Symmetries and Integrability

TL;DR

This survey explores finite-dimensional integrable systems with Hamiltonian group actions, focusing on noncommutative integrability, collective motions, and reductions, including generalizations of classical rigid body problems.

Contribution

It provides a comprehensive overview of integrability in Hamiltonian systems with symmetry, emphasizing noncommutative integrability and reduction techniques.

Findings

Analysis of G-invariant systems and their integrability

Study of collective motions and reduced integrability

Generalizations of the Hess–Appel'rot problem

Abstract

This is a survey on finite-dimensional integrable dynamical systems related to Hamiltonian $G$-actions. Within a framework of noncommutative integrability we study integrability of $G$-invariant systems, collective motions and reduced integrability. We also consider reductions of the Hamiltonian flows restricted to their invariant submanifolds generalizing classical Hess--Appel'rot case of a heavy rigid body motion.