Driven cofactor systems and Hamilton-Jacobi separability

TL;DR

This paper advances the understanding of driven cofactor systems by detailing the construction of quadratic integrals and showing how canonical transformations can simplify the Hamilton-Jacobi problem to a Stäckel-type system.

Contribution

It provides a detailed algorithmic method for constructing quadratic integrals and demonstrates how to reduce the Hamilton-Jacobi problem to an autonomous Stäckel system.

Findings

Explicit construction of quadratic integrals for driven cofactor systems

Canonical transformations reduce time-dependent problems to autonomous Stäckel systems

Enhanced geometric understanding of partially decoupling differential equations

Abstract

This is a continuation of the work initiated in a previous paper on so-called driven cofactor systems, which are partially decoupling second-order differential equations of a special kind. The main purpose in that paper 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 Hamilton-Jacobi problem of the (a priori time-dependent) driven part of the system into that of an equivalent autonomous system of St\"ackel type.