The Lie-Poisson Structure of the Symmetry Reduced Regularised n-Body   Problem
  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. 
Suntharan Arunasalam and Holger R. Dullin and Diana M.H. Nguyen
math.DS math-ph math.MP