Adler-Kostant-Symes systems as Lagrangian gauge theories

TL;DR

This paper develops a Lagrangian gauge theory framework for Adler-Kostant-Symes integrable systems, extending previous Hamiltonian reduction results to a Lagrangian setting for Lie algebras with invariant scalar products.

Contribution

It introduces a Lagrangian formulation with gauge symmetry that reproduces Adler-Kostant-Symes systems via Dirac's algorithm, generalizing earlier Hamiltonian reduction approaches.

Findings

Constructed a Lagrangian with gauge symmetry for AKS systems.

Demonstrated the Lagrangian reproduces AKS systems through Dirac's algorithm.

Extended reduction techniques to a Lagrangian formalism for Lie algebras with invariant scalar products.

Abstract

It is well known that the integrable Hamiltonian systems defined by the Adler-Kostant-Symes construction correspond via Hamiltonian reduction to systems on cotangent bundles of Lie groups. Generalizing previous results on Toda systems, here a Lagrangian version of the reduction procedure is exhibited for those cases for which the underlying Lie algebra admits an invariant scalar product. This is achieved by constructing a Lagrangian with gauge symmetry in such a way that, by means of the Dirac algorithm, this Lagrangian reproduces the Adler-Kostant-Symes system whose Hamiltonian is the quadratic form associated with the scalar product on the Lie algebra.