ELLIPTIC BILLIARDS AND HYPERELLIPTIC FUNCTIONS

The geometrical properties of the elliptical billiard system are relat ed to Poncelet's theorem. This theorem states that if a polygon is ins cribed in a conic and circumscribed about a second conic, every point of the former conic is a vertex of a polygon with the same number of s ides and the same perimeter. Chang and Friedberg have extended this th eorem to three and higher dimensions. They have shown that the geometr ical properties of the hyperelliptic billiard system are related to th e algebraic character of a Poincare map in the phase space. The geomet rical and algebraic properties of the system can be understood in term s of the analytical structure of the equations of motion. These equati ons form a complete system of Abelian integrals. The integrability of the physical system is reflected by the topology of the Riemann surfac es associated to these integrals. The algebraic properties are connect ed with the existence of addition formulas for hyperelliptic functions . The main purpose of this study is to establish such a connection, an d to provide an algebraic proof of Poncelet's theorem in three and hig her dimensions.