Point mass dynamics on spherical hypersurfaces

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. Due 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‖r_j‖ 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).