Symmetries and conservation laws for differential equations of mathematical physics:
Gespeichert in:
| Weitere Verfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English Russian |
| Veröffentlicht: |
Providence, Rhode Island
American Math. Soc.
1999
|
| Schriftenreihe: | Translations of mathematical monographs
182 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. [323] - 328. -Aus dem Russ. übers. |
| Beschreibung: | XIV, 333 S. graph. Darst. |
| ISBN: | 082180958X |
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| 245 | 1 | 0 | |a Symmetries and conservation laws for differential equations of mathematical physics |c A. V. Bocharov ... I. S. Krasil'shchik (ed.) ... |
| 264 | 1 | |a Providence, Rhode Island |b American Math. Soc. |c 1999 | |
| 300 | |a XIV, 333 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Translations of mathematical monographs |v 182 | |
| 500 | |a Literaturverz. S. [323] - 328. -Aus dem Russ. übers. | ||
| 650 | 4 | |a Calcul quantifié | |
| 650 | 4 | |a Elément contact | |
| 650 | 4 | |a Equation Burgers | |
| 650 | 4 | |a Equation Korteweg-De Vries | |
| 650 | 7 | |a Equations différentielles - Solutions numériques |2 ram | |
| 650 | 4 | |a Lois de conservation (Mathématiques) | |
| 650 | 7 | |a Lois de conservation (Mathématiques) |2 ram | |
| 650 | 4 | |a Méthode Lagrange-Charpit | |
| 650 | 4 | |a Physique mathématique | |
| 650 | 7 | |a Physique mathématique |2 ram | |
| 650 | 4 | |a Revêtement | |
| 650 | 4 | |a Symétrie | |
| 650 | 7 | |a Symétrie |2 ram | |
| 650 | 4 | |a Théorème Noether | |
| 650 | 4 | |a Équations différentielles - Solutions numériques | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Conservation laws (Mathematics) | |
| 650 | 4 | |a Differential equations |x Numerical solutions | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 4 | |a Symmetry | |
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Datensatz im Suchindex
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| adam_text | Contents
Preface ix
Chapter 1. Ordinary Differential Equations 1
1. Ordinary differential equations from the geometric viewpoint 1
2. Ordinary differential equations of arbitrary order 6
3. Symmetries of distributions 10
4. Some applications of symmetry theory to integration of distributions 17
4.1. Distributions and characteristic symmetries 17
4.2. Symmetries and dynamical systems 18
4.3. Distributions and noncharacteristic symmetries 20
4.4. Integration of equations by quadratures 21
5. Generating functions 29
6. How to search for equations integrable by quadratures: an example
of using symmetries 33
Chapter 2. First Order Equations 37
1. Contact transformations 37
1.1. Contact elements and the Cartan distribution 37
1.2. Contact transformations 42
1.3. Clairaut equation and its integrals 47
1.4. Contact manifolds in mechanics 49
2. Infinitesimal contact transformations and characteristic fields 50
2.1. Infinitesimal contact transformations 50
2.2. Infinitesimal symmetries of equations 54
2.3. Characteristic vector fields and integration of first order
equations 56
2.4. Symmetries and first integrals 59
3. Complete integrals of first order differential equations 60
3.1. Complete integrals: a coordinate approach 61
3.2. The construction of complete integrals using symmetry algebras 61
3.3. Complete integrals: an invariant approach 64
3.4. The Lagrange Charpit method 66
Chapter 3. The Theory of Classical Symmetries 69
1. Equations and the Cartan distribution 69
2. Jet manifolds and the Cartan distribution 72
2.1. Geometric definition of the jet spaces 73
2.2. The Cartan distribution 75
2.3. Integral manifolds of the Cartan distribution 79
3. Lie transformations 83
vii
viii CONTENTS
3.1. Finite Lie transformations 84
3.2. Lie fields 89
4. Classical symmetries of equations 92
4.1. Defining equations 92
4.2. Invariant solutions and reproduction of solutions 94
5. Examples of computations 96
5.1. The Burgers equation 96
5.2. The Korteweg de Vries equation 98
5.3. The Khokhlov Zabolotskaya equation 99
5.3.1. Physically meaningful symmetries 100
5.3.2. Invariant solutions 101
5.4. The Kadomtsev Pogutse equations 102
5.4.1. Computation of symmetries 103
5.4.2. Invariant solutions 104
5.4.3. Reproduction of solutions 106
6. Factorization of equations by symmetries 108
