An ellipsoidal billiard with a quadratic potential

Abstract

There exists an in infite hierarchy of integrable generalizations of the geodesic flow on an n -di- mensional ellipsoid.hese generalizations describe the motion of a point in the force fields of certain polynomial potentials.In the limit as one of semiaxes of the ellipsoidtends to zero,one obtains inte- grable mappings corresponding to billiards with polynomial potentials inside an (n+1)-dimensional ellipsoid. In this paper, for the first time we give explicit expressions for the ellipsoidal billiard with a quadratic (Hooke)potential,its representation in Lax form,and a theta function solution.We also indicate the generating function of the restriction of the potential billiard map to a level set of an energy type integral. The methodwe use to obtain theta function solutions is different from those applied earlier and is based on the calculation of limit values of meromorphic functions on generalized Jacobians.