Minimal P-cyclic periodic brake orbits in Hamiltonian systems

<p style='text-indent:20px;'>We study the Maslov-type index theory under a special Lagrangian boundary condition, namely <inline-formula><tex-math id="M2">\begin{document}$ (P,L_0) $\end{document}</tex-math></inline-formula> boundary condition, which comes naturally in the study of brake orbits of <inline-formula><tex-math id="M3">\begin{document}$ P $\end{document}</tex-math></inline-formula>-invariant Hamiltonian systems with certain orthogonal symplectic matrix <inline-formula><tex-math id="M4">\begin{document}$ P $\end{document}</tex-math></inline-formula> satisfying <inline-formula><tex-math id="M5">\begin{document}$ P^p = {\rm Id} $\end{document}</tex-math></inline-formula>. In this paper, we give some new iteration inequalities of Maslov-type index under this boundary condition. As applications, we consider the minimal cyclic period problems for brake orbits of first order nonlinear autonomous reversible <inline-formula><tex-math id="M6">\begin{document}$ P $\end{document}</tex-math></inline-formula>-symmetric Hamiltonian systems. We prove that if <inline-formula><tex-math id="M7">\begin{document}$ P = R(\theta_1)\diamond\ldots\diamond R(\theta_n) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \theta_i\in [0,\pi] $\end{document}</tex-math></inline-formula> for each <inline-formula><tex-math id="M9">\begin{document}$ 1\leq i\leq n $\end{document}</tex-math></inline-formula>, and the Hamiltonian is strictly convex and superquadratic at zero and infinity, then for each <inline-formula><tex-math id="M10">\begin{document}$ T&gt;0 $\end{document}</tex-math></inline-formula>, the Hamiltonian system possesses a nonconstant <inline-formula><tex-math id="M11">\begin{document}$ P $\end{document}</tex-math></inline-formula>-cyclic brake orbit with minimal <inline-formula><tex-math id="M12">\begin{document}$ P $\end{document}</tex-math></inline-formula>-cyclic period belonging to <inline-formula><tex-math id="M13">\begin{document}$ \{pT, \frac{pT}{p+1}\} $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M14">\begin{document}$ P = R\left(-\frac{2\pi}{p}\right)^{\diamond n} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ p $\end{document}</tex-math></inline-formula> large enough, the system possesses a nonconstant <inline-formula><tex-math id="M16">\begin{document}$ P $\end{document}</tex-math></inline-formula>-cyclic brake orbit with prescribed minimal <inline-formula><tex-math id="M17">\begin{document}$ P $\end{document}</tex-math></inline-formula>-cyclic period.</p>