A theorem for a class of motions in a spherical geometry

Abstract

Considering the motion, at classical level, of a point particle constrained to move under the influence of a conservative centre of force, on a N-sphere S(N), embedded in a Euclidean (N+1)-dimensional space, the author has shown that, if the motion on S(N) is obtained from the motion on a tangent N-plane Pi (N) by means of a central projection, then the equations of motion of S(N) can be obtained from those on the Euclidean Pi (N) by a local reparameterisation of time. Some consequences of the theorem are also discussed.