We prove for the $N$-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level $h>0$ of the motion can also be chosen arbitrarily. Our approach is based on the construction of global viscosity solutions for the Hamilton-Jacobi equation $H(x,d_x u)=h$. We prove that these solutions are fixed points of the associated Lax-Oleinik semigroup. The presented results can also be viewed as a new application of Marchal's Theorem, whose main use in recent literature has been to prove the existence of periodic orbits.
"Viscosity solutions and hyperbolic motions: a new PDE method for the $N$-body problem." Ann. of Math. (2) 192 (2) 499 - 550, September 2020. https://doi.org/10.4007/annals.2020.192.2.5