Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1999
|
| Edition: | 2. ed. |
| Series: | Texts in applied mathematics
17 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XVIII, 582 S. graph. Darst. |
| ISBN: | 038798643X 9781441931436 |
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MARC
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| 245 | 1 | 0 | |a Introduction to mechanics and symmetry |b a basic exposition of classical mechanical systems |c Jerrold E. Marsden ; Tudor S. Ratiu |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1999 | |
| 300 | |a XVIII, 582 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text |
Contents
Preface ix
About the Authors xiii
1 Introduction and Overview 1
1.1 Lagrangian and Hamiltonian Formalisms 1
1.2 The Rigid Body 6
1.3 Lie Poisson Brackets, Poisson Manifolds, Momentum Maps 9
1.4 The Heavy Top 16
1.5 Incompressible Fluids 18
1.6 The Maxwell Vlasov System 22
1.7 Nonlinear Stability 29
1.8 Bifurcation 43
1.9 The Poincare Melnikov Method 47
1.10 Resonances, Geometric Phases, and Control 50
2 Hamiltonian Systems on Linear Symplectic Spaces 61
2.1 Introduction 61
2.2 Symplectic Forms on Vector Spaces 66
2.3 Canonical Transformations, or Symplectic Maps 69
2.4 The General Hamilton Equations 74
2.5 When Are Equations Hamiltonian? 77
2.6 Hamiltonian Flows 80
2.7 Poisson Brackets 82
xvi Contents
2.8 A Particle in a Rotating Hoop 87
2.9 The Poincare Melnikov Method 94
3 An Introduction to Infinite Dimensional Systems 105
3.1 Lagrange's and Hamilton's Equations for Field Theory . . 105
3.2 Examples: Hamilton's Equations 107
3.3 Examples: Poisson Brackets and Conserved Quantities . . 115
4 Manifolds, Vector Fields, and Differential Forms 121
4.1 Manifolds 121
4.2 Differential Forms 129
4.3 The Lie Derivative 137
4.4 Stokes' Theorem 141
5 Hamiltonian Systems on Symplectic Manifolds 147
5.1 Symplectic Manifolds 147
5.2 Symplectic Transformations 150
5.3 Complex Structures and Kahler Manifolds 152
5.4 Hamiltonian Systems 157
5.5 Poisson Brackets on Symplectic Manifolds 160
6 Cotangent Bundles 165
6.1 The Linear Case 165
6.2 The Nonlinear Case 167
6.3 Cotangent Lifts 170
6.4 Lifts of Actions 173
6.5 Generating Functions 174
6.6 Fiber Translations and Magnetic Terms 176
6.7 A Particle in a Magnetic Field 178
7 Lagrangian Mechanics 181
7.1 Hamilton's Principle of Critical Action 181
7.2 The Legendre Transform 183
7.3 Euler Lagrange Equations 185
7.4 Hyperregular Lagrangians and Hamiltonians 188
7.5 Geodesies 195
7.6 The Kaluza Klein Approach to Charged Particles 200
7.7 Motion in a Potential Field 202
7.8 The Lagrange d'Alembert Principle 205
7.9 The Hamilton Jacobi Equation 210
8 Variational Principles, Constraints, Rotating Systems 219
8.1 A Return to Variational Principles 219
8.2 The Geometry of Variational Principles 226
8.3 Constrained Systems 234
Contents xvii
8.4 Constrained Motion in a Potential Field 238
8.5 Dirac Constraints 242
8.6 Centrifugal and Coriolis Forces 248
8.7 The Geometric Phase for a Particle in a Hoop 253
8.8 Moving Systems 257
8.9 Routh Reduction 260
9 An Introduction to Lie Groups 265
9.1 Basic Definitions and Properties 267
9.2 Some Classical Lie Groups 283
9.3 Actions of Lie Groups 309
10 Poisson Manifolds 327
10.1 The Definition of Poisson Manifolds 327
10.2 Hamiltonian Vector Fields and Casimir Functions 333
10.3 Properties of Hamiltonian Flows 338
10.4 The Poisson Tensor 340
10.5 Quotients of Poisson Manifolds 349
10.6 The Schouten Bracket 353
10.7 Generalities on Lie Poisson Structures 360
11 Momentum Maps 365
11.1 Canonical Actions and Their Infinitesimal Generators . . 365
11.2 Momentum Maps 367
11.3 An Algebraic Definition of the Momentum Map 370
11.4 Conservation of Momentum Maps 372
11.5 Equivariance of Momentum Maps 378
12 Computation and Properties of Momentum Maps 383
12.1 Momentum Maps on Cotangent Bundles 383
12.2 Examples of Momentum Maps 389
12.3 Equivariance and Infinitesimal Equivariance 396
12.4 Equivariant Momentum Maps Are Poisson 403
12.5 Poisson Automorphisms 412
12.6 Momentum Maps and Casimir Functions 413
13 Lie—Poisson and Euler—Poincare Reduction 417
13.1 The Lie Poisson Reduction Theorem 417
13.2 Proof of the Lie Poisson Reduction Theorem for GL^n) . 420
13.3 Lie Poisson Reduction Using Momentum Functions . 421
13.4 Reduction and Reconstruction of Dynamics 423
13.5 The Euler Poincare Equations 432
13.6 The Lagrange Poincare Equations 442
xviii Contents
14 Coadjoint Orbits 445
14.1 Examples of Coadjoint Orbits 446
14.2 Tangent Vectors to Coadjoint Orbits 453
14.3 The Symplectic Structure on Coadjoint Orbits 455
14.4 The Orbit Bracket via Restriction of the Lie Poisson Bracket461
