Non-collision periodic solutions of prescribed energy problem for a class of singular Hamiltonian systems

Abstract

We study the existence of non-collision periodic solutions with prescribed energy for the following singular Hamiltonian systems: $$ \cases \ddot q+\nabla V(q)=0, \\ \displaystyle \frac{1}{2}|\dot q|^2+V(q)=H. \endcases $$ In particular for the potential $V(q)\sim -1/\text{\rm dist} (q,D)^\alpha$, where the singular set $D$ is a non-empty compact subset of $\mathbb R^N$, we prove the existence of a non-collision periodic solution for all $H> 0$ and $\alpha\in (0,2)$.