6.1. Second order equations in two independent variables 110
7. Extrinsic and intrinsic symmetries 116
Chapter 4. Higher Symmetries 123
1. Spaces of infinite jets and basic differential geometric structures on
them 123
1.1. The manifolds Jx(n) 124
1.2. Smooth functions on Joc(tt) 124
1.3. Prolongations of differential operators 128
1.4. Vector fields on Joc(tt) 131
1.5. Differential forms on Jx(ir) 134
1.6. The horizontal de Rham complex 135
1.7. Distributions on J°°(n) and their automorphisms 137
2. The Cartan distribution on Joc(tt) and its infinitesimal
automorphisms 138
2.1. The Cartan distribution 139
2.2. Integral manifolds 141
2.3. A computational experiment 142
2.4. Evolutionary derivations 144
2.5. Jacobi brackets 148
2.6. Comparison with Lie fields 148
2.7. Linearizations 150
3. Infinitely prolonged equations and the theory of higher symmetries 154
3.1. Prolongations 154
3.2. Infinitely prolonged equations 156
3.3. Higher symmetries 158
3.4. Extrinsic and intrinsic higher symmetries 161
3.5. Defining equations for higher symmetries 162
4. Examples of computation 164
4.1. Preparatory remarks 164
4.2. The Burgers and heat equations 167
4.3. The plasticity equations 174
4.4. Transformation of symmetries under change of variables 177
CONTENTS ix
4.5. Ordinary differential equations 178
Chapter 5. Conservation Laws 185
1. Introduction: What are conservation laws? 185
2. The C spectral sequence 187
2.1. The definition of the C spectral sequence 187
2.2. The term Eo 188
2.3. The term E : preparatory results 189
2.4. Generalizations 193
2.5. The term Ex for J°°(7r) 194
2.6. The term E in the general case 198
2.7. Conservation laws and generating functions 201
2.8. Euler Lagrange equations 202
2.9. Hamiltonian formalism on Joc(tt) 203
3. Computation of conservation laws 206
3.1. Basic results 206
3.2. Examples 208
4. Symmetries and conservation laws 214
4.1. The Noether theorem 214
4.2. Hamiltonian equations 216
Chapter 6. Nonlocal Symmetries 221
1. Coverings 221
1.1. First examples 221
1.2. Definition of coverings 224
1.3. Coverings in the category of differential equations 224
1.4. Examples of coverings 225
1.5. Coordinates 226
1.6. Basic concepts of covering theory 227
1.7. Coverings and connections 231
1.8. The horizontal de Rham complex and nonlocal conservation
laws 231
1.9. Covering equations 232
1.10. Horizontal de Rham cohomology and coverings 234
1.11. Backlund transformations 236
2. Examples of computations: coverings 238
2.1. Coverings over the Burgers equation 239
2.2. Coverings over the Korteweg de Vries equation 242
2.3. Coverings over the equation ut = {B(u)ux)x 245
2.4. Covering over the / Gordon equation 245
2.5. Coverings of the equation uxx + uyy = p{u) 246
3. Nonlocal symmetries 249
3.1. Definition of nonlocal symmetries 249
3.2. How to find nonlocal symmetries? 249
4. Examples of computation:
nonlocal symmetries of the Burgers equation 251
5. The problem of symmetry reconstruction 257
5.1. Universal Abelian covering 257
5.2. Symmetries in the universal Abelian covering 258
x CONTENTS
5.3. Nonlocal symmetries for equations admitting a recursion
operator 258
5.4. Example: nonlocal symmetries of the Korteweg de Vries
equation 259
5.5. Master symmetries 260
5.6. Examples 261
5.7. General problem of nonlocal symmetry reconstruction 262
5.8. Kiso s construction 263
5.9. Construction of the covering t$ 264
5.10. The universal property of the symmetry ST 265
6. Symmetries of integro differential equations 266
6.1. Transformation of integro differential equations to boundary
differential form 266
6.2. Spaces of (fc, Q) jets 271
6.3. Boundary differential operators 275
6.4. The Cartan distribution on J°°(7r;^) 279
6.5. ^ invariant symmetries of the Cartan distribution on J°°(7r; Q) 284
6.6. Higher symmetries of boundary differential equations 287
6.7. Examples 290
Appendix. From Symmetries of Partial Differential Equations Towards
Secondary ( Quantized ) Calculus 301
Introduction 301
1. From symmetries to concepts 302
2. Troubled times of quantum field theory 303
3. Linguization of the Bohr correspondence principle 304
4. Differential equations are diffieties 306
5. Secondary ( quantized ) functions 308
6. Higher order scalar secondary ( quantized ) differential operators 310