14.5 The Special Linear Group of the Plane 467
14.6 The Euclidean Group of the Plane 469
14.7 The Euclidean Group of Three Space 474
15 The Free Rigid Body 483
15.1 Material, Spatial, and Body Coordinates 483
15.2 The Lagrangian of the Free Rigid Body 485
15.3 The Lagrangian and Hamiltonian in Body Representation 487
15.4 Kinematics on Lie Groups 491
15.5 Poinsot's Theorem 492
15.6 Euler Angles 495
15.7 The Hamiltonian of the Free Rigid Body 497
15.8 The Analytical Solution of the Free Rigid Body Problem . 500
15.9 Rigid Body Stability 505
15.10 Heavy Top Stability 509
15.11 The Rigid Body and the Pendulum 514
References 521
Index 555 |
| any_adam_object | 1 |
| author | Marsden, Jerrold E. 1942-2010 Ratiu, Tudor S. |
| author_GND | (DE-588)124171141 |
| author_facet | Marsden, Jerrold E. 1942-2010 Ratiu, Tudor S. |
| author_role | aut aut |
| author_sort | Marsden, Jerrold E. 1942-2010 |
| author_variant | j e m je jem t s r ts tsr |
| building | Verbundindex |
| bvnumber | BV012569505 |
| callnumber-first | Q - Science |
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| callnumber-raw | QA808 |
| callnumber-search | QA808 |
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| callnumber-subject | QA - Mathematics |
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| ctrlnum | (OCoLC)245743805 (DE-599)BVBBV012569505 |
| dewey-full | 531 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 531 - Classical mechanics |
| dewey-raw | 531 |
| dewey-search | 531 |
| dewey-sort | 3531 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| edition | 2. ed. |
| format | Book |
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| spelling | Marsden, Jerrold E. 1942-2010 (DE-588)124171141 aut Introduction to mechanics and symmetry a basic exposition of classical mechanical systems Jerrold E. Marsden ; Tudor S. Ratiu 2. ed. New York [u.a.] Springer 1999 XVIII, 582 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 17 Hier auch später erschienene, unveränderte Nachdrucke Theoretische Mechanik - Symmetrie Mechanics, Analytic Symmetry (Physics) Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Symmetrie (DE-588)4058724-1 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf Mechanik (DE-588)4038168-7 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 s Symmetrie (DE-588)4058724-1 s DE-604 Mechanik (DE-588)4038168-7 s Hamiltonsches System (DE-588)4139943-2 s 1\p DE-604 Ratiu, Tudor S. aut Texts in applied mathematics 17 (DE-604)BV002476038 17 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008535566&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Marsden, Jerrold E. 1942-2010 Ratiu, Tudor S. Introduction to mechanics and symmetry a basic exposition of classical mechanical systems Texts in applied mathematics Theoretische Mechanik - Symmetrie Mechanics, Analytic Symmetry (Physics) Hamiltonsches System (DE-588)4139943-2 gnd Symmetrie (DE-588)4058724-1 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Mechanik (DE-588)4038168-7 gnd |
| subject_GND | (DE-588)4139943-2 (DE-588)4058724-1 (DE-588)4185100-6 (DE-588)4038168-7 |
| title | Introduction to mechanics and symmetry a basic exposition of classical mechanical systems |
| title_auth | Introduction to mechanics and symmetry a basic exposition of classical mechanical systems |
| title_exact_search | Introduction to mechanics and symmetry a basic exposition of classical mechanical systems |
| title_full | Introduction to mechanics and symmetry a basic exposition of classical mechanical systems Jerrold E. Marsden ; Tudor S. Ratiu |
| title_fullStr | Introduction to mechanics and symmetry a basic exposition of classical mechanical systems Jerrold E. Marsden ; Tudor S. Ratiu |
| title_full_unstemmed | Introduction to mechanics and symmetry a basic exposition of classical mechanical systems Jerrold E. Marsden ; Tudor S. Ratiu |
| title_short | Introduction to mechanics and symmetry |
| title_sort | introduction to mechanics and symmetry a basic exposition of classical mechanical systems |
| title_sub | a basic exposition of classical mechanical systems |
| topic | Theoretische Mechanik - Symmetrie Mechanics, Analytic Symmetry (Physics) Hamiltonsches System (DE-588)4139943-2 gnd Symmetrie (DE-588)4058724-1 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Mechanik (DE-588)4038168-7 gnd |
| topic_facet | Theoretische Mechanik - Symmetrie Mechanics, Analytic Symmetry (Physics) Hamiltonsches System Symmetrie Theoretische Mechanik Mechanik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008535566&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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