  The classical n-body problem in d-dimensional space is invariant under the
Galilean symmetry group. We reduce by this symmetry group using the method of
polynomial invariants. As a result we obtain a reduced system with a
Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The
reduction preserves the natural form of the Hamiltonian as a sum of kinetic
energy that depends on velocities only and a potential that depends on
positions only. Hence we proceed to construct a Poisson integrator for the
reduced n-body problem using a splitting method.
