Geometric Reductions, Dynamics and Controls for Hamiltonian System with Symmetry

TL;DR

This survey explores recent advances in symmetric reductions and Hamilton-Jacobi theory for various Hamiltonian systems, emphasizing their geometric structures and control aspects.

Contribution

It provides a comprehensive overview of recent developments in symmetric reductions and Hamilton-Jacobi theory for controlled Hamiltonian systems.

Findings

Revealed internal relationships between geometrical structures and system controls.

Summarized progress in nonholonomic and magnetic Hamiltonian systems.

Connected phase space geometry with control dynamics.

Abstract

This is a survey article, from the viewpoint of the completeness of the Marsden- Weinstein reduction, to introduce briefly some recent developments of the symmetric reductions and Hamilton-Jacobi theory of the regular controlled Hamiltonian systems, the nonholonomic controlled Hamiltonian systems and the controlled magnetic Hamiltonian systems. These research reveal the deeply internal relationships of the geometrical structures of phase spaces, the nonholonomic constraint, the dynamical vector fields and the controls of these systems.