On reciprocal equivalence of St\"ackel systems

TL;DR

This paper explores the relationships between different St"ackel systems, demonstrating that they can be classified into equivalence classes connected by St"ackel transforms, and provides explicit formulas relating their solutions.

Contribution

It establishes the classification of St"ackel systems into equivalence classes and derives explicit solution relations, simplifying existing proofs.

Findings

All St"ackel systems of the same dimension form equivalence classes.

Any two geodesic St"ackel systems are St"ackel equivalent.

Provided explicit formulas relating solutions of St"ackel-related systems.

Abstract

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