 | León M. de. Methods of differential geometry in classical field theories: k-symplectic and k-cosymplectic approaches / M. de Leon, M.Salgato, S.Vilarino. - Singapore: World Scientific, 2016. - xiii, 207 p.: ill. - Bibliogr.: p.195-204. - Ind.: p.205-207. - ISBN 978-981-4699-75-4
Шифр: (И-В31-L55) 02
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Introduction .................................................... v
List of Tables ............................................... xiii
A review of Hamiltonian and Lagrangian mechanics ................ 1
1 Hamiltonian and Lagrangian Mechanics ....................... 3
1.1 Hamiltonian mechanics ...................................... 3
1.1.1 Algebraic preliminaries ............................. 3
1.1.2 Canonical forms on the cotangent bundle ............. 4
1.1.3 Hamilton equations .................................. 6
1.2 Lagrangian mechanics ....................................... 6
1.2.1 Geometric preliminaries ............................. 7
1.2.2 Second order differential equations ................. 8
1.2.3 Euler-Lagrange equations ............................ 9
1.3 Legendre transformation ................................... 12
1.4 Non-autonomous Hamiltonian and Lagrangian mechanics ....... 13
1.4.1 Hamiltonian mechanics .............................. 13
1.4.2 Lagrangian mechanics ............................... 14
fc-symplectic formulation of classical field theories ..... 17
2 k-symplectic Geometry ..................................... 21
2.1 The cotangent bundle of k1-covelocities ................... 21
2.2 fc-symplectic geometry .................................... 24
2.2.1 k-symplectic vector spaces ......................... 25
2.2.2 k-symplectic manifolds ............................. 27
3 fc-symplectic Formalism ................................... 31
3.1 fc-vector fields and integral sections .................... 31
3.2 fc-symplectic Hamiltonian equation ........................ 34
3.3 Example: electrostatic equations .......................... 39
4 Hamiltonian Classical Field Theory ........................ 41
4.1 Variational approach ...................................... 41
4.1.1 Prolongation of diffeomorphism and vector fields ... 42
4.1.2 Variational principle .............................. 44
4.2 Hamilton-De Donder-Weyl equations ......................... 48
5 Hamilton-Jacobi Theory in k-symplectic Field Theories ..... 51
5.1 The Hamilton-Jacobi equation .............................. 52
5.2 Example: the vibrating string problem ..................... 57
6 Lagrangian Classical Field Theories ....................... 59
6.1 The tangent bundle of k1-velocities ....................... 59
6.1.1 Geometric elements ................................. 61
6.1.2 Prolongation of vector fields ...................... 64
6.1.3 First prolongation of maps ......................... 65
6.2 Variational principle for the Euler-Lagrange equations .... 65
6.3 Euler-Lagrange field equations: k-symplectic version ...... 68
6.3.1 Poincaré-Cartan forms on the tangent bundle of
k1-velocities ...................................... 68
6.3.2 Second order partial differential equations on
Tk1Q ............................................... 69
6.3.3 Euler-Lagrange field equations ..................... 72
6.4 fc-symplectic Legendre transformation ..................... 74
7 Examples .................................................. 79
7.1 Electrostatic equations ................................... 80
7.2 Wave equation ............................................. 81
7.3 Laplace's equations ....................................... 83
7.4 Sine-Gordon equation ...................................... 85
7.5 Ginzburg-Landau's equation ................................ 86
7.6 k-symplectic quadratic systems ............................ 88
7.7 Navier's equations ........................................ 91
7.8 Equation of minimal surfaces .............................. 92
7.9 The massive scalar field .................................. 93
fc-cosymplectic formulation of classical field theories ... 95
8 fc-cosymplectic Geometry .................................. 99
8.1 The stable cotangent bundle of k1-covelocities  k ×
(Tk1)*Q .................................................... 99
8.2 fc-cosymplectic geometry ................................. 102
8.2.1 k-cosymplectic vector spaces ...................... 102
8.2.2 k-cosymplectic manifolds .......................... 103
9 k-cosymplectic Formalism ................................. 105
9.1 k-cosymplectic Hamiltonian equations ..................... 105
9.2 Example: massive scalar field ............................ 109
10 Hamiltonian Classical Field Theories ..................... 111
10.1 Variational approach ..................................... 111
10.1.1 Prolongation of vector fields ..................... 112
10.1.2 Variational principle ............................. 114
10.2 Hamilton-De Donder-Weyl equations: k-cosymplectic
approach ................................................. 120
11 Hamilton-Jacobi Equation ................................. 123
11.1 The Hamilton-Jacobi equation ............................. 124
11.2 Examples ................................................. 129
12 Lagrangian Classical Field Theories ...................... 133
12.1 The stable tangent bundle of k1-covelocities k × Tk1Q ... 133
12.1.1 Canonical tensor fields ........................... 136
12.1.2 Prolongation of diffeomorphism and vector fields .. 137
12.1.3 k-vector fields and SOPDES ........................ 139
12.2 Variational principle .................................... 141
12.3 k-cosymplectic version of Euler-Lagrange field
equations ................................................ 145
12.3.1 Poincare-Cartan forms on k × Tk1Q ................ 146
12.3.2 k-cosymplectic Euler-Lagrange equation ............ 147
12.4 k-cosymplectic Legendre transformation ................... 149
13 Examples ................................................. 155
13.1 Electrostatic equations .................................. 155
13.2 The massive scalar field ................................. 156
13.3 Harmonic maps ............................................ 157
13.4 Electromagnetic field in vacuum: Maxwell's equations ..... 159
14 k-symplectic Systems versus Autonomous k-cosymplectic
Systems .................................................. 165
Relationship between k-symplectic and k-cosymplectic
approaches and the multisymplectic formalism ............. 171
15 Multisymplectic Formalism ................................ 173
15.1 First order jet bundles .................................. 174
15.2 Multisymplectic Hamiltonian formalism .................... 175
15.2.1 Multimomentum bundles ............................. 175
15.2.2 Hamiltonian systems ............................... 176
15.2.3 Relation with the k-cosymplectic Hamiltonian
formalism ......................................... 178
15.3 Multisymplectic Lagrangian formalism ..................... 180
15.3.1 Multisymplectic Lagrangian systems ................ 180
15.3.2 Relation with the k-cosymplectic Lagrangian
formalism ......................................... 182
15.4 Correspondences .......................................... 184
Appendix A Symplectic Manifolds .............................. 187
Appendix В Cosymplectic Manifolds ............................ 189
Appendix С Glossary of Symbols ............................... 193
Bibliography .................................................. 195
Index ......................................................... 205
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