7. Secondary ( quantized ) differential forms 312
8. Quantization or singularity propagation? Heisenberg or Schrodinger? 314
9. Geometric singularities of solutions of partial differential equations 316
10. Wave and geometric optics and other examples 320
10.1. E characteristic equations 320
10.2. Maxwell s equations and geometric optics 320
10.3. On the complementary equations 321
10.4. Alternative singularities via the homogenization trick 322
10.5. ^^ characteristic equations 322
Bibliography 323
Index 329
|
| adam_txt |
Contents
Preface ix
Chapter 1. Ordinary Differential Equations 1
1. Ordinary differential equations from the geometric viewpoint 1
2. Ordinary differential equations of arbitrary order 6
3. Symmetries of distributions 10
4. Some applications of symmetry theory to integration of distributions 17
4.1. Distributions and characteristic symmetries 17
4.2. Symmetries and dynamical systems 18
4.3. Distributions and noncharacteristic symmetries 20
4.4. Integration of equations by quadratures 21
5. Generating functions 29
6. How to search for equations integrable by quadratures: an example
of using symmetries 33
Chapter 2. First Order Equations 37
1. Contact transformations 37
1.1. Contact elements and the Cartan distribution 37
1.2. Contact transformations 42
1.3. Clairaut equation and its integrals 47
1.4. Contact manifolds in mechanics 49
2. Infinitesimal contact transformations and characteristic fields 50
2.1. Infinitesimal contact transformations 50
2.2. Infinitesimal symmetries of equations 54
2.3. Characteristic vector fields and integration of first order
equations 56
2.4. Symmetries and first integrals 59
3. Complete integrals of first order differential equations 60
3.1. Complete integrals: a coordinate approach 61
3.2. The construction of complete integrals using symmetry algebras 61
3.3. Complete integrals: an invariant approach 64
3.4. The Lagrange Charpit method 66
Chapter 3. The Theory of Classical Symmetries 69
1. Equations and the Cartan distribution 69
2. Jet manifolds and the Cartan distribution 72
2.1. Geometric definition of the jet spaces 73
2.2. The Cartan distribution 75
2.3. Integral manifolds of the Cartan distribution 79
3. Lie transformations 83
vii
viii CONTENTS
3.1. Finite Lie transformations 84
3.2. Lie fields 89
4. Classical symmetries of equations 92
4.1. Defining equations 92
4.2. Invariant solutions and reproduction of solutions 94
5. Examples of computations 96
5.1. The Burgers equation 96
5.2. The Korteweg de Vries equation 98
5.3. The Khokhlov Zabolotskaya equation 99
5.3.1. "Physically meaningful" symmetries 100
5.3.2. Invariant solutions 101
5.4. The Kadomtsev Pogutse equations 102
5.4.1. Computation of symmetries 103
5.4.2. Invariant solutions 104
5.4.3. Reproduction of solutions 106
6. Factorization of equations by symmetries 108
6.1. Second order equations in two independent variables 110
7. Extrinsic and intrinsic symmetries 116
Chapter 4. Higher Symmetries 123
1. Spaces of infinite jets and basic differential geometric structures on
them 123
1.1. The manifolds Jx(n) 124
1.2. Smooth functions on Joc(tt) 124
1.3. Prolongations of differential operators 128
1.4. Vector fields on Joc(tt) 131
1.5. Differential forms on Jx(ir) 134
1.6. The horizontal de Rham complex 135
1.7. Distributions on J°°(n) and their automorphisms 137
2. The Cartan distribution on Joc(tt) and its infinitesimal
automorphisms 138
2.1. The Cartan distribution 139
2.2. Integral manifolds 141
2.3. A computational experiment 142
2.4. Evolutionary derivations 144
2.5. Jacobi brackets 148
2.6. Comparison with Lie fields 148
2.7. Linearizations 150
3. Infinitely prolonged equations and the theory of higher symmetries 154
3.1. Prolongations 154
3.2. Infinitely prolonged equations 156
3.3. Higher symmetries 158
3.4. Extrinsic and intrinsic higher symmetries 161
3.5. Defining equations for higher symmetries 162
4. Examples of computation 164
4.1. Preparatory remarks 164
4.2. The Burgers and heat equations 167
4.3. The plasticity equations 174
4.4. Transformation of symmetries under change of variables 177
CONTENTS ix
4.5. Ordinary differential equations 178
Chapter 5. Conservation Laws 185
1. Introduction: What are conservation laws? 185
2. The C spectral sequence 187
