We consider the existence of periodic solutions of a Hamiltonian system q ̈ + ∇V(q) = 0 (HS) such that 1 2 | q ̇ (t)| 2+ V(q(t)) = H for all t, where q ∈ R N ( N ≥ 3), H <0 is a given number, V( q) ∈ C 2( R N \\{0}, R) is a potential with a singularity, and ∇ V/( q) denotes its gradient. We consider a potential V( q) which behaves like −l/| q| α (α ∈ (0, 2)). In particular, in case V( q) satisfies ∇V(q) q ≤ −α 1V(q) for all q ∈ R N\\{0}, and ∇ V( q) q ≤ − α 2 V( q) for all 0 < | q| ≤ R 0 for α 1, α 2 ∈ (0, 2), R 0 > 0, we prove the existence of a generalized solution that may enter the singularity 0. Moreover, under the assumption V(q) = − 1 |q| α + W(q), where 0 < α < 2 and | q| α W( q), | q| α + 1 ∇ W( q), | q| α + 2 ∇ 2 W( q) → 0 as | q| → 0, we estimate the number of collisions of generalized solutions. In particular, we get the existence of a classical ( non-collision) solution of (HS) for α ∈ (l, 2) when N ≥ 4 and for α ∈ (4/3, 2) when N = 3.