Isospectral Flow and Liouville-Arnold Integration in Loop Algebras

TL;DR

This paper demonstrates how classical integrable Hamiltonian systems can be understood through isospectral flows in loop algebras, using spectral invariants and Liouville-Arnold methods for explicit integration.

Contribution

It introduces a unified framework for integrable systems in loop algebras, including spectral curve invariants, spectral Darboux coordinates, and separation of variables techniques.

Findings

Examples include Neumann oscillator, nonlinear Schrödinger, sine-Gordon systems

Spectral curves are invariant and facilitate integration

Liouville-Arnold method yields explicit solutions via Abel maps

Abstract

A number of examples of Hamiltonian systems that are integrable by classical means are cast within the framework of isospectral flows in loop algebras. These include: the Neumann oscillator, the cubically nonlinear Schr\"odinger systems and the sine-Gordon equation. Each system has an associated invariant spectral curve and may be integrated via the Liouville-Arnold technique. The linearizing map is the Abel map to the associated Jacobi variety, which is deduced through separation of variables in hyperellipsoidal coordinates. More generally, a family of moment maps is derived, identifying certain finite dimensional symplectic manifolds with rational coadjoint orbits of loop algebras. Integrable Hamiltonians are obtained by restriction of elements of the ring of spectral invariants to the image of these moment maps. The isospectral property follows from the Adler-Kostant-Symes theorem, and gives rise to invariant spectral curves. {\it Spectral Darboux coordinates} are introduced on rational coadjoint orbits, generalizing the hyperellipsoidal coordinates to higher rank cases. Applying the Liouville-Arnold integration technique, the Liouville generating function is expressed in completely separated form as an abelian integral, implying the Abel map linearization in the general case.