Separation of Variables and the Geometry of Jacobians

TL;DR

This survey explores how separation of variables relates to the geometry of Jacobians in algebraically integrable Hamiltonian systems, highlighting a natural geometric class that includes many classical cases.

Contribution

It identifies a natural geometric class of integrable systems with separations linked to Jacobians of Riemann surfaces, unifying many known cases.

Findings

Separation of variables can be understood through the geometry of Jacobians.

Many classical integrable systems fall into this geometric class.

The approach provides a unified geometric framework for these systems.

Abstract

This survey examines separation of variables for algebraically integrable Hamiltonian systems whose tori are Jacobians of Riemann surfaces. For these cases there is a natural class of systems which admit separations in a nice geometric sense. This class includes many of the well-known cases.