Non-Hamiltonian systems separable by Hamilton–Jacobi method

Abstract

We show that with every separable calssical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These system are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.

Bi-Hamiltonian representation of Stäckel systems

“…This class, which will hereinafter be referred to as the Benenti class, includes systems generated by conformal Killing tensors [19]- [22], as well as bi-cofactor systems, generated by a pair of conformal Killing tensors [23]- [27]. Here the functions f i define the Stäckel metric while the functions γ i define a separable potential.…”

Driven cofactor systems and Hamilton–Jacobi separability

“…Note in passing that there exists a different technique for obtaining a Hamiltonian representation out of a quasi-Hamiltonian one: roughly, it consists of absorbing the overall factor by a change of timescale, which has the disadvantage, however, that the definition of the new time makes sense only along (the unknown) solutions of the system. This technique is well documented in [3], for example, and was extensively used in the context of cofactor systems in [11]. We believe that the line of approach we adopt here offers more insight in understanding the delicate aspects of the driven nature of our cofactor system.…”

Geometric characterization of driven cofactor systems

“…Cofactor pair systems constitute an interesting subcase: the non-conservative system (g, µ) then has a double cofactor representation, leading to two modified Poisson tensors which are compatible and a hierarchy of first integrals which are in involution with respect to both Poisson structures [3]. For a recent contribution to the subject, see [10].…”