Generalized St\"ackel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems

TL;DR

This paper introduces a multiparameter generalization of the Stäckel transform that preserves integrability properties of finite-dimensional systems and relates different Hamiltonian systems via reciprocal transformations.

Contribution

It develops a generalized Stäckel transform applicable to a broad class of integrable systems and explores its properties and implications for system equivalences.

Findings

Generalized Stäckel transform preserves Liouville and superintegrability.

Reciprocal transformations relate different integrable systems.

Hamiltonians with different separation curves can be connected through the transform.

Abstract

We present a multiparameter generalization of the St\"ackel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized St\"ackel transform preserves the Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation. Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized St\"ackel transform.