A geometric setting for Hamiltonian perturbation theory
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Bibliographic Information
A geometric setting for Hamiltonian perturbation theory
（Memoirs of the American Mathematical Society, no. 727）
American Mathematical Society, 2001
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"September 2001, volume 153, number 727 (third of 5 numbers)"
Includes bibliographical references (p. 110-112)
Description and Table of Contents
Description
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.
Table of Contents
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.
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