Department of Mathematics, University of Turin,

Via Carlo Alberto 10, 10123 Torino, ITALY

E-mail: chanu@dm.unito.it

**Separation of variables in the Hamilton-Jacobi equation
for fixed values of the energy**

**Abstract:**

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.