbounded potential wells.

Applied variational methods to singular Hamiltonian systems.

Abstract

By using the variational minimizing method with a special constraint and the direct variational minimizing method without constraint, we study second order Hamiltonian systems with a singular potential $V\in C^2(R^n\backslash O,R)$ and $V\in C^1(R^2\backslash O,R)$ which may have an unbounded potential well, and prove the existence of non-trivial periodic solutions with a prescribed energy. Our results can be regarded as some complements of the well-known Theorems of Benci-Gluck-Ziller-Hayashi and Ambrosetti-Coti Zelati and so on.