Dirac method and symplectic submanifolds in the cotangent bundle of a factorizable Lie group

TL;DR

This paper explores symplectic submanifolds within the cotangent bundle of factorizable Lie groups, utilizing the Dirac method to analyze their structure, symmetries, and integrability, connecting to the AKS theory.

Contribution

It introduces a framework for studying symplectic submanifolds with second class constraints in cotangent bundles of factorizable Lie groups, linking Dirac brackets to integrability.

Findings

Derived fundamental Dirac brackets for these submanifolds

Identified symmetries and collective dynamics

Connected the structure to AKS integrability theory

Abstract

In this work we study some symplectic submanifolds in the cotangent bundle of a factorizable Lie group defined by second class constraints. By applying the Dirac method, we study many issues of these spaces as fundamental Dirac brackets, symmetries, and collective dynamics. This last item allows to study integrability as inherited from a system on the whole cotangent bundle, leading in a natural way to the AKS theory for integrable systems.