Electronic ISBN:  9781470403201 
Product Code:  MEMO/153/727.E 
List Price:  $57.00 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 153; 2001; 112 ppMSC: Primary 70; 37;
The perturbation theory of noncommutatively integrable systems is revisited from the point of view of nonAbelian symmetry groups. Using a coordinate system intrinsic to the geometry of the symmetry, we generalize and geometrize wellknown 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 coadjoint orbits. The geometric framework adopted is described explicitly in examples, including the EulerPoinsot rigid body.
ReadershipGraduate students and research mathematicians interested in topology and algebraic geometry.

Table of Contents

Chapters

Introduction

Part 1. Dynamics

1. Lietheoretic preliminaries

2. Actiongroup coordinates

3. On the existence of actiongroup coordinates

4. Naive averaging

5. An abstract formulation of Nekhoroshev’s theorem

6. Applying the abstract Nekhoroshev theorem to actiongroup coordinates

7. Nekhoroshevtype estimates for momentum maps

Part 2. Geometry

8. On Hamiltonian $G$spaces with regular momenta

9. Actiongroup coordinates as a symplectic crosssection

10. Constructing actiongroup coordinates

11. The axisymmetric EulerPoinsot rigid body

12. Passing from dynamic integrability to geometric integrability

13. Concluding remarks


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The perturbation theory of noncommutatively integrable systems is revisited from the point of view of nonAbelian symmetry groups. Using a coordinate system intrinsic to the geometry of the symmetry, we generalize and geometrize wellknown 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 coadjoint orbits. The geometric framework adopted is described explicitly in examples, including the EulerPoinsot rigid body.
Graduate students and research mathematicians interested in topology and algebraic geometry.

Chapters

Introduction

Part 1. Dynamics

1. Lietheoretic preliminaries

2. Actiongroup coordinates

3. On the existence of actiongroup coordinates

4. Naive averaging

5. An abstract formulation of Nekhoroshev’s theorem

6. Applying the abstract Nekhoroshev theorem to actiongroup coordinates

7. Nekhoroshevtype estimates for momentum maps

Part 2. Geometry

8. On Hamiltonian $G$spaces with regular momenta

9. Actiongroup coordinates as a symplectic crosssection

10. Constructing actiongroup coordinates

11. The axisymmetric EulerPoinsot rigid body

12. Passing from dynamic integrability to geometric integrability

13. Concluding remarks