New classes of quadratically integrable systems with velocity dependent potentials: non-subgroup type cases

TL;DR

This paper explores new classes of three-dimensional quadratic integrable systems with velocity-dependent potentials, revealing non-separable cases with generalized integrals, and introduces two novel integrable systems with elliptic cylindrical integrals.

Contribution

It identifies and analyzes non-subgroup type integrable systems with velocity-dependent potentials, discovering two new non-separable integrable systems with elliptic cylindrical integrals.

Findings

Two new integrable systems with velocity-dependent potentials and elliptic cylindrical integrals.

Most other cases studied correspond to known systems with standard integrals.

In the zero vector potential limit, systems reduce to free motion, exhibiting separability in all orthogonal coordinates.

Abstract

We study quadratic integrability of systems with velocity dependent potentials in three-dimensional Euclidean space. Unlike in the case with only scalar potential, quadratic integrability with velocity dependent potentials does not imply separability in the configuration space. The leading order terms in the pairs of commuting integrals can either generalize or have no relation to the forms leading to separation in the absence of a vector potential. We call such pairs of integrals generalized, to distinguish them from the standard ones, which would correspond to separation. Here we focus on three cases of generalized non-subgroup type integrals, namely elliptic cylindrical, prolate / oblate spheroidal and circular parabolic integrals, together with one case not related to any coordinate system. We find two new integrable systems, non-separable in the configuration space, both with generalized elliptic cylindrical integrals. In the other cases, all systems found were already known and possess standard pairs of integrals. In the limit of vanishing vector potential, both systems reduce to free motion and therefore separate in every orthogonal coordinate system.