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

This paper investigates the symmetry reduction of the regularized n-body problem. The three body problem, regularized 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 generalize this result to the n-body problem for n > 3.