Differential geometry with applications to mechanics and physics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Dekker
2001
|
| Series: | Pure and applied mathematics
237 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 454 S. graph. Darst. |
| ISBN: | 0824703855 |
Staff View
MARC
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| 245 | 1 | 0 | |a Differential geometry with applications to mechanics and physics |c Yves Talpaert |
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
Preface v
Lecture 0. TOPOLOGY AND DIFFERENTIAL CALCULUS REQUIREMENTS 1
1. Topology 1
1.1 Topological space 1
1.2 Topological space basis 2
1.3 Haussdorff space 4
1.4 Homeomorphism 5
1.5 Connected spaces 6
1.6 Compact spaces 6
1.7 Partition of unity 7
2. Differential calculus in Banach spaces 8
2.1 Banachspace 8
2.2 Differential calculus in Banach spaces 10
2.3 Differentiation of R into Banach 17
2.4 Differentiation of R into jR 19
2.5 Differentiation of R into R 22
3. Exercises 30
Lecture 1. MANIFOLDS 37
Introduction 37
1. Differentiable manifolds 40
1.1 Chart and local coordinates 40
1.2 Differentiable manifold structure 41
1.3 Differentiable manifolds 43
2. Differentiable mappings 50
2.1 Generalities on differentiable mappings 50
2.2 Particular differentiable mappings 55
2.3 Pull back of function 57
3. Submanifolds 59
3.1 Submanifolds ofR 59
3.2 Submanifold of manifold 64
4. Exercises 6S
vii
viii Contents
Lecture 2. TANGENT VECTOR SPACE 71
1. Tangent vector 71
1.1 Tangent curves 71
1.2 Tangent vector 74
2. Tangent space 80
2.1 Definitionof a tangent space 80
2.2 Basis of tangent space 81
2.3 Change of basis 82
3. Differential at a point 83
3.1 Definitions 84
3.2 The image in local coordinates 85
3.3 Diferential of a function 86
4. Exercises 87
Lecture 3. TANGENT BUNDLE VECTOR FIELD ONE PARAMETER
GROUP LIE ALGEBRA 91
Introduction 91
1. Tangent bundle 93
1.1 Natural manifold TM 93
1.2 Extension and commutative diagram 94
2. Vector field on manifold 96
2.1 Definitions 96
2.2 Properties of vector fields 96
3. Lie algebra structure 97
3.1 Bracket 97
3.2 Liealgebra 100
3.3 Lie derivative 101
4. One parameter group of diffeomorphisms 102
4.1 Differential equations in Banach 102
4.2 One parameter group of diffeomorphisms 104
5. Exercises Ill
Lecture 4. COTANGENT BUNDLE VECTOR BUNDLE OF TENSORS 125
1. Cotangent bundle and covector field 125
1.1 1 form 125
1.2 Cotangent bundle 129
1.3 Field of covectors 130
Contents ix
2. Tensor algebra 130
2.1 Tensor at a point and tensor algebra 130
2.2 Tensor fields and tensor algebra 137
3. Exercises 144
Lectures. EXTERIOR DIFFERENTIAL FORMS 153
1. Exterior form at a point 153
1.1 Definition of a/7 form 153
1.2 Exterior product of 1 forms 155
1.3 Expression of a p form 156
1.4 Exterior product of forms 158
1.5 Exterior algebra 159
2. Differential forms on a manifold 162
2.1 Exterior algebra (Grassmann algebra) 162
2.2 Changeofbasis 165
3. Pull back of a differential form 167
3.1 Definition and representation 167
3.2 Pull back properties 168
4. Exterior differentiation 170
4.1 Definition 170
4.2 Exterior differential and pull back 173
5. Orientable manifolds 174
6. Exercises 178
Lecture 6. LIE DERIVATIVE LIE GROUP 185
1. Lie derivative 186
1.1 First presentation of Lie derivative 186
1.2 Alternative interpretation of Lie derivative 195
2. Inner product and Lie derivative 199
2.1 Definition and properties 199
2.2 Fundamental theorem 201
3. Frobenius theorem 204
4. Exterior differential systems 207
4.1 Generalities 207
4.2 Pfaff systems and Frobenius theorem 208
x Contents
5. Invariance of tensor fields 211
5.1 Definitions 211
5.2 Invariance of differential forms 212
5.3 Lie algebra 214
6. Lie group and algebra 214
6.1 Lie group definition 215
6.2 Lie algebra of Lie group 215
6.3 Invariant differential forms on G 217
6.4 One parameter subgroup of a Lie group 218
7. Exercises 224
Lecture 7. INTEGRATION OF FORMS: Stokes Theorem, Cohomology and
Integral Invariants 235
1. n form integration on /i manifold 235
1.1 Integration definition 235
1.2 Pull back of a form and integral evaluation 237
2. Integral over a chain 239
2.1 Integral over a chain element 239
2.2 Integral over a chain 239
3. Stokes theorem 240
3.1 Stokes formula for a closed p interval 240
3.2 Stokes formula for a chain 242
4. An introduction to cohomology theory 243
4.1 Closed and exact forms Cohomology 243
4.2 Poincare lemma 244
4.3 Cycle Boundary Homology 247
5. Integral invariants 248
5.1 Absolute integral invariant 248
5.2 Relative integral invariant 252
6. Exercises 253
Lecture 8. RIEMANNIAN GEOMETRY 257
1. Riemannian manifolds 257
1.1 Metric tensor and manifolds 257
