A geometric setting for Hamiltonian perturbation theory
Author(s)
Bibliographic Information
A geometric setting for Hamiltonian perturbation theory
（Memoirs of the American Mathematical Society, no. 727）
American Mathematical Society, 2001
Available at 15 libraries
Note
"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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