Coupling constant metamorphosis as an integrability-preserving transformation for general finite-dimensional dynamical systems and ODEs

TL;DR

This paper extends the coupling constant metamorphosis to general finite-dimensional dynamical systems and ODEs, providing conditions for integrability preservation and formulas relating solutions.

Contribution

It introduces a generalized transform that preserves integrability in broader dynamical systems beyond Hamiltonian cases.

Findings

Transform interchanges integrals of motion with parameters.

Conditions identified for integrability preservation.

Provides formulas linking solutions before and after transformation.

Abstract

In the present paper we extend the multiparameter coupling constant metamorphosis, also known as the generalized St\"ackel transform, from Hamiltonian dynamical systems to general finite-dimensional dynamical systems and ODEs. This transform interchanges the values of integrals of motion with the parameters these integrals depend on but leaves the phase space coordinates intact. Sufficient conditions under which the transformation in question preserves integrability and a simple formula relating the solutions of the original system to those of the transformed one are given.