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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.
Curved N-body problem; Spaces of constant curvature; Differential equations; First integrals