Point mass dynamics on spherical hypersurfaces

Abstract

The equations of motion are derived for a system of point masses on the (hyper)surface Sn of a sphere embedded in Rn+1 for any dimension n > 1. Owing to the symmetry of the surface, the equations take a particularly simple form when using the Cartesian coordinates of Rn+1. The constraint that the distance of the jth mass ∥rj∥ from the origin remains constant (i.e. each mass remains on the surface) is automatically satisfied by the equations of motion. Moreover, the equations are a Hamiltonian system with a conserved energy as well as a host of conserved angular momenta. Several examples are illustrated in dimensions n = 2 (the sphere) and n = 3 (the glome). This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.