Classical Mechanics: Hamiltonian and Lagrangian formalism
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2010
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| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XII, 308 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783642140365 |
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IMAGE 1
CONTENTS
1 SKETCH OF LAGRANGIAN FORMALISM 1
1.1 NEWTON'S EQUATION 1
1.2 GALILEAN TRANSFORMATIONS: PRINCIPLE OF GALILEAN RELATIVITY 8 1.3
POINCARE AND LORENTZ TRANSFORMATIONS: THE PRINCIPLE OF SPECIAL
RELATIVITY 13
1.4 PRINCIPLE OF LEAST ACTION 23
1.5 VARIATIONAL ANALYSIS 24
1.6 GENERALIZED COORDINATES, COORDINATE TRANSFORMATIONS AND SYMMETRIES
OF AN ACTION 29
1.7 EXAMPLES OF CONTINUOUS (FIELD) SYSTEMS 36
1.8 ACTION OF A CONSTRAINED SYSTEM: THE RECIPE 44
1.9 ACTION OF A CONSTRAINED SYSTEM: JUSTIFICATION OF THE RECIPE 51 1.10
DESCRIPTION OF CONSTRAINED SYSTEM BY SINGULAR ACTION 52 1.11 KINETIC
VERSUS POTENTIAL ENERGY: FORCELESS MECHANICS OF HERTZ 54
1.12 ELECTROMAGNETIC FIELD IN LAGRANGIAN FORMALISM 56
1.12.1 MAXWELL EQUATIONS 56
1.12.2 NONSINGULAR LAGRANGIAN ACTION OF ELECTRODYNAMICS 59 1.12.3
MANIFESTLY POINCARE-INVARIANT FORMULATION IN TERMS OF A SINGULAR
LAGRANGIAN ACTION 63
1.12.4 NOTION OF LOCAL (GAUGE) SYMMETRY 65
1.12.5 LORENTZ TRANSFORMATIONS OF THREE-DIMENSIONAL POTENTIAL: ROLE OF
GAUGE SYMMETRY 68
1.12.6 RELATIVISTIC PARTICLE ON ELECTROMAGNETIC BACKGROUND 69 1.12.7
POINCARE TRANSFORMATIONS OF ELECTRIC AND MAGNETIC FIELDS . 72
2 HAMILTONIAN FORMALISM 77
2.1 DERIVATION OF HAMILTONIAN EQUATIONS 77
2.1.1 PRELIMINARIES 77
2.1.2 FROM LAGRANGIAN TO HAMILTONIAN EQUATIONS 79
2.1.3 SHORT PRESCRIPTION FOR HAMILTONIZATION PROCEDURE, PHYSICAL
INTERPRETATION OF HAMILTONIAN 83
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1002961696
DIGITALISIERT DURCH
IMAGE 2
CONTENTS
2.1.4 INVERSE PROBLEM: FROM HAMILTONIAN TO LAGRANGIAN FORMULATION 85
2.2 POISSON BRACKET AND SYMPLECTIC MATRIX 85
2.3 GENERAL SOLUTION TO HAMILTONIAN EQUATIONS 87
2.4 PICTURE OF MOTION IN PHASE SPACE 91
2.5 CONSERVED QUANTITIES AND THE POISSON BRACKET 93
2.6 PHASE SPACE TRANSFORMATIONS AND HAMILTONIAN EQUATIONS 96 2.7
DEFINITION OF CANONICAL TRANSFORMATION 100
2.8 GENERALIZED HAMILTONIAN EQUATIONS: EXAMPLE OF NON- CANONICAL POISSON
BRACKET 102
2.9 HAMILTONIAN ACTION FUNCTIONAL 106
2.10 SCHROEDINGER EQUATION AS THE HAMILTONIAN SYSTEM 107
2.10.1 LAGRANGIAN ACTION ASSOCIATED WITH THE SCHROEDINGER EQUATION 108
2.10.2 PROBABILITY AS A CONSERVED CHARGE VIA THE NOETHER THEOREM 111
2.11 HAMILTONIZATION PROCEDURE IN TERMS OF FIRST-ORDER ACTION FUNCTIONAL
113 2.12 HAMILTONIZATION OF A THEORY WITH HIGHER-ORDER DERIVATIVES 114
2.12.1 FIRST-ORDER TRICK 114
2.12.2 OSTROGRADSKY METHOD 116
CANONICAL TRANSFORMATIONS OF TWO-DIMENSIONAL PHASE SPACE 119 3.1
TIME-INDEPENDENT CANONICAL TRANSFORMATIONS 119
3.1.1 TIME-INDEPENDENT CANONICAL TRANSFORMATIONS AND SYMPLECTIC MATRIX
119
3.1.2 GENERATING FUNCTION 121
3.2 TIME-DEPENDENT CANONICAL TRANSFORMATIONS 123
3.2.1 CANONICAL TRANSFORMATIONS AND SYMPLECTIC MATRIX 123 3.2.2
GENERATING FUNCTION 125
PROPERTIES OF CANONICAL TRANSFORMATIONS 127
4.1 INVARIANCE OF THE POISSON BRACKET (SYMPLECTIC MATRIX) 128 4.2
