The non-existence of centre-of-mass and linear-momentum integrals in the   curved N-body problem
  We provide a class of orbits in the curved N-body problem for which no point that could play the role of the centre of mass is fixed or moves uniformly along a geodesic. This proves that the equations of motion lack centre-of-mass and linear-momentum integrals. 
Florin Diacu
math.DS math-ph math.MP