The non-existence of centre-of-mass and linear-momentum integrals in the curved N-body problem

TL;DR

This paper demonstrates that in the curved N-body problem, there are specific orbits where no fixed or uniformly moving point can serve as a center of mass, proving the absence of such integrals.

Contribution

It establishes the non-existence of centre-of-mass and linear-momentum integrals in the curved N-body problem through explicit orbit examples.

Findings

No point can serve as a fixed or uniformly moving center of mass.

The equations of motion lack centre-of-mass and linear-momentum integrals.

Provides a class of orbits illustrating this non-existence.

Abstract

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.