INFINITESIMAL CANONICAL TRANSFORMATIONS: HAMILTONIAN AS A GENERATOR OF
EVOLUTION 133
4.3 GENERATING FUNCTION OF CANONICAL TRANSFORMATION 136
4.3.1 FREE CANONICAL TRANSFORMATION AND ITS FUNCTION F(Q', P', R) 136
4.3.2 GENERATING FUNCTION S(Q, Q',Z) 137
4.4 EXAMPLES OF CANONICAL TRANSFORMATIONS 140
4.4.1 EVOLUTION AS A CANONICAL TRANSFORMATION: INVARIANCE OF PHASE-SPACE
VOLUME 140
4.4.2 CANONICAL TRANSFORMATIONS IN PERTURBATION THEORY 143 4.4.3
COORDINATES ADJUSTED TO A SURFACE 144
4.5 TRANSFORMATION PROPERTIES OF THE HAMILTONIAN ACTION 145 4.6 SUMMARY:
EQUIVALENT DEFINITIONS FOR CANONICAL TRANSFORMATION . . . 146 4.7
HAMILTON-JACOBI EQUATION 147
4.8 ACTION FUNCTIONAL AS A GENERATING FUNCTION OF EVOLUTION 151
IMAGE 3
CONTENTS
INTEGRAL INVARIANTS 155
5.1 POINCARE-CARTAN INTEGRAL INVARIANT 155
5.1.1 PRELIMINARIES 155
5.1.2 LINE INTEGRAL OF A VECTOR FIELD, HAMILTONIAN ACTION,
POINCARE-CARTAN AND POINCARE INTEGRAL INVARIANTS 157 5.1.3 INVARIANCE OF
THE POINCARE-CARTAN INTEGRAL 159
5.2 UNIVERSAL INTEGRAL INVARIANT OF POINCARE 162
POTENTIAL MOTION IN A GEOMETRIC SETTING 167
6.1 ANALYSIS OF TRAJECTORIES AND THE PRINCIPLE OF MAUPERTUIS 167 6.1.1
TRAJECTORY: SEPARATION OF KINEMATICS FROM DYNAMICS 168 6.1.2 EQUATIONS
FOR TRAJECTORY IN THE HAMILTONIAN FORMULATION .170 6.1.3 THE PRINCIPLE
OF MAUPERTUIS FOR TRAJECTORIES 171 6.1.4 LAGRANGIAN ACTION FOR
TRAJECTORIES 172
6.2 DESCRIPTION OF A POTENTIAL MOTION IN TERMS OF A PAIR OF RIEMANN
SPACES 174
6.3 SOME NOTIONS OF RIEMANN GEOMETRY 178
6.3.1 RIEMANN SPACE 178
6.3.2 COVARIANT DERIVATIVE AND RIEMANN CONNECTION 183 6.3.3 PARALLEL
TRANSPORT: NOTIONS OF COVARIANCE AND COORDINATE INDEPENDENCE 185
6.4 DEFINITION OF COVARIANT DERIVATIVE THROUGH PARALLEL TRANSPORT:
FORMAL SOLUTION TO THE PARALLEL TRANSPORT EQUATION 189 6.5 THE GEODESIC
LINE AND ITS REPARAMETRIZATION COVARIANT EQUATION .191 6.6 EXAMPLE: A
SURFACE EMBEDDED IN EUCLIDEAN SPACE 193 6.7 SHORTEST LINE AND GEODESIC
LINE: ONE MORE EXAMPLE OF A SINGULAR
ACTION 196
6.8 FORMAL GEOMETRIZATION OF MECHANICS 200
TRANSFORMATIONS, SYMMETRIES AND NOETHER THEOREM 203 7.1 THE NOTION OF
INVARIANT ACTION FUNCTIONAL 203
7.2 COORDINATE TRANSFORMATION, INDUCED TRANSFORMATION OF FUNCTIONS AND
SYMMETRIES OF AN ACTION 206
7.3 EXAMPLES OF INVARIANT ACTIONS, GALILEO GROUP 211
7.4 POINCARE GROUP, RELATIVISTIC PARTICLE 214
7.5 SYMMETRIES OF EQUATIONS OF MOTION 215
7.6 NOETHER THEOREM 218
7.7 INFINITESIMAL SYMMETRIES 220
7.8 DISCUSSION OF THE NOETHER THEOREM 223
7.9 USE OF NOETHER CHARGES FOR REDUCTION OF THE ORDER OF EQUATIONS OF
MOTION 224
7.10 EXAMPLES 225
IMAGE 4
XII CONTENTS
7.11 SYMMETRIES OF HAMILTONIAN ACTION 228
7.11.1 INFINITESIMAL SYMMETRIES GIVEN BY CANONICAL TRANSFORMATIONS 228
7.11.2 STRUCTURE OF INFINITESIMAL SYMMETRY OF A GENERAL FORM . . . 230
7.11.3 HAMILTONIAN VERSUS LAGRANGIAN GLOBAL SYMMETRY 234
8 HAMILTONIAN FORMALISM FOR SINGULAR THEORIES 237
8.1 HAMILTONIZATION OF A SINGULAR THEORY: THE RECIPE 238
8.1.1 TWO BASIC EXAMPLES 238
8.1.2 DIRAC PROCEDURE 242
8.2 JUSTIFICATION OF THE HAMILTONIZATION RECIPE 247
8.2.1 CONFIGURATION-VELOCITY SPACE 247
8.2.2 HAMILTONIZATION 249
8.2.3 COMPARISON WITH THE DIRAC RECIPE 252
8.3 CLASSIFICATION OF CONSTRAINTS 254
8.4 COMMENT ON THE PHYSICAL INTERPRETATION OF A SINGULAR THEORY 255 8.5
