Notes on a Theorem of Benci-Gluck-Ziller-Hayashi

TL;DR

This paper employs constrained variational methods to establish the existence of periodic solutions with prescribed energy in certain Hamiltonian systems, extending previous theorems by Benci-Gluck-Ziller and Hayashi.

Contribution

It introduces a new approach for finding periodic solutions in Hamiltonian systems with unbounded potential wells, complementing existing theorems.

Findings

Existence of periodic solutions with prescribed energy established.

Applicable to Hamiltonian systems with unbounded potential wells.

Extends classical results by Benci-Gluck-Ziller and Hayashi.

Abstract

We use constrained variational minimizing methods to study the existence of periodic solutions with a prescribed energy for a class of second order Hamiltonian systems with a $C^2$ potential function which may have an unbounded potential well. Our result can be regarded as complementary to the well-known theorem of Benci-Gluck-Ziller and Hayashi.