Driven cofactor systems and Hamilton–Jacobi separability

Abstract

Abstract. This is a continuation of the work initiated in [18] on so-called driven cofactor systems, which are partially decoupling second-order differential equations of a special kind. The main purpose in [18] was to obtain an intrinsic, geometrical characterization of such systems, and to explain the basic underlying concepts in a brief note. In the present paper we address the more intricate part of the theory. It involves in the first place understanding all details of an algorithmic construction of quadratic integrals and their involutivity. It secondly requires explaining the subtle way in which suitably constructed canonical transformations reduce the HamiltonJacobi problem of the (a priori time-dependent) driven part of the system into that of an equivalent autonomous system of Stäckel type.

Lifting geometric objects to the dual of the first jet bundle of a bundle fibred over R

“…Special conformal Killing tensors further play a significant role in related work, such as the study of a certain bi-differential calculus [9], and the intrinsic characterization and generalization in [8] of what Lundmark called Newtonian systems of cofactor type (see [19] and [20]). We recently also succeeded in providing a full geometrical description of so-called driven cofactor systems in [26].…”