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A Geometric Setting for Hamiltonian Perturbation Theory
Anthony D. Blaom, Burwood, Victoria, Australia
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Memoirs of the American Mathematical Society
2001; 112 pp; softcover
Volume: 153
ISBN-10: 0-8218-2720-0
ISBN-13: 978-0-8218-2720-8
List Price: US$57 Individual Members: US$34.20
Institutional Members: US\$45.60
Order Code: MEMO/153/727

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.

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

• Introduction
Part 1. Dynamics
• 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
Part 2. Geometry
• 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