The Lie-Poisson Structure of the Symmetry Reduced Regularised n-Body Problem

TL;DR

This paper explores the symmetry reduction of the regularised n-body problem, revealing a Lie-Poisson structure isomorphic to u(3,3) for three bodies and generalizing to more, providing new insights into the problem's geometric structure.

Contribution

It demonstrates that the symmetry-reduced regularised n-body problem has a Lie-Poisson structure isomorphic to u(3,3), extending previous results from the three-body case to n>3.

Findings

The space of quadratic invariants is closed under the symmetry group action.

The Hamiltonian can be expressed in terms of quadratic invariants.

The Lie-Poisson structure is isomorphic to u(3,3).

Abstract

This paper investigates the symmetry reduction of the regularised n-body problem. The three body problem, regularised through quaternions, is examined in detail. We show that for a suitably chosen symmetry group action the space of quadratic invariants is closed and the Hamiltonian can be written in terms of the quadratic invariants. The corresponding Lie-Poisson structure is isomorphic to the Lie algebra u(3,3). Finally, we generalise this result to the n-body problem for n>3.