2.1. The definition of the C spectral sequence 187
2.2. The term Eo 188
2.3. The term E\: preparatory results 189
2.4. Generalizations 193
2.5. The term Ex for J°°(7r) 194
2.6. The term E\ in the general case 198
2.7. Conservation laws and generating functions 201
2.8. Euler Lagrange equations 202
2.9. Hamiltonian formalism on Joc(tt) 203
3. Computation of conservation laws 206
3.1. Basic results 206
3.2. Examples 208
4. Symmetries and conservation laws 214
4.1. The Noether theorem 214
4.2. Hamiltonian equations 216
Chapter 6. Nonlocal Symmetries 221
1. Coverings 221
1.1. First examples 221
1.2. Definition of coverings 224
1.3. Coverings in the category of differential equations 224
1.4. Examples of coverings 225
1.5. Coordinates 226
1.6. Basic concepts of covering theory 227
1.7. Coverings and connections 231
1.8. The horizontal de Rham complex and nonlocal conservation
laws 231
1.9. Covering equations 232
1.10. Horizontal de Rham cohomology and coverings 234
1.11. Backlund transformations 236
2. Examples of computations: coverings 238
2.1. Coverings over the Burgers equation 239
2.2. Coverings over the Korteweg de Vries equation 242
2.3. Coverings over the equation ut = {B(u)ux)x 245
2.4. Covering over the / Gordon equation 245
2.5. Coverings of the equation uxx + uyy = p{u) 246
3. Nonlocal symmetries 249
3.1. Definition of nonlocal symmetries 249
3.2. How to find nonlocal symmetries? 249
4. Examples of computation:
nonlocal symmetries of the Burgers equation 251
5. The problem of symmetry reconstruction 257
5.1. Universal Abelian covering 257
5.2. Symmetries in the universal Abelian covering 258
x CONTENTS
5.3. Nonlocal symmetries for equations admitting a recursion
operator 258
5.4. Example: nonlocal symmetries of the Korteweg de Vries
equation 259
5.5. Master symmetries 260
5.6. Examples 261
5.7. General problem of nonlocal symmetry reconstruction 262
5.8. Kiso's construction 263
5.9. Construction of the covering t$ 264
5.10. The universal property of the symmetry ST 265
6. Symmetries of integro differential equations 266
6.1. Transformation of integro differential equations to boundary
differential form 266
6.2. Spaces of (fc, Q) jets 271
6.3. Boundary differential operators 275
6.4. The Cartan distribution on J°°(7r;^) 279
6.5. ^ invariant symmetries of the Cartan distribution on J°°(7r; Q) 284
6.6. Higher symmetries of boundary differential equations 287
6.7. Examples 290
Appendix. From Symmetries of Partial Differential Equations Towards
Secondary ("Quantized") Calculus 301
Introduction 301
1. From symmetries to concepts 302
2. "Troubled times" of quantum field theory 303
3. "Linguization" of the Bohr correspondence principle 304
4. Differential equations are diffieties 306
5. Secondary ("quantized") functions 308
6. Higher order scalar secondary ("quantized") differential operators 310
7. Secondary ("quantized") differential forms 312
8. Quantization or singularity propagation? Heisenberg or Schrodinger? 314
9. Geometric singularities of solutions of partial differential equations 316
10. Wave and geometric optics and other examples 320
10.1. E characteristic equations 320
10.2. Maxwell's equations and geometric optics 320
10.3. On the complementary equations 321
10.4. Alternative singularities via the homogenization trick 322
10.5. ^^ characteristic equations 322
Bibliography 323
Index 329 |
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| id | DE-604.BV022122438 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:16:19Z |
| indexdate | 2024-07-09T20:50:58Z |
| institution | BVB |
| isbn | 082180958X |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015337143 |
| oclc_num | 40473732 |
| open_access_boolean | |
| owner | DE-706 DE-188 DE-11 |
| owner_facet | DE-706 DE-188 DE-11 |
