Minimal $P$-symmetric period problem of first-order autonomous Hamiltonian Systems

TL;DR

This paper investigates the minimal $P$-symmetric period problem in autonomous Hamiltonian systems, establishing conditions under which solutions with specific symmetric periods exist, extending the understanding of periodic solutions in symplectic geometry.

Contribution

It provides new results on the existence of minimal $P$-symmetric periodic solutions under Rabinowitz's conditions, for certain symplectic matrices $P$.

Findings

Existence of $k au$-periodic solutions with minimal $P$-symmetric period.

Conditions on Hamiltonian $H$ ensuring symmetric periodic solutions.

Extension of minimal period conjecture in the context of symplectic matrices.

Abstract

Let $P\in Sp(2n)$ satisfying $P^{k}=I_{2n}$, we consider the minimal $P$-symmetric period problem of the autonomous nonlinear Hamiltonian system \begin{equation*} \dot x(t) = JH^{\prime}(x(t)). \end{equation*} For some symplectic matrices $P$, we show that for any $\tau>0$ the above Hamiltonian system possesses a $k\tau$ periodic solution $x$ with $k\tau$ being its minimal $P$-symmetric period provided $H$ satisfies the Rabinowitz's conditions on the minimal period conjecture, together with that $H$ is convex and $H(Px)=H(x)$.