THEORY WITH SECOND-CLASS CONSTRAINTS: DIRAC BRACKET 259 8.6 EXAMPLES OF
THEORIES WITH SECOND-CLASS CONSTRAINTS 262 8.6.1 MECHANICS WITH
KINEMATIC CONSTRAINTS 262
8.6.2 SINGULAR LAGRANGIAN ACTION UNDERLYING THE SCHROEDINGER EQUATION 264
8.7 EXAMPLES OF THEORIES WITH FIRST-CLASS CONSTRAINTS 266
8.7.1 ELECTRODYNAMICS 266
8.7.2 SEMICLASSICAL MODEL FOR DESCRIPTION OF NON RELATIVISTIC SPIN 268
8.8 LOCAL SYMMETRIES AND CONSTRAINTS 274
8.9 LOCAL SYMMETRY DOES NOT IMPLY A CONSERVED CHARGE 281 8.10 FORMALISM
OF EXTENDED LAGRANGIAN 281
8.11 LOCAL SYMMETRIES OF THE EXTENDED LAGRANGIAN: DIRAC CONJECTURE . .
286 8.12 LOCAL SYMMETRIES OF THE INITIAL LAGRANGIAN 290
8.13 CONVERSION OF SECOND-CLASS CONSTRAINTS BY DEFORMATION OF LAGRANGIAN
LOCAL SYMMETRIES 293
8.13.1 CONVERSION IN A THEORY WITH HIDDEN SO(L, 4) GLOBAL SYMMETRY 296
8.13.2 CLASSICAL MECHANICS SUBJECT TO KINEMATIC CONSTRAINTS AS A GAUGE
THEORY 298
8.13.3 CONVERSION IN MAXWELL-PROCA LAGRANGIAN FOR MASSIVE VECTOR FIELD
301
BIBLIOGRAPHY 303
INDEX 305 |
| any_adam_object | 1 |
| author | Deriglazov, Alexei |
| author_facet | Deriglazov, Alexei |
| author_role | aut |
| author_sort | Deriglazov, Alexei |
| author_variant | a d ad |
| building | Verbundindex |
| bvnumber | BV036684289 |
| classification_rvk | UF 1000 |
| ctrlnum | (OCoLC)705858958 (DE-599)DNB1002961696 |
| dewey-full | 531.01515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 531 - Classical mechanics |
| dewey-raw | 531.01515 |
| dewey-search | 531.01515 |
| dewey-sort | 3531.01515 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-20T10:47:11Z |
| institution | BVB |
| isbn | 9783642140365 |
| language | English |
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| physical | XII, 308 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2010 |
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| publisher | Springer |
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| spelling | Deriglazov, Alexei Verfasser aut Classical Mechanics Hamiltonian and Lagrangian formalism Alexei Deriglazov Berlin [u.a.] Springer 2010 XII, 308 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3484434&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020603086&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Deriglazov, Alexei Classical Mechanics Hamiltonian and Lagrangian formalism Theoretische Mechanik (DE-588)4185100-6 gnd |
| subject_GND | (DE-588)4185100-6 |
| title | Classical Mechanics Hamiltonian and Lagrangian formalism |
| title_auth | Classical Mechanics Hamiltonian and Lagrangian formalism |
| title_exact_search | Classical Mechanics Hamiltonian and Lagrangian formalism |
| title_full | Classical Mechanics Hamiltonian and Lagrangian formalism Alexei Deriglazov |
| title_fullStr | Classical Mechanics Hamiltonian and Lagrangian formalism Alexei Deriglazov |
| title_full_unstemmed | Classical Mechanics Hamiltonian and Lagrangian formalism Alexei Deriglazov |
| title_short | Classical Mechanics |
| title_sort | classical mechanics hamiltonian and lagrangian formalism |
| title_sub | Hamiltonian and Lagrangian formalism |
| topic | Theoretische Mechanik (DE-588)4185100-6 gnd |
| topic_facet | Theoretische Mechanik |
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