On integrable potential perturbations of the Jacobi problem for the geodesies on the ellipsoid

Abstract

All integrable mechanical systems describing motion of a particle on an ellipsoid surface in three-dimensional space are described in the class of the Loran polynomial potentials. Two countable families of the basic solutions are obtained. Explicit formulae are given. The limit, when the considered system goes into the billiard system within an ellipse, is analysed, and the results are compared with those obtained previously, in relation to the billiard system.