The Appell hypergeometric functions and classical separable mechanical systems

TL;DR

This paper explores the connection between Appell hypergeometric functions and classical separable mechanical systems, revealing new formulas for wider classes of separable potentials in integrable billiard systems and geodesic problems.

Contribution

It establishes a novel relationship between hypergeometric functions and separability in classical mechanics, providing new formulas for separable potentials involving Appell functions.

Findings

Expressed separable potential perturbations via Appell hypergeometric functions.

Extended formulas to wider classes of separable potentials.

Maintained constant number of hypergeometric variables for symmetric ellipsoids.

Abstract

A relationship between two old mathematical subjects is observed: the theory of hypergeometric functions and the separability in classical mechanics. Separable potential perturbations of the integrable billiard systems and the Jacobi problem for geodesics on an ellipsoid are expressed through the Appell hypergeometric functions $F_4$ of two variables. Even when the number of degrees of freedom increases, if an ellipsoid is symmetric, the number of variables in the hypergeometric functions does not. Wider classes of separable potentials are given by the obtained new formulae automaticaly.