Geometric mechanics on Riemannian manifolds: applications to partial differential equations
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2005
|
| Schriftenreihe: | Applied and numerical harmonic analysis
|
| Schlagworte: | |
| Online-Zugang: | BTU01 TUM01 UBA01 UBR01 UBT01 UPA01 Volltext Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 271 - 273 |
| Beschreibung: | 1 Online-Ressource (XV, 278 S.) graph. Darst. |
| ISBN: | 0817643540 9780817644215 |
| DOI: | 10.1007/b138771 |
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| 245 | 1 | 0 | |a Geometric mechanics on Riemannian manifolds |b applications to partial differential equations |c Ovidiu Calin ; Der-Chen Chang |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2005 | |
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Datensatz im Suchindex
| _version_ | 1804136320175439872 |
|---|---|
| adam_text | Contents
Preface xiii
1 Introductory Chapter 1
1.1 Manifolds 1
1.2 Tangent vectors 3
1.3 The Differential of a Map 5
1.4 The Lie bracket 6
1.5 One forms 7
1.6 Tensors 8
1.7 Riemannian Manifolds 9
1.8 Linear Connections 10
1.9 The Volume element 12
1.10 Exercises 13
2 Laplace Operators on Riemannian Manifolds 17
2.1 Gradient vector field; Divergence and Laplacian 17
2.2 Applications 22
2.2.0.1 Pluri harmonic functions 22
2.2.0.2 Uniqueness for solution of the Cauchy problem
for the heat operator 23
2.3 The Hessian and applications 24
2.3.0.3 An application to the heat equation with
convection on compact manifolds 28
2.4 Exercises 29
3 Lagrangian Formalism on Riemannian Manifolds 33
3.1 A simple example 33
3.2 The pendulum equation 34
3.3 Euler Lagrange equations on Riemannian manifolds 38
3.4 Laplace s Equation A/ = 0 41
3.5 A geometrical interpretation for a A operator 42
viii Contents
3.6 Poisson s equation 43
3.7 Geodesies 44
3.8 The natural Lagrangian on manifolds 45
3.8.0.4 Momentum and Work 46
3.8.0.5 Force and Newton s Equation 47
3.9 A geometrical interpretation for the potential U 50
3.10 Exercises 52
4 Harmonic Maps from a Lagrangian Viewpoint 55
4.1 Introduction to harmonic maps 55
4.1.1 The energy density 56
4.1.2 Harmonic maps using Lagrangian formalism 57
4.2 D Alembert principle on Riemannian manifolds 61
4.3 Exercises 64
5 Conservation Theorems 67
5.1 Noether s Theorem 67
5.2 The role of Killing vector fields 70
5.3 The Energy Momentum tensor 74
5.3.1 Definition of Energy Momentum 75
5.3.2 Einstein tensor 77
5.3.3 Field equations 79
5.3.4 Divergence of the energy momentum tensor 83
5.3.5 Conservation Theorems 85
5.3.6 Applications of the conservation theorems 88
5.4 Exercises 96
6 Hamiltonian Formalism 97
6.1 Momenta vector fields. Hamiltonian 97
6.2 Hamilton s system of equations 99
6.3 Harmonic functions 100
6.4 Geodesies 101
6.5 Harmonic maps 103
6.6 Poincare half plane 106
6.7 Exercises 109
7 Hamilton Jacobi Theory 113
7.1 Hamilton Jacobi equation in the case of natural Lagrangian 113
7.2 The action function on Riemannian manifolds 117
7.2.0.1 Hamilton Jacobi for conservative systems 120
7.2.1 Action for an arbitrary Lagrangian 120
7.2.2 Examples 122
7.3 The Eiconal Equation on Riemannian Manifolds 127
7.4 Applications of Eiconal equation 130
7.4.1 Fundamental solution for the Laplace Beltrami operator ... 130
Contents ix
7.4.2 Fundamental Singularity for the Laplacian 131
7.4.3 Laplacian momenta on a compact manifold 132
7.4.4 Minimizing geodesies 132
