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 $\alpha\in (0,2)$.