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

摘要

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