Transforming St\"{a}ckel Hamiltonians of Benenti type to polynomial form

TL;DR

This paper introduces two canonical transformations that convert Stäckel Hamiltonians of Benenti type into polynomial form, providing new proofs and geometric insights in both Viète and Newton coordinates.

Contribution

It presents a new proof that the Newton coordinate transformation yields polynomial Hamiltonians and details the geometric structure in both coordinate systems.

Findings

Transformation to Newton coordinates results in polynomial Hamiltonians.

Geometric structures are characterized in both Viète and Newton coordinates.

The paper clarifies the role of these transformations in integrable systems.

Abstract

In this paper we discuss two canonical transformations that turn St\"{a}ckel separable Hamiltonians of Benenti type into polynomial form: transformation to Vi\`ete coordinates and transformation to Newton coordinates. Transformation to Newton coordinates has been applied to these systems only very recently and in this paper we present a new proof that this transformation indeed leads to polynomial form of St\"{a}ckel Hamiltonians of Benenti type. Moreover we present all geometric ingredients of these Hamiltonians in both Vi\`ete and Newton coordinates.