The Geometry of Higher-Order Lagrange Spaces: Applications to Mechanics and Physics (Fundamental Theories of Physics)

Contents......Page 4
Preface......Page 10
1 Lagrange Spaces of Order 1......Page 16
1.1 The Manifold Osc^1 M. Sprays......Page 17
1.2 Nonlinear Connections......Page 22
1.3 N-Linear Connections......Page 27
1.4 d-Tensors of Torsion and Curvature......Page 31
1.5 Parallelism. Structure Equations......Page 36
1.6 Lagrange Spaces......Page 39
1.7 Variational Problem......Page 40
1.8 A Noether Theorem......Page 43
1.9 Canonical Nonlinear Connection of a Lagrange Space......Page 45
1.10 Canonical Metrical Connection......Page 47
1.11 Finsler Spaces......Page 50
1.12 Generalized Lagrange Spaces......Page 52
1.13 Almost Kählerian Model of the Space L^n......Page 55
1.14 Problems......Page 57
2.1 The Fibre Bundle Osc^2 M......Page 60
2.2 Vertical Distributions. Liouville Vector Fields......Page 64
2.3 2-Tangent Structure. 2-Sprays......Page 67
2.4 Nonlinear Connections......Page 71
2.5 J-Vertical Distributions......Page 75
2.6 The Dual Coefficients of a Nonlinear Connection......Page 77
2.7 Determination of a Nonlinear Connection from a 2-Spray.......Page 83
2.8 The Almost Product Structure P. The Almost n-Contact Structure F......Page 85
2.9 Problems......Page 89
3.1 The Algebra of d-Tensor Fields......Page 90
3.2 N-Linear Connection on Osc^2 M......Page 93
3.3 The Coefficients of the N-Linear Connections......Page 96
3.4 d-Tensors of Torsion......Page 101
3.5 d-Tensors of Curvature......Page 105
3.6 Structure Equations of an N-Linear Connection......Page 115
3.7 Problems......Page 121
4 Lagrangians of Second Order. Variational Problem. Nöther Type Theorems.......Page 122
4.1 Lagrangians of Second Order. Zermelo Conditions......Page 123
4.2 Variational Problem......Page 125
4.3 Operators 1I_v, 2I_v, d_V/dt......Page 128
4.4 Craig-Synge Covectors......Page 131
4.5 The Energies 1E_c(L), 2E_c(L)......Page 134
4.6 Nöther Theorems......Page 138
4.7 Jacobi-Ostrogradski Momenta......Page 141
4.8 Regular Lagrangians.  Canonical Nonlinear Connections......Page 145
4.9 Prolongation to Osc^2 M of the Riemannian Structures......Page 149
4.10 Prolongation to Osc^2 M of the Finslerian and Lagrangian Structures......Page 151
4.11 Problems......Page 154
5.1 The Notion of Lagrange Space of Order 2......Page 156
5.2 Euler-Lagrange Equations of a Lagrange Space L^{(2)n}......Page 159
5.3 Canonical Nonlinear Connections......Page 161
5.4 Canonical Metrical N-Connections......Page 165
5.5 Problems......Page 167
6.1 The Notion of k-Osculator Bundle......Page 168
6.2 Vertical Distributions. Liouville Vector Fields......Page 172
6.3 k-Tangent Structure; k-Sprays......Page 175
6.4 Nonlinear Connections......Page 178
6.5 J-Vertical Distributions......Page 181
6.6 The Dual Coeficients of a Nonlinear Connection N on Osc^k M......Page 183
6.7 The Determination of a Nonlinear Connection from a k-Spray.  The Structures P and F......Page 189
6.8 Problems......Page 192
7.1 d-Tensors Algebra......Page 194
7.2 N-Linear Connection......Page 197
7.3 N-Linear Connections in Adapted Basis......Page 200
7.4 d-Tensors of Torsion......Page 203
7.5 d-Tensors of Curvature......Page 206
7.6 The d-Tensors of Curvature in Adapted Basis......Page 208
7.7 The Structure Equations......Page 214
7.8 Problems......Page 217
8.1 Lagrangians of Order k. Zermello Conditions......Page 218
8.2 Variational Problem......Page 220
8.3 Operators d_V/dt, I^1_V, ..., I^k_V......Page 223
8.4 Craig-Synge Covectors......Page 225
8.5 Energies of Higher Order......Page 227
8.6 Noether Theorems......Page 230
8.7 Jacobi-Ostrogradski Momenta......Page 234
8.8 Regular Lagrangians. Canonical Nonlinear Connection......Page 238
8.9 Problems......Page 241
9.1 Prolongation to Osc^k M of the Riemannian Structures......Page 242
9.2 Prolongation to the k-Osculator Bundle of the Finslerian Structures......Page 248
9.3 Prolongation to Osc^k M of a Lagrangian Structure......Page 253
9.4 Remarkable Regular Lagrangians of Order k......Page 254
9.5 Problems......Page 257
10 Higher Order Lagrange Spaces......Page 258
10.1 The Definition of the Space L^{(k)n}......Page 259
10.2 Canonical Metrical N-Connections......Page 264
10.3 The Riemannian (k-1)n-Contact Model of the Space L^{(k)n}......Page 269
10.4 The Gravitational and Electromagnetic Fields in L^{(k)n}......Page 271
10.5 The Generalized Lagrange Spaces of Order k......Page 275
10.6 Problems......Page 278
11.1 Submanifolds in Osc^k M......Page 280
11.2 Subspaces in the Lagrange Space of Order k, L^{(k)}M......Page 283
11.3 Induced Nonlinear Connection......Page 285
11.4 The Relative Covariant Derivative......Page 289
11.5 The Gauss-Weingarten Formulae......Page 294
11.6 The Gauss-Codazzi Equations......Page 297
11.7 Problems......Page 299
12 Gauge Theory in the Higher Order Lagrange Spaces......Page 300
12.1 Gauge Transformations in Principal Bundles......Page 301
12.2 Gauge k-Osculator Bundles......Page 302
12.3 The Local Representation of Gauge Transformations......Page 305
12.4 Gauge d-Tensor Fields.  Gauge Nonlinear Connections......Page 310
12.5 Gauge N-Linear Connections and Gauge h- and v^\alpha-Covariant Derivatives......Page 314
12.6 Einstein-Yang-Mills Equations......Page 316
12.7 Gauge Invariance of the Lagrangians of Order k......Page 319
References......Page 324
Index......Page 346