We study solutions of the Newtonian n-body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as t→ + ∞ or as t→ - ∞. In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium points in a boundary manifold “at infinity”. We show that the flow near these manifolds can be analytically linearized and use this to give a new proof of Chazy’s classical asymptotic formulas. We also address the scattering problem, namely: for solutions which are hyperbolic in both forward and backward time, how are the limiting equilibrium points related? After proving some basic theorems about this scattering relation, we use perturbations of our manifold at infinity to study scattering “near infinity”, that is, when the bodies stay far apart and interact only weakly.
Bibliographical noteFunding Information:
This work was started at MSRI during their semester on Hamiltonian Systems in the Fall of 2019 and all of us would like to acknowledge their support and the wonderful working environment at MSRI. We would like to acknowledge useful discussions with other MSRI members attending a seminar on scattering at MSRI, notably Tere M. Seara, Amadeu Delshams, Jim Meiss, and Pau Martin. Montgomery would like to acknowledge useful discussions with Andreas Knauf and Maciej Zworski. Moeckel would like to acknowledge support from NSF Grant DMS-1712656, NCTS in Hsinchu, Taiwan, Université Paris-Dauphine and IMCCE, Paris. Guowei Yu thanks useful discussion with Xijun Hu and Yuwei Ou, and the support of Nankai Zhide Foundation. Nathan Duignan thanks the support of the Australian government’s Endeavour Fellowship.