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This paper we consider for the N-body problem with potential 1/r{\alpha} (0 <{\alpha} < 1) the existence of hyperbolic motions for any prescribed limitshape and any given initial configuration of the bodies. Here E is theEuclidean space where the bodies moving and is the norm induced by the innerproduct. The energy level h > 0 of the motion can also be chosen arbitrarily.We use the global viscosity solutions for the Hamilton-Jacobi equation H (x,dxu) = h and geodesics.

A N-body problem with weak force potential through Hamilton-Jacobi equation approach