The Lie–Poisson structure of the symmetry reduced regularizedn-body problem

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.

Symmetry reduction for central force problems

“…Consider a particle moving in three-dimensions subject to a conservative central force. It has position  Î q 3 , momentum  Î p 3 , and total (kinetic plus potential) energy…”