Hyperbolic-like solutions for singular Hamiltonian systems

We study the existence of unbounded solutions of singular Hamiltonian systems: q̈ + ∇V(q)=0, (*) where V(q) ∼ -1/|q|^α is a potential with a singularity. For a class of singular potentials with a strong force α > 2, we show the existence of at least one hyperbolic-like solutions. More precisely, for given H > 0 and θ₊, θ_- ε S^N-1, we find a solution q(t) of (*) satisfying 1/2 |q̇|² + V(q) = H, |q(t) |→ as t → ±∞ lim_t→±∞ q(t)/|q(t)|=θ±.