The N-Body Problem on Coadjoint Orbits

We show (Theorem 3) that the symplectic reduction of the spatial $n$-body problem at non-zero angular momentum is a singular symplectic space consisting of two symplectic strata, one for spatial motions and the other for planar motions. Each stratum is realized as coadjoint orbit in the dual of the Lie algebra of the linear symplectic group $Sp(2n-2)$. The planar stratum arises as the frontier upon taking the closure of the spatial stratum. We reduce by going to center-of-mass coordinates to reduce by translations and boosts and then performing symplectic reduction with respect to the orthogonal group $O(3)$. The theorem is a special case of a general theorem (Theorem 2) which holds for the $n$-body problem in any dimension $d$. This theorem follows largely from a ``Poisson reduction'' theorem, Theorem 1. We achieve our reduction theorems by combining the Howe dual pair perspective of reduction espoused by Lerman-Montgomery-Sjamaar with a normal form arising from a symplectic singular value decomposition due to Xu. We begin the paper by showing how Poisson reduction by the Galilean group rewrites Newton's equations for the $n$-body problem as a Lax pair. In section 6.4 we show that this Lax pair representation of the $n$-body equations is equivalent to the Albouy-Chenciner representation in terms of symmetric matrices.