Global Weinstein Type Theorem on Multiple Rotating Periodic Solutions for Hamiltonian Systems

TL;DR

This paper proves the existence of multiple rotating periodic solutions in convex Hamiltonian systems, extending Weinstein's theorem to multiple solutions with various symmetries, using a new index theory.

Contribution

It introduces a new index for rotating periodic orbits and establishes conditions for multiple solutions on convex energy surfaces.

Findings

Existence of at least n geometrically distinct rotating solutions.

If a symmetric energy surface has a nonsymmetric periodic solution, it has infinitely many.

The results extend Weinstein's theorem to multiple solutions with different symmetries.

Abstract

This paper concerns the existence of multiple rotating periodic solutions for $2n$ dimensional convex Hamiltonian systems. For the symplectic orthogonal matrix $Q$, the rotating periodic solution has the form of $z(t+T)=Qz(t)$, which might be periodic, anti-periodic, subharmonic or quasi-periodic according to the structure of $Q$. It is proved that there exist at least $n$ geometrically distinct rotating periodic solutions on a given $Q$ invariant convex energy surface under a pinching condition. As a result, it is proved that if the symmetric energy surface admits a nonsymmetric periodic solution, it has infinitely many periodic orbits. In order to prove the result, we introduce a new index on rotating periodic orbits.