The Lie-Poisson structure of the reduced n-body problem

TL;DR

This paper derives a Lie-Poisson structure for the reduced n-body problem in any dimension, enabling the construction of specialized integrators that preserve geometric properties of the system.

Contribution

It introduces a polynomial invariant-based reduction of the n-body problem, revealing a universal Lie-Poisson structure isomorphic to sp(2n-2), regardless of spatial dimension.

Findings

Reduced system has a Lie-Poisson structure isomorphic to sp(2n-2)

Preserves Hamiltonian form with kinetic and potential energy

Constructed a Poisson integrator using splitting methods

Abstract

The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a result we obtain a reduced system with a Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.