The N-body problem in d-dimension space has symmetry group SE(d).

Centre of mass reduction leads to a system with SO(d) symmetry acting

diagonally on positions and momenta. For N=3, d=4 reduction of the SO(4)

symmetry is complicated because the tensor of inertia is non-invertible.

The fully reduced system has 4 degrees of freedom and a Hamiltonian that 

is not polynomial in the momenta. The most surprising property of the 

reduced Hamiltonian is that it has equilibria that are minima.