Notes on the degenerate integrability of reduced systems obtained from the master systems of free motion on cotangent bundles of compact Lie groups

TL;DR

This paper investigates the degenerate integrability of reduced systems derived from free motion on cotangent bundles of compact Lie groups, showing that such integrability properties are preserved under Hamiltonian reduction.

Contribution

It demonstrates that the degenerate integrability of the master system on $T^*G$ is inherited by the reduced system on the principal orbit type quotient space.

Findings

Degenerate integrability is preserved after reduction.

The proof applies to other Hamiltonian reduction examples.

Reduced systems inherit integrability properties from master systems.

Abstract

The reduction of the `master system' of free motion on the cotangent bundle $T^*G$ of a compact, connected and simply connected, semisimple Lie group is considered using the conjugation action of $G$. It is proved that the restriction of the reduced system to the smooth component of the quotient space $T^*G/G$, given by the principal orbit type, inherits the degenerate integrability of the master system. The proof can be generalized easily to other interesting examples of Hamiltonian reduction.