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

We study the existence of non-collision periodic solutions with prescribed energy for the following singular Hamiltonian systems: {(q) double overdot +del V(q)=0 {1/2 vertical bar(q) over dot vertical bar(2) + V(q)=H In particular for the potential V(q) similar to -1/dist (q, D)(alpha), where the singular set D is a non-empty compact subset of R-N, we prove the existence of a non-collision periodic solution for all H > 0 and alpha is an element of (0,2).