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The Geometry of Lagrange Spaces : Theory and Applications

By: R. Miron, Mihai Anastasiei

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304 Pages

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Differential-geometric methods are gaining increasing importance in the understanding of a wide range of fundamental natural phenomena. Very often, the starting point for such studies is a variational problem formulated for a convenient Lagrangian. From a formal point of view, a Lagrangian is a smooth real function defined on the total space of the tangent bundle to a manifold satisfying some regularity conditions. The main purpose of this book is to present: (a) an extensive discussion of the geometry of the total space of a vector bundle; (b) a detailed exposition of Lagrange geometry; and (c) a description of the most important applications. New methods are described for construction geometrical models for applications.
The various chapters consider topics such as fibre and vector bundles, the Einstein equations, generalized Einstein--Yang--Mills equations, the geometry of the total space of a tangent bundle, Finsler and Lagrange spaces