7.5 Exercises 134
8 Minimal Hypersurfaces 137
8.1 The Curl tensor 137
8.2 Application to minimal hypersurfaces 140
8.3 Helmholtz decomposition 145
8.3.0.1 The non compact case 146
8.4 Exercises 146
9 Radially Symmetric Spaces 149
9.1 Existence and uniqueness of geodesies 149
9.2 Geodesic spheres 153
9.3 A radially non symmetric space 158
9.4 The Heisenberg group 160
9.4.1 The left invariant metric 160
9.4.1.1 The Euler Lagrange system 162
9.4.2 The classical action 169
9.4.3 The complex action 171
9.4.4 The volume function at the origin 172
9.5 Exercises 173
10 Fundamental Solutions for Heat Operators with Potentials 175
10.1 The heat operator on Riemannian manifolds 175
10.1.1 The case of compact manifolds 176
10.2 Heat kernel on radially symmetric spaces 178
10.3 Heat kernel for the Casimir operator 181
10.4 Heat kernel for operators with potential 182
10.4.1 The kernel of 3, d2 ± b2x2 182
10.4.2 The kernel of d, £3* ±a2 x 2 187
10.4.3 Fourier transform method 191
10.4.3.1 Fundamental solution with singularity at the origin 191
10.4.3.2 Isotropic case: kj¦ = k for all j 198
10.4.3.3 Partial inverse and projection to the kernel 199
10.4.3.4 Fundamental solution with singularity at an
arbitrary point y 201
10.5 Heat kernel on radially symmetric spaces with potential 205
10.6 The case of the quartic potential 207
10.7 The kernel of the operator 9, d2 U(x) 212
10.7.1 The linear potential 215
10.8 Propagators for Schrodinger s equation in the one dimensional case 216
10.8.1 Free quantum particle 216
10.8.2 Quantum particle in a linear potential 217
x Contents
10.8.3 Linear harmonic quantum oscillator 218
10.9 Propagators for Schrodinger s equation in the n dimensional case .. 219
10.10 The operator P = d, d* U(x)dx 220
10.10.1 The linear potential 221
10.10.2 The quadratic potential 223
10.10.3 The kernel of d, d2x U(x)dx 224
10.10.4 The square root potential 226
10.10.5 The constant potential case U(x) = a, with a e R 228
10.10.6 The exponential potential 229
10.10.7 Physical interpretation 232
10.11 Exercises 234
11 Fundamental Solutions for Elliptic Operators 237
11.1 Fundamental solutions for Laplace operators 237
11.2 The transport operator 237
11.3 Properties of the transport operator 238
11.4 The homogeneous transport equation 240
11.5 The non homogeneous transport equation 241
11.6 Fundamental solution 242
11.7 The parametrix 246
11.8 Solving the system (E) 248
11.9 Exercises 250
12 Mechanical Curves 251
12.1 The areal velocity 251
12.1.0.1 Areal velocity as an angular momentum 252
12.2 The circular motion 252
12.3 The astroid 256
12.3.0.2 Noether s Theorem 257
12.3.0.3 The first integral of energy 259
12.3.0.4 Physical interpretation 259
12.4 The cycloid 259
12.4.0.5 Solving the Euler Lagrange system (12.4.28) 260
12.4.0.6 The total energy 262
12.4.0.7 Galileo s law 262
12.5 Curves that minimize a potential 263
12.5.0.8 The gravitational potential 265
12.5.0.9 Minimal surfaces 265
12.5.0.10 The brachistochrone curve 265
12.5.0.11 Coloumb potential 267
12.5.0.12 Physical interpretation 268
12.5.1 Hamiltonian approach 268
12.5.2 Hamiltonian system 268
12.6 Exercises 269
Contents xi
Bibliography 271
Index 275
|
| adam_txt |
Contents
Preface xiii
1 Introductory Chapter 1
1.1 Manifolds 1
1.2 Tangent vectors 3
1.3 The Differential of a Map 5
1.4 The Lie bracket 6
1.5 One forms 7
1.6 Tensors 8
1.7 Riemannian Manifolds 9
