Leon M. de. Methods of differential geometry in classical field theories: k-symplectic and k-cosymplectic approaches (Singapore, 2016 ). - ОГЛАВЛЕНИЕ / CONTENTS
Навигация
Архив выставки новых поступлений | Отечественные поступления | Иностранные поступления | Сиглы
ОбложкаLeón 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
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
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 fig.2k ×
     (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 fig.2k × 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 fig.2k × 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

Архив выставки новых поступлений | Отечественные поступления | Иностранные поступления | Сиглы
 

[О библиотеке | Академгородок | Новости | Выставки | Ресурсы | Библиография | Партнеры | ИнфоЛоция | Поиск | English]
  Пожелания и письма: www@prometeus.nsc.ru
© 1997-2018 Отделение ГПНТБ СО РАН (Новосибирск)
Статистика доступов: архив | текущая статистика
 

Документ изменен: Thu Apr 5 17:11:28 2018. Размер: 11,942 bytes.
Посещение N 150 c 07.11.2017