Flat coordinates of flat Stäckel systems

In this article we explicitly construct Stäckel separable systems in separation coordinates with the help of separation curve as introduced by Sklyanin. Further, we construct explicit transformation bewteen separable and ‡at coordinates for ‡at Stäckel systems. We also exploit the geometric structre of these systems in the obtained ‡at coordinates. These coordinates generalize the well known generalized elliptic and generalized parabolic coordinates introduced by C.G. Jacobi. Keywords and phrases: Hamiltonian systems, completely integrable systems, Stäckel systems, Hamilton-Jacobi theory, separable potentials 1 Introduction The search for explicit form ‡at coordinates for metrics that we a priori know are ‡at is not easy. This article is devoted to search for ‡at coordinates for the so called Stäckel systems [1]. The Stäckel systems are roughly speaking (for more precise de…nition, see below) Hamiltonian systems separable in the sense of Hamilton-Jacobi theory by a pointwise transformation to orthogonal coordinates. As such, they are of great importance in theory of classical integrable systems. In this paper we construct separable ‡at systems of Stäckel type directly from scratch i.e. from an appropriate separation curve (or an appropriate set of separation relations [2]) and then …nd ‡at coordinates for (almost) all ‡at Stäckel systems of Benenti type. We also establish the signature of metric tensors of these systems. Further, we present an explicit form of many important geometric objects connected to these ‡at Stäckel systems (namely metric tensors, Killing tensors and separable potentials) in these new coordinates. Thus, we end up with separable ‡at Hamiltonians written in ‡at coordinates of respective pseudo-Euclidian metrics. Our construction encompasses two known cases: Jacobi elliptic coordinates (introduced in [3] and fully described in [4]) and Jacobi parabolic coordinates and also one of the less known cases considered recently by Blaszak and Sergyeyev in [5] (but with no degeneration of coordinate systems). Such ‡at representation of Stäckel systems can be then used for their quantization. What is interesting, when considered systems are additionally maximally superintegrable, i.e. there exist extra n 1 global constants of motion commuting with Hamiltonian, then the related stationary Schrödinger equation can be solved directly in our ‡at coordinates [6]. The paper is organized as follows. In Section 2 we remind basic facts about Stäckel systems and in particular about Stäckel systems of Benenti type. In our approach we construct Stäckel systems directly in their separation coordinates using an appropriate separation relations (separation curve). In Section