 | Abraham R. Foundations of mechanics / R.Abraham, J.E.Marsden; with the assistance of T.Rauiu, R.Cushman. - 2nd ed. - Reading: Benjamin, 1978. - xxiv, 826 p.: ill. - Bibliogr.: p.759-789. - Ind.: p.791-806. - Suppl.: p.809-826. - Ref.: p.824-826. - ISBN 978-0-8218-4438-0
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Preface to the AMS Chelsea Edition ........................... xiii
Preface to the Second Edition .................................. xv
Preface to the First Edition ................................. xvii
Introduction .................................................. xix
Preview ..................................................... xxiii
PART I PRELIMINARIES ............................................ 1
Chapter 1 Differential Theory .................................. 3
1.1 Topology ................................................... 3
Exercises ................................................. 17
1.2 Finite-Dimensional Banach Spaces .......................... 17
Exercises ................................................. 19
1.3 Local Differential Calculus ............................... 20
Exercises ................................................. 30
1.4 Manifolds and Mappings .................................... 31
Exercises ................................................. 36
1.5 Vector Bundles ............................................ 37
Exercises ................................................. 41
1.6 The Tangent Bundle ........................................ 42
Exercises ................................................. 50
1.7 Tensors ................................................... 52
Exercises ................................................. 59
Chapter 2 Calculus on Manifolds ............................... 60
2.1 Vector Fields as Dynamical Systems ........................ 60
Exercises ................................................. 78
2.2 Vector Fields as Differential Operators ................... 78
Exercises ................................................. 98
2.3 Exterior Algebra ......................................... 101
Exercises ................................................ 109
2.4 Cartan's Calculus of Differential Forms .................. 109
Exercises ................................................ 121
2.5 Orientable Manifolds ..................................... 122
Exercises ................................................ 131
2.6 Integration on Manifolds ................................. 131
Exercises ................................................ 143
2.7 Some Riemannian Geometry ................................. 144
Exercises ................................................ 156
PART II ANALYTICAL DYNAMICS ................................... 159
Chapter 3 Hamiltonian and Lagrangian Systems ................. 161
3.1 Symplectic Algebra ....................................... 161
Exercises ................................................ 174
3.2 Symplectic Geometry ...................................... 174
Exercises ................................................ 185
3.3 Hamiltonian Vector Fields and Poisson Brackets ........... 187
Exercises ................................................ 199
3.4 Integral Invariants, Energy Surfaces, and Stability ...... 201
Exercises ................................................ 208
3.5 Lagrangian Systems ....................................... 208
Exercises ................................................ 217
3.6 The Legendre Transformation .............................. 218
Exercises ................................................ 223
3.7 Mechanics on Riemannian Manifolds ........................ 224
Exercises ................................................ 244
3.8 Variational Principles in Mechanics ...................... 246
Exercises ................................................ 252
Chapter 4 Hamiltonian Systems with Symmetry .................. 253
4.1 Lie Groups and Group Actions ............................. 253
Exercises ................................................ 271
4.2 The Momentum Mapping ..................................... 276
Exercises ................................................ 295
4.3 Reduction of Phase Spaces with Symmetry .................. 298
Exercises ................................................ 309
4.4 Hamiltonian Systems on Lie Groups and the Rigid Body ..... 311
Exercises ................................................ 338
4.5 The Topology of Simple Mechanical Systems ................ 338
Exercises ................................................ 359
4.6 The Topology of the Rigid Body ........................... 360
Exercises ................................................ 368
Chapter 5 Hamilton-Jacobi Theory and Mathematical Physics .... 370
5.1 Time-Dependent Systems ................................... 370
Exercises ................................................ 378
5.2 Canonical Transformations and Hamilton-Jacobi Theory ..... 379
Exercises ................................................ 400
5.3 Lagrangian Submanifolds .................................. 402
Exercises ................................................ 420
5.4 Quantization ............................................. 425
5.5 Introduction to Infinite-Dimensional Hamiltonian
Systems .................................................. 453
Exercises ................................................ 486
5.6 Introduction to Nonlinear Oscillations ................... 489
PART III AN OUTLINE OF QUALITATIVE DYNAMICS ................... 507
Chapter 6 Topological Dynamics ............................... 509
6.1 Limit and Minimal Sets ................................... 509
Exercises ................................................ 513
6.2 Recurrence ............................................... 513
Exercises ................................................ 515
6.3 Stability ................................................ 515
Exercises ................................................ 519
Chapter 7 Differentiable Dynamics ............................ 520
7.1 Critical Elements ........................................ 520
Exercises ................................................ 525
7.2 Stable Manifolds ......................................... 525
Exercises ................................................ 531
7.3 Generic Properties ....................................... 531
Exercises ................................................ 534
7.4 Structural Stability ..................................... 534
Exercises ................................................ 536
7.5 Absolute Stability and Axiom A ........................... 536
Exercises ................................................ 542
7.6 Bifurcations of Generic Arcs ............................. 543
Exercises ................................................ 548
7.7 A Zoo of Stable Bifurcations ............................. 548
7.8 Experimental Dynamics .................................... 570
Chapter 8 Hamiltonian Dynamics ............................... 572
8.1 Critical Elements ........................................ 572
8.2 Orbit Cylinders .......................................... 576
Exercises ................................................ 579
8.3 Stability of Orbits ...................................... 579
Exercises ................................................ 587
8.4 Generic Properties ....................................... 587
8.5 Structural Stability ..................................... 592
8.6 A Zoo of Stable Bifurcations ............................. 595
8.7 The General Pathology .................................... 606
8.8 Experimental Mechanics ................................... 610
РАRТ IV CELESTIAL MECHANICS ................................... 617
Chapter 9 The Two-Body Problem ............................... 619
9.1 Models for Two Bodies .................................... 619
Exercises ................................................ 624
9.2 Elliptic Orbits and Kepler Elements ...................... 624
9.3 The Delaunay Variables ................................... 631
9.4 Lagrange Brackets of Kepler Elements ..................... 635
9.5 Whittaker's Method ....................................... 638
9.6 Poincare Variables ....................................... 647
Exercises ................................................ 652
9.7 Summary of Models ........................................ 652
Exercise ................................................. 656
9.8 Topology of the Two-Body Problem ......................... 656
Chapter 10 The Three-Body Problem ............................. 663
10.1 Models for Three Bodies .................................. 663
Exercises ................................................ 673
10.2 Critical Points in the Restricted Three-Body Problem ..... 675
Exercises ................................................ 687
10.3 Closed Orbits in the Restricted Three-Body Problem ....... 688
Exercises ................................................ 699
10.4 Topology of the Planar n-Body Problem .................... 699
Appendix
The General Theory of Dynamical Systems and Classical
Mechanics by A.N. Kolmogorov ............................... 741
Bibliography .................................................. 759
Index ......................................................... 791
Glossary of Symbols ........................................... 807
Errata ........................................................ 809
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