A universal Hamilton-Jacobi equation for second-order ODEs

A universal version of the Hamilton-Jacobi equation on R x TM arises from t he Liouville-Arnol'd theorem for a completely integrable system on a finite -dimensional manifold M. We give necessary and sufficient conditions for su ch complete integrability to imply a canonical separability of both this un iversal Hamilton-Jacobi equation and its traditional counterpart. The geode sic case is particularly interesting. We show that these conditions also ap ply for systems of second-order ordinary differential equations (contact Bo ws) which are not Euler-Lagrange. The Kerr metric, the Toda lattice and a c ompletely integrable contact flow are given as examples.