Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems

TL;DR

This paper establishes minimal period estimates for brake orbits in nonlinear symmetric Hamiltonian systems, proving existence of periodic brake orbits with specific minimal periods under super-quadratic convex conditions.

Contribution

It introduces new minimal period bounds for brake orbits in symmetric Hamiltonian systems with super-quadratic convex Hamiltonians.

Findings

Existence of τ-periodic brake orbits with minimal period τ or τ/2.

Brake orbits exist for all positive τ under the given conditions.

The results apply to Hamiltonians satisfying symmetry and convexity assumptions.

Abstract

In this paper, we consider the minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. We prove that if the Hamiltonian function $H\in C^2(\Bbb R^{2n}, \Bbb R)$ is super-quadratic and convex, for every number $\tau>0$, there exists at least one $\tau$-periodic brake orbit $(\tau,x)$ with minimal period $\tau$ or $\tau/2$ provided $H(Nx)=H(x)$.