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A Geometric Setting for Hamiltonian Perturbation Theory
 
Anthony D. Blaom Burwood, Victoria, Australia
Front Cover for A Geometric Setting for Hamiltonian Perturbation Theory
Available Formats:
Electronic ISBN: 978-1-4704-0320-1
Product Code: MEMO/153/727.E
List Price: $57.00
MAA Member Price: $51.30
AMS Member Price: $34.20
Front Cover for A Geometric Setting for Hamiltonian Perturbation Theory
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  • Front Cover for A Geometric Setting for Hamiltonian Perturbation Theory
  • Back Cover for A Geometric Setting for Hamiltonian Perturbation Theory
A Geometric Setting for Hamiltonian Perturbation Theory
Anthony D. Blaom Burwood, Victoria, Australia
Available Formats:
Electronic ISBN:  978-1-4704-0320-1
Product Code:  MEMO/153/727.E
List Price: $57.00
MAA Member Price: $51.30
AMS Member Price: $34.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1532001; 112 pp
    MSC: Primary 70; 37;

    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
     
     
    • Chapters
    • Introduction
    • Part 1. Dynamics
    • 1. Lie-theoretic preliminaries
    • 2. Action-group coordinates
    • 3. On the existence of action-group coordinates
    • 4. Naive averaging
    • 5. An abstract formulation of Nekhoroshev’s theorem
    • 6. Applying the abstract Nekhoroshev theorem to action-group coordinates
    • 7. Nekhoroshev-type estimates for momentum maps
    • Part 2. Geometry
    • 8. On Hamiltonian $G$-spaces with regular momenta
    • 9. Action-group coordinates as a symplectic cross-section
    • 10. Constructing action-group coordinates
    • 11. The axisymmetric Euler-Poinsot rigid body
    • 12. Passing from dynamic integrability to geometric integrability
    • 13. Concluding remarks
  • Requests
     
     
    Review Copy – for reviewers who would like to review an AMS book
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1532001; 112 pp
MSC: Primary 70; 37;

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.

  • Chapters
  • Introduction
  • Part 1. Dynamics
  • 1. Lie-theoretic preliminaries
  • 2. Action-group coordinates
  • 3. On the existence of action-group coordinates
  • 4. Naive averaging
  • 5. An abstract formulation of Nekhoroshev’s theorem
  • 6. Applying the abstract Nekhoroshev theorem to action-group coordinates
  • 7. Nekhoroshev-type estimates for momentum maps
  • Part 2. Geometry
  • 8. On Hamiltonian $G$-spaces with regular momenta
  • 9. Action-group coordinates as a symplectic cross-section
  • 10. Constructing action-group coordinates
  • 11. The axisymmetric Euler-Poinsot rigid body
  • 12. Passing from dynamic integrability to geometric integrability
  • 13. Concluding remarks
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.