| physical | XIV, 333 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spelling | Simmetrii i zakony sochranenija uravnenij matematiceskoj fiziki Symmetries and conservation laws for differential equations of mathematical physics A. V. Bocharov ... I. S. Krasil'shchik (ed.) ... Providence, Rhode Island American Math. Soc. 1999 XIV, 333 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 182 Literaturverz. S. [323] - 328. -Aus dem Russ. übers. Calcul quantifié Elément contact Equation Burgers Equation Korteweg-De Vries Equations différentielles - Solutions numériques ram Lois de conservation (Mathématiques) Lois de conservation (Mathématiques) ram Méthode Lagrange-Charpit Physique mathématique Physique mathématique ram Revêtement Symétrie Symétrie ram Théorème Noether Équations différentielles - Solutions numériques Mathematische Physik Conservation laws (Mathematics) Differential equations Numerical solutions Mathematical physics Symmetry Theoretische Physik (DE-588)4117202-4 gnd rswk-swf Erhaltungssatz (DE-588)4131214-4 gnd rswk-swf Symmetrie (DE-588)4058724-1 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s DE-604 Symmetrie (DE-588)4058724-1 s Theoretische Physik (DE-588)4117202-4 s Erhaltungssatz (DE-588)4131214-4 s Krasil'ščik, Iosif S. edt Bocarov, A. V. edt Translations of mathematical monographs 182 (DE-604)BV000002394 182 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015337143&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Symmetries and conservation laws for differential equations of mathematical physics Translations of mathematical monographs Calcul quantifié Elément contact Equation Burgers Equation Korteweg-De Vries Equations différentielles - Solutions numériques ram Lois de conservation (Mathématiques) Lois de conservation (Mathématiques) ram Méthode Lagrange-Charpit Physique mathématique Physique mathématique ram Revêtement Symétrie Symétrie ram Théorème Noether Équations différentielles - Solutions numériques Mathematische Physik Conservation laws (Mathematics) Differential equations Numerical solutions Mathematical physics Symmetry Theoretische Physik (DE-588)4117202-4 gnd Erhaltungssatz (DE-588)4131214-4 gnd Symmetrie (DE-588)4058724-1 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4117202-4 (DE-588)4131214-4 (DE-588)4058724-1 (DE-588)4012249-9 |
| title | Symmetries and conservation laws for differential equations of mathematical physics |
| title_alt | Simmetrii i zakony sochranenija uravnenij matematiceskoj fiziki |
| title_auth | Symmetries and conservation laws for differential equations of mathematical physics |
| title_exact_search | Symmetries and conservation laws for differential equations of mathematical physics |
| title_exact_search_txtP | Symmetries and conservation laws for differential equations of mathematical physics |
| title_full | Symmetries and conservation laws for differential equations of mathematical physics A. V. Bocharov ... I. S. Krasil'shchik (ed.) ... |
| title_fullStr | Symmetries and conservation laws for differential equations of mathematical physics A. V. Bocharov ... I. S. Krasil'shchik (ed.) ... |
| title_full_unstemmed | Symmetries and conservation laws for differential equations of mathematical physics A. V. Bocharov ... I. S. Krasil'shchik (ed.) ... |
| title_short | Symmetries and conservation laws for differential equations of mathematical physics |
| title_sort | symmetries and conservation laws for differential equations of mathematical physics |
| topic | Calcul quantifié Elément contact Equation Burgers Equation Korteweg-De Vries Equations différentielles - Solutions numériques ram Lois de conservation (Mathématiques) Lois de conservation (Mathématiques) ram Méthode Lagrange-Charpit Physique mathématique Physique mathématique ram Revêtement Symétrie Symétrie ram Théorème Noether Équations différentielles - Solutions numériques Mathematische Physik Conservation laws (Mathematics) Differential equations Numerical solutions Mathematical physics Symmetry Theoretische Physik (DE-588)4117202-4 gnd Erhaltungssatz (DE-588)4131214-4 gnd Symmetrie (DE-588)4058724-1 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Calcul quantifié Elément contact Equation Burgers Equation Korteweg-De Vries Equations différentielles - Solutions numériques Lois de conservation (Mathématiques) Méthode Lagrange-Charpit Physique mathématique Revêtement Symétrie Théorème Noether Équations différentielles - Solutions numériques Mathematische Physik Conservation laws (Mathematics) Differential equations Numerical solutions Mathematical physics Symmetry Theoretische Physik Erhaltungssatz Symmetrie Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015337143&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000002394 |
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