The perturbation theory of non-commutatively integrable systems is revisited from the point of view of non-Abelian symmetry groups. Using a coordinate system intrinsic to the geometry of the symmetry, we generalize and geometrize well-known estimates of Nekhoroshev (1977), in a class of systems having almost $G$-invariant Hamiltonians. These estimates are shown to have a natural interpretation in terms of momentum maps and co-adjoint orbits. The geometric framework adopted is described explicitly in examples, including the Euler-Poinsot rigid body.

Readership

Graduate students and research mathematicians interested in topology and algebraic geometry.

Table of Contents

• Introduction

• Lie-Theoretic preliminaries
• Action-group coordinates
• On the existence of action-group coordinates
• Naive averaging
• An abstract formulation of Nekhoroshev's theorem
• Applying the abstract Nekhoroshev's theorem to action-group coordinates
• Nekhoroshev-type estimates for momentum maps

• On Hamiltonian $G$-spaces with regular momenta
• Action-group coordinates as a symplectic cross-section
• Constructing action-group coordinates
• The axisymmetric Euler-Poinsot rigid body
• Passing from dynamic integrability to geometric integrability
• Concluding remarks
• Appendix A. Proof of the Nekhoroshev-Lochak theorem
• Appendix B. Proof the ${\mathcal W}$ is a slice
• Appendix C. Proof of the extension lemma
• Appendix D. An application of converting dynamic integrability into geometric integrability: The Euler-Poinsot rigid body revisited
• Appendix E. Dual pairs, leaf correspondence, and symplectic reduction
• Bibliography