Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems

Abstract

In this paper, for any positive integer $n$, we study the Maslov-type index theory of $i_{L_0}$, $i_{L_1}$ and $i_{\sqrt{-1}}^{L_0}$ with $L_0=\{0\}\times {\mathbb{R}}^n\subset {\mathbb{R}}^{2n}$ and $L_1={\mathbb{R}}^n\times \{0\}\subset {\mathbb{R}}^{2n}$. As applications we study the minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltonian systems. For first order nonlinear autonomous reversible Hamiltonian systems in ${\mathbb{R}}^{2n}$, which are semipositive, and superquadratic at zero and infinity, we prove that for any $T>0$, the considered Hamiltonian systems possesses a nonconstant $T$ periodic brake orbit $X_T$ with minimal period no less than $\frac{T}{2n+2}$. Furthermore if ${\int }_0^{T}H{"}_{22}(x_T(t))dt$ is positive definite, then the minimal period of $x_T$ belongs to $\{T,\frac{T}{2}\}$. Moreover, if the Hamiltonian system is even, we prove that for any $T>0$, the considered even semipositive Hamiltonian systems possesses a nonconstant symmetric brake orbit with minimal period belonging to $\{T,\frac{T}{3}\}$

arXiv comment63 pages; MSC classes symmetric, brake orbit, semipositive and reversible, Maslov-type index, minimal period, Hamiltonian systems