Global structure and geodesics for Koenigs superintegrable systems

Abstract

Starting from the framework defined by Matveev and Shevchishin we derive the local and the global structure for the four types of super-integrable Koenigs metrics. These dynamical systems are always defined on non-compact manifolds, namely R 2 and H 2 . The study of their geodesic flows is made easier using their linear and quadratic integrals. Using Carter (or minimal) quantization we show that the formal superintegrability is preserved at the quantum level and in two cases, for which all of the geodesics are closed, it is even possible to compute the discrete spectrum of the quantum hamiltonian.

Projectively equivalent 2-dimensional superintegrable systems with projective symmetries

“…These references discuss a particular class of systems without potential. Reference [62], on the other hand, studies (Darboux-)Koenigs systems with potential, from a global perspective. Our approach is local and includes a potential, while retaining a high degree of generality.…”

Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

“…is nothing but Koenigs metric of type 3 as it is written in Theorem 7 of [15] (setting ξ = 0) which was shown to be globally defined on the manifold M ∼ = H 2 . This Koenigs metric of type 3 suggests the following generalization to SI systems with integrals of any even degree larger than 4:…”

“…As a first check let us consider the simplest case n = 1 for which the integrals are merely quadratic: we should recover one of the Koenigs metrics [15].…”

“…Their analysis, as we will show later on, can be generalized to integrals of any degree, starting from degree 2. In this last simple case it was proved in [15] that one recovers Koenigs metrics [8] studied and generalized in [10]. Koenigs where H, S 1 , S 2 are of second degree in the momenta.…”

Superintegrable geodesic flows on the hyperbolic plane

“…In fact a prior work by Koenigs [2], popularized and generalized in [3] and [4], involving a completely different analysis, had established that for a quadratic extra integral only the three issues stated above were possible. Unfortunately the geodesic flow of Koenigs superintegrable (SI) systems never meet S 2 as emphasized in [8].…”