A universal Hamilton-Jacobi equation for second-order ODEs

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