Existence of Hyperbolic Motions to a Class of Hamiltonians and Generalized N-Body System via a Geometric Approach

For the classical N -body problem in R d with d ≧ 2 , Maderna–Venturelli in their remarkable paper (Ann Math 192:499–550, 2020) proved the existence of hyperbolic motions with any positive energy constant, starting from any configuration and along any non-collision configuration. Their original proof relies on the long time behavior of solutions by Chazy 1922 and Marchal-Saari 1976, on the Hölder estimate for Mañé’s potential by Maderna 2012, and on the weak KAM theory. We give a new and completely different proof for the above existence of hyperbolic motions. The central idea is that, via some geometric observation, we build up uniform estimates for Euclidean length and angle of geodesics of Mañé’s potential starting from a given configuration and ending at the ray along a given non-collision configuration. Moreover, our geometric approach works for Hamiltonians 1 2 ‖ p ‖ 2 - F ( x ) , where F ( x ) ≧ 0 is lower semicontinuous and decreases very slowly to 0 faraway from collisions. We therefore obtain the existence of hyperbolic motions to such Hamiltonians with any positive energy constant, starting from any admissible configuration and along any non-collision configuration. Consequently, for several important potentials F ∈ C 2 ( Ω ) , we get similar existence of hyperbolic motions to the generalized N -body system x ¨ = ∇ x F ( x ) , which is an extension of Maderna–Venturelli [Ann Math 2020].