On Reciprocal Equivalence of Stäckel Systems

Abstract

In this paper we ivestigate Stäckel transforms between di¤erent classes of parameter-dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel-related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.

Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians

“…Let us now find whether the functionsh m r also constitute a Poisson algebra. As it has been demonstrated in [3] {h r ,h s } = n i,j=1…”

“…In Section 4 we prove a theorem (Theorem 4.1) which establishes an explicit relation between Poisson algebras of h i and Lie-algebras of the corresponding non-homogeneous hydrodynamic vector fields (1.1). In Section 5 we exploit the notion of Stäckel transform [3,4,8,14,21] and analyze which of our systems h i can be mapped by this transform to new systemsh i in such a way that the Hamiltoniansh i also constitute an algebra. In this way we obtain new non-homogeneous hydrodynamic equations.…”

Classical and Quantum Superintegrability of Stäckel Systems

“…In [22] the authors introduced a multiparameter generalization of this transform. This idea has been further developed in [8] and later in [4].…”

“…. ,h 2n−1 ) where the first n commuting Hamiltoniansh r are defined by (see [4]) the following separation relations whereh r for r = 1, . .…”

Integrable quantum Stäckel systems

“…It is also important to note that all other classes of Stäckel systems considered in the present paper, for which we proved that quantum integrability does not survive, are not independent from Benenti class. In fact, all remaining classes of Stäckel systems (3.1) are related to the Benenti class by multi-parameter generalized Stäckel transforms at the level of Hamiltonians and by the so-called reciprocal transformations at the level of equations of motion [27,28,29].…”