The Appell hypergeometric functions and classical separable mechanical systems

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.