1.2 Canonical isomorphism and conjugate tensor 262
1.3 Orthonormal bases 266
1.4 Hyperbolic manifold and special relativity 267
Contents xi
1.5 Killing vector field 274
1.6 Volume 275
1.7 The Hodge operator and adjoint 277
1.8 Special relativity and Maxwell equations 280
1.9 Induced metric and isometry 283
2. Affine connection 285
2.1 Affine connection definition 285
2.2 Christoffel symbols 286
2.3 Interpretation of the covariant derivative 288
2.4 Torsion 291
2.5 Levi Civita (or Riemannian) connection 291
2.6 Gradient Divergence Laplace operators 293
3. Geodesic and Euler equation 300
4. Curvatures Ricci tensor Bianchi identity Einstein equations ... 302
4.1 Curvature tensor 302
4.2 Ricci tensor 305
4.3 Bianchi identity 308
4.4 Einstein equations 309
5. Exercises 310
Lecture 9 LAGRANGE AND HAMILTON MECHANICS 325
1. Classical mechanics spaces and metric 325
1.1 Generalized coordinates and spaces 325
1.2 Kinetic energy and Riemannian manifold 327
2. Hamilton principle, Motion equations, Phase space 329
2.1 Lagrangian 329
2.2 Principleof least action 329
2.3 Lagrange equations 331
2.4 Canonical equations of Hamilton 332
2.5 Phasespace 337
3. D Alembert Lagrange principle Lagrange equations 338
3.1 D Alembert Lagrange principle 338
3.2 Lagrange equations 340
3.3 Euler Noether theorem 341
3.4 Motion equations on Riemannian manifolds 343
4. Canonical transformations and integral invariants 344
4.1 Diffeomorphisms on phase spacetime 344
4.2 Integral invariants 346
4.3 Integral invariants and canonical transformations 348
xii Contents
4.4 Liouville theorem 352
5. The TV body problem and a problem of statistical mechanics 352
5.1 TV body problem and fundamental equations 353
5.2 A problem of statistical mechanics 358
6. Isolating integrals 369
6.1 Definition and examples 369
6.2 Jeanstheorem 372
6.3 Stellar trajectories in the galaxy 373
6.4 The third integral 375
6.5 Invariant curve and third integral existence 379
7. Exercises 381
Lecture 10. SYMPLECTIC GEOMETRY Hamilton Jacobi Mechanics 385
Preliminaries 385
1. Symplectic geometry 388
1.1 Darboux theorem and symplectic matrix 388
1.2 Canonical isomorphism 391
1.3 Poisson bracket of one forms 393
1.4 Poisson bracket of functions 396
1.5 Symplectic mapping and canonical transformation 399
2. Canonical transformations in mechanics 404
2.1 Hamilton vector field 404
2.2 Canonical transformations Lagrange brackets 408
2.3 Generating functions 412
3. Hamilton Jacobi equation 415
3.1 Hamilton Jacobi equation and Jacobi theorem 415
3.2 Separability 419
4. A variational principle of analytical mechanics 422
4.1 Variational principle (with one degree of freedom) 423
4.2 Variational principle (with « degrees of freedom) 427
5. Exercises 429
Bibliography 443
Glossary 445
Contents xiii
Index 449
|
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| spelling | Talpaert, Yves Verfasser aut Lec̨ons et applications de géométrie différentiale et de mécanique analytique Differential geometry with applications to mechanics and physics Yves Talpaert New York [u.a.] Dekker 2001 XIII, 454 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 237 Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Mathematische Physik (DE-588)4037952-8 s DE-604 Pure and applied mathematics 237 (DE-604)BV000001885 237 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009135926&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Talpaert, Yves Differential geometry with applications to mechanics and physics Pure and applied mathematics Mathematische Physik (DE-588)4037952-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4012248-7 |
| title | Differential geometry with applications to mechanics and physics |
| title_alt | Lec̨ons et applications de géométrie différentiale et de mécanique analytique |
| title_auth | Differential geometry with applications to mechanics and physics |
| title_exact_search | Differential geometry with applications to mechanics and physics |
| title_full | Differential geometry with applications to mechanics and physics Yves Talpaert |
| title_fullStr | Differential geometry with applications to mechanics and physics Yves Talpaert |
| title_full_unstemmed | Differential geometry with applications to mechanics and physics Yves Talpaert |
| title_short | Differential geometry with applications to mechanics and physics |
| title_sort | differential geometry with applications to mechanics and physics |
| topic | Mathematische Physik (DE-588)4037952-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Mathematische Physik Differentialgeometrie |
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