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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$54
Individual Members: US$32.40
Institutional Members: US$43.20
Order Code: MEMO/153/727
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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
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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