Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems

Abstract

The Darboux-Koenigs metrics in 2D are an important class of conformally flat, non-constant curvature metrics with a single Killing vector and a pair of quadratic Killing tensors. In [arXiv:1804.06904] it was shown how to derive these by using the conformal symmetries of the 2D Euclidean metric. In this paper we consider the conformal symmetries of the 3D Euclidean metric and similarly derive a large family of conformally flat metrics possessing between 1 and 3 Killing vectors (and therefore not constant curvature), together with a number of quadratic Killing tensors. We refer to these as generalised Darboux-Koenigs metrics. We thus construct multi-parameter families of super-integrable systems in 3 degrees of freedom. Restricting the parameters increases the isometry algebra, which enables us to fully determine the Poisson algebra of first integrals. This larger algebra of isometries is then used to reduce from 3 to 2 degrees of freedom, obtaining Darboux-Koenigs kinetic energies with potential functions, which are specific cases of the known super-integrable potentials.

Superintegrable systems on 3 dimensional conformally flat spaces

“…In later sections we seek particular cases of the Hamiltonian functions of Table 4 which admit quadratic integrals of the type described below. We use the method introduced in [2], and used in [5], to construct quadratic invariants out of conformal invariants.…”

“…Remark 5.1 (Comparison with [5]) In [5] we discuss the equivalent system with algebra e 3 , h 1 , f 3 , so to compare formula we must transform all formulae in accordance with ι 13 .…”

“…which (taking Poisson brackets with the isometries) generates another three quadratic integrals. The resulting Poisson algebra is equivalent (under ι 23 and a small change of notation) to that shown in Table 3 of [5]. The algebra has rank 5 and defines a maximally superintegrable system.…”

“…In [2], the method was illustrated by reproducing some well known, non-constant curvature systems in 2 degrees of freedom, such as the Darboux-Koenigs systems [6][7][8], with one linear and two quadratic integrals, and a case from the classification [10] of systems with one linear and a cubic integral. In [5], we applied this method to non-constant curvature systems in 3 degrees of freedom, seeking systems with one linear and three quadratic integrals (since we were interested in maximally superintegrable systems). We found several 3 parameter systems, corresponding to inequivalent choices of Killing vector (first order integral), but, without further restriction, the full Poisson algebra is very difficult to determine.…”

“…In Section 5 we consider the three inequivalent isometry algebras of dimensions 3 and present all superintegrable systems in our class. There are six examples, but several of these already appeared in [5], so we only give brief details, except where we have made significant simplifications or where the case is new.…”