  By using the variational minimizing method with a special constraint and the
direct variational minimizing method without constraint, we study second order
Hamiltonian systems with a singular potential $V\in C^2(R^n\backslash O,R)$ and
$V\in C^1(R^2\backslash O,R)$ which may have an unbounded potential well, and
prove the existence of non-trivial periodic solutions with a prescribed energy.
Our results can be regarded as some complements of the well-known Theorems of
Benci-Gluck-Ziller-Hayashi and Ambrosetti-Coti Zelati and so on.
