Stackel separability for Newton systems of cofactor type

Abstract

A conservative Newton system (d/dt)^2 q = -grad V(q) in R^n is called separable when the Hamilton--Jacobi equation for the natural Hamiltonian H = (1/2) p^2 + V(q) can be solved through separation of variables in some curvilinear coordinates. If these coordinates are orhogonal, the Newton system admits n first integrals, which all have separable Stackel form with quadratic dependence on p. We study here separability of the more general class of Newton systems (d/dt)^2 q = -cof G^(-1) grad 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 Stackel separable Hamiltonian system and solved by qudratures, providing a solution to the original system. The separation coordinates, which are defined as characteristic roots of a linear pencil (G - mu 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 of Jacobi, Liouville, Stackel, Levi-Civita, Eisenhart, Benenti, Kalnins and Miller.