AKS systems and Lepage equivalent problems

TL;DR

This paper introduces a new variational formulation for AKS integrable systems, linking the Lagrangian and Hamiltonian descriptions and providing a novel perspective on their structure.

Contribution

It presents a novel variational approach to AKS systems and constructs their Hamiltonian formulation, bridging gaps in the existing theoretical framework.

Findings

New variational formulation for AKS systems

Established equivalence with traditional Hamiltonian approach

Extended understanding of AKS systems' geometric structure

Abstract

The integrable systems known as "AKS systems" admit a natural formulation in terms of a Hamiltonian picture. The Lagrangian side of these systems are far less known; a version in these terms can be found in a work of Feher et al. The purpose of these notes in to provide a novel description of AKS systems in terms of a variational problem different from the usual in mechanics. Additionally, and using techniques borrowed from an article of M. Gotay, it was possible to build the Hamiltonian side of this variational problem, allowing us to establish the equivalence with the usual approach to these integrable systems.