A prescribed energy problem for a singular Hamiltonian system with a weak force (Q2367746)

We consider the existence of periodic solutions of a Hamiltonian system \(\ddot q+\nabla V(q)=0\) such that \(1/2\mid \dot q(t)\mid^ 2+V(q(t))=H\) for all \(t\), where \(q\in \mathbb{R}^ N(N\geq 3)\), \(H<0\) is a given number, \(V(q)\) is a potential with a singularity, and \(\nabla V(q)\) denotes its gradient. In a particular case we prove the existence of a generalized solution that may enter the singularity 0. Moreover, under some assumption we estimate the number of collisions of generalized solutions and get the existence of a classical (non-collision) solution.