A conservative Newton system ¨q = -∇V(q) in Rnis called separable when the Hamilton-Jacobi equation for the Natural Hamiltonian H = ½p2+ V (q) can be solved through separation of variables in some curvilinear coordinates. If these coordinates are orthogonal, the Newton system admits n first integrals, which all have separable Stäckel form with quadratic dependence on p.
We study here separability of the more general class of Newton systems ¨q = - (cof G)-1∇W(q) that admit n quadratic first integrals. We prove that a related system with the same integrals can be transformed through a non-canonical transformation into a Stäckel separable Hamiltonian system and solved by quadratures, providing a solution to the original system.
The separation coordinates, which are defined as characteristic roots of a linear pencil G - μ~G of elliptic coordinates matrices, generalize the well known elliptic and parabolic coordinates. Examples of such new coordinates in two and three dimensions are given.
These results extend, in a new direction, the classical separability theory for natural Hamiltonians developed in the works if Jacobi, Liouville, Stäckel, Levi-Civita, Eisenhart, Bebenti, Kalnins and Miller.