Separation of variables in the Hamilton-Jacobi equation for fixed values of the energy
The theory of the separation of variables for Hamilton-Jacobi equations with a fixed value of the energy is revisited, and based on an extension of the Levi-Civita separability conditions involving Lagrangian multipliers. We show that this kind of separation is equivalent to the ordinary separation of the image of the original Hamiltonian under a generalized Jacobi-Maupertuis transformation. The general results are applied to the case of natural Hamiltonians in orthogonal coordinates. The separation is then related to conditions which extend those of St\"ackel and it is characterized by the existence of conformal Killing two-tensors of a special kind.
This is a joint work with Sergio Benenti and Giovanni Rastelli.