1.8 Linear Connections 10
1.9 The Volume element 12
1.10 Exercises 13
2 Laplace Operators on Riemannian Manifolds 17
2.1 Gradient vector field; Divergence and Laplacian 17
2.2 Applications 22
2.2.0.1 Pluri harmonic functions 22
2.2.0.2 Uniqueness for solution of the Cauchy problem
for the heat operator 23
2.3 The Hessian and applications 24
2.3.0.3 An application to the heat equation with
convection on compact manifolds 28
2.4 Exercises 29
3 Lagrangian Formalism on Riemannian Manifolds 33
3.1 A simple example 33
3.2 The pendulum equation 34
3.3 Euler Lagrange equations on Riemannian manifolds 38
3.4 Laplace's Equation A/ = 0 41
3.5 A geometrical interpretation for a A operator 42
viii Contents
3.6 Poisson's equation 43
3.7 Geodesies 44
3.8 The natural Lagrangian on manifolds 45
3.8.0.4 Momentum and Work 46
3.8.0.5 Force and Newton's Equation 47
3.9 A geometrical interpretation for the potential U 50
3.10 Exercises 52
4 Harmonic Maps from a Lagrangian Viewpoint 55
4.1 Introduction to harmonic maps 55
4.1.1 The energy density 56
4.1.2 Harmonic maps using Lagrangian formalism 57
4.2 D'Alembert principle on Riemannian manifolds 61
4.3 Exercises 64
5 Conservation Theorems 67
5.1 Noether's Theorem 67
5.2 The role of Killing vector fields 70
5.3 The Energy Momentum tensor 74
5.3.1 Definition of Energy Momentum 75
5.3.2 Einstein tensor 77
5.3.3 Field equations 79
5.3.4 Divergence of the energy momentum tensor 83
5.3.5 Conservation Theorems 85
5.3.6 Applications of the conservation theorems 88
5.4 Exercises 96
6 Hamiltonian Formalism 97
6.1 Momenta vector fields. Hamiltonian 97
6.2 Hamilton's system of equations 99
6.3 Harmonic functions 100
6.4 Geodesies 101
6.5 Harmonic maps 103
6.6 Poincare half plane 106
6.7 Exercises 109
7 Hamilton Jacobi Theory 113
7.1 Hamilton Jacobi equation in the case of natural Lagrangian 113
7.2 The action function on Riemannian manifolds 117
7.2.0.1 Hamilton Jacobi for conservative systems 120
7.2.1 Action for an arbitrary Lagrangian 120
7.2.2 Examples 122
7.3 The Eiconal Equation on Riemannian Manifolds 127
7.4 Applications of Eiconal equation 130
7.4.1 Fundamental solution for the Laplace Beltrami operator . 130
Contents ix
7.4.2 Fundamental Singularity for the Laplacian 131
7.4.3 Laplacian momenta on a compact manifold 132
7.4.4 Minimizing geodesies 132
7.5 Exercises 134
8 Minimal Hypersurfaces 137
8.1 The Curl tensor 137
8.2 Application to minimal hypersurfaces 140
8.3 Helmholtz decomposition 145
8.3.0.1 The non compact case 146
8.4 Exercises 146
9 Radially Symmetric Spaces 149
9.1 Existence and uniqueness of geodesies 149
9.2 Geodesic spheres 153
9.3 A radially non symmetric space 158
9.4 The Heisenberg group 160
9.4.1 The left invariant metric 160
9.4.1.1 The Euler Lagrange system 162
9.4.2 The classical action 169
9.4.3 The complex action 171
9.4.4 The volume function at the origin 172
9.5 Exercises 173
10 Fundamental Solutions for Heat Operators with Potentials 175
10.1 The heat operator on Riemannian manifolds 175
10.1.1 The case of compact manifolds 176
10.2 Heat kernel on radially symmetric spaces 178
10.3 Heat kernel for the Casimir operator 181
10.4 Heat kernel for operators with potential 182
10.4.1 The kernel of 3, d2 ± b2x2 182
10.4.2 The kernel of d, £3* ±a2\x\2 187
10.4.3 Fourier transform method 191
10.4.3.1 Fundamental solution with singularity at the origin 191
10.4.3.2 Isotropic case: kj¦ = k for all j 198
