Abraham R. Foundations of mechanics (Reading, 1978). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаAbraham 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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Оглавление / Contents
 
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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