Simultaneous separation for the Neumann and Chaplygin systems

TL;DR

This paper demonstrates that the Neumann and Chaplygin systems on the sphere are simultaneously separable using elliptic coordinates and Backlund transformations, revealing new links between different integrable systems.

Contribution

It introduces a method to obtain separable variables for multiple systems via Backlund transformations, unifying their integrability properties.

Findings

Neumann and Chaplygin systems are separable in elliptic coordinates.

Backlund transformations relate different integrable systems and their separable variables.

New analogs of hetero Backlund transformations are proposed for various Hamilton-Jacobi equations.

Abstract

The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Backlund transformation. We also prove that after similar Backlund transformations other curvilinear coordinates on the sphere and on the plane become variables of separations for the system with quartic potential, for the Henon-Heiles system and for the Kowalevski top. It allows us to say about some analog of the hetero Backlund transformations relating different Hamilton-Jacobi equations.