10.4.3.3 Partial inverse and projection to the kernel 199
10.4.3.4 Fundamental solution with singularity at an
arbitrary point y 201
10.5 Heat kernel on radially symmetric spaces with potential 205
10.6 The case of the quartic potential 207
10.7 The kernel of the operator 9, d2 U(x) 212
10.7.1 The linear potential 215
10.8 Propagators for Schrodinger's equation in the one dimensional case 216
10.8.1 Free quantum particle 216
10.8.2 Quantum particle in a linear potential 217
x Contents
10.8.3 Linear harmonic quantum oscillator 218
10.9 Propagators for Schrodinger's equation in the n dimensional case . 219
10.10 The operator P = d, d* U(x)dx 220
10.10.1 The linear potential 221
10.10.2 The quadratic potential 223
10.10.3 The kernel of d, d2x U(x)dx 224
10.10.4 The square root potential 226
10.10.5 The constant potential case U(x) = a, with a e R 228
10.10.6 The exponential potential 229
10.10.7 Physical interpretation 232
10.11 Exercises 234
11 Fundamental Solutions for Elliptic Operators 237
11.1 Fundamental solutions for Laplace operators 237
11.2 The transport operator 237
11.3 Properties of the transport operator 238
11.4 The homogeneous transport equation 240
11.5 The non homogeneous transport equation 241
11.6 Fundamental solution 242
11.7 The parametrix 246
11.8 Solving the system (E) 248
11.9 Exercises 250
12 Mechanical Curves 251
12.1 The areal velocity 251
12.1.0.1 Areal velocity as an angular momentum 252
12.2 The circular motion 252
12.3 The astroid 256
12.3.0.2 Noether's Theorem 257
12.3.0.3 The first integral of energy 259
12.3.0.4 Physical interpretation 259
12.4 The cycloid 259
12.4.0.5 Solving the Euler Lagrange system (12.4.28) 260
12.4.0.6 The total energy 262
12.4.0.7 Galileo's law 262
12.5 Curves that minimize a potential 263
12.5.0.8 The gravitational potential 265
12.5.0.9 Minimal surfaces 265
12.5.0.10 The brachistochrone curve 265
12.5.0.11 Coloumb potential 267
12.5.0.12 Physical interpretation 268
12.5.1 Hamiltonian approach 268
12.5.2 Hamiltonian system 268
12.6 Exercises 269
Contents xi
Bibliography 271
Index 275 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Calin, Ovidiu L. 1971- Chang, Der-Chen |
| author_GND | (DE-588)128990554 (DE-588)128990562 |
| author_facet | Calin, Ovidiu L. 1971- Chang, Der-Chen |
| author_role | aut aut |
| author_sort | Calin, Ovidiu L. 1971- |
| author_variant | o l c ol olc d c c dcc |
| building | Verbundindex |
| bvnumber | BV022302035 |
| classification_rvk | SK 370 |
| classification_tum | MAT 000 |
| collection | ZDB-2-SMA |
| ctrlnum | (OCoLC)249489403 (DE-599)BVBBV022302035 |
| dewey-full | 516.373 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.373 |
| dewey-search | 516.373 |
| dewey-sort | 3516.373 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| doi_str_mv | 10.1007/b138771 |
| format | Electronic eBook |
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| id | DE-604.BV022302035 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:55:26Z |
| indexdate | 2024-07-09T20:54:31Z |
| institution | BVB |
| isbn | 0817643540 9780817644215 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015512011 |
| oclc_num | 249489403 |
| open_access_boolean | |
| owner | DE-739 DE-355 DE-BY-UBR DE-634 DE-91 DE-BY-TUM DE-384 DE-703 DE-83 |
| owner_facet | DE-739 DE-355 DE-BY-UBR DE-634 DE-91 DE-BY-TUM DE-384 DE-703 DE-83 |
| physical | 1 Online-Ressource (XV, 278 S.) graph. Darst. |
| psigel | ZDB-2-SMA |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Applied and numerical harmonic analysis |
| spelling | Calin, Ovidiu L. 1971- Verfasser (DE-588)128990554 aut Geometric mechanics on Riemannian manifolds applications to partial differential equations Ovidiu Calin ; Der-Chen Chang Boston [u.a.] Birkhäuser 2005 1 Online-Ressource (XV, 278 S.) graph. Darst. txt rdacontent c rdamedia cr rdacarrier Applied and numerical harmonic analysis Literaturverz. S. 271 - 273 Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Globale Riemannsche Geometrie (DE-588)4157622-6 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 s Theoretische Mechanik (DE-588)4185100-6 s Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Riemannscher Raum (DE-588)4128295-4 s 1\p DE-604 Globale Riemannsche Geometrie (DE-588)4157622-6 s 2\p DE-604 Chang, Der-Chen Verfasser (DE-588)128990562 aut https://doi.org/10.1007/b138771 Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015512011&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Calin, Ovidiu L. 1971- Chang, Der-Chen Geometric mechanics on Riemannian manifolds applications to partial differential equations Partielle Differentialgleichung (DE-588)4044779-0 gnd Riemannscher Raum (DE-588)4128295-4 gnd Globale Riemannsche Geometrie (DE-588)4157622-6 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd Theoretische Mechanik (DE-588)4185100-6 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4128295-4 (DE-588)4157622-6 (DE-588)4128462-8 (DE-588)4185100-6 |
| title | Geometric mechanics on Riemannian manifolds applications to partial differential equations |
| title_auth | Geometric mechanics on Riemannian manifolds applications to partial differential equations |
| title_exact_search | Geometric mechanics on Riemannian manifolds applications to partial differential equations |
| title_exact_search_txtP | Geometric mechanics on Riemannian manifolds applications to partial differential equations |
| title_full | Geometric mechanics on Riemannian manifolds applications to partial differential equations Ovidiu Calin ; Der-Chen Chang |
| title_fullStr | Geometric mechanics on Riemannian manifolds applications to partial differential equations Ovidiu Calin ; Der-Chen Chang |
| title_full_unstemmed | Geometric mechanics on Riemannian manifolds applications to partial differential equations Ovidiu Calin ; Der-Chen Chang |
| title_short | Geometric mechanics on Riemannian manifolds |
| title_sort | geometric mechanics on riemannian manifolds applications to partial differential equations |
| title_sub | applications to partial differential equations |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd Riemannscher Raum (DE-588)4128295-4 gnd Globale Riemannsche Geometrie (DE-588)4157622-6 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd Theoretische Mechanik (DE-588)4185100-6 gnd |
| topic_facet | Partielle Differentialgleichung Riemannscher Raum Globale Riemannsche Geometrie Riemannsche Geometrie Theoretische Mechanik |
| url | https://doi.org/10.1007/b138771 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015512011&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT calinovidiul geometricmechanicsonriemannianmanifoldsapplicationstopartialdifferentialequations AT changderchen geometricmechanicsonriemannianmanifoldsapplicationstopartialdifferentialequations |