| eBook 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 |
| eBook 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 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 153; 2001; 112 ppMSC: 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.
ReadershipGraduate students and research mathematicians interested in topology and algebraic geometry.
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Table of Contents
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Chapters
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Introduction
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Part 1. Dynamics
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1. Lie-theoretic preliminaries
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2. Action-group coordinates
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3. On the existence of action-group coordinates
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4. Naive averaging
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5. An abstract formulation of Nekhoroshev’s theorem
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6. Applying the abstract Nekhoroshev theorem to action-group coordinates
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7. Nekhoroshev-type estimates for momentum maps
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Part 2. Geometry
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8. On Hamiltonian $G$-spaces with regular momenta
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9. Action-group coordinates as a symplectic cross-section
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10. Constructing action-group coordinates
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11. The axisymmetric Euler-Poinsot rigid body
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12. Passing from dynamic integrability to geometric integrability
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13. Concluding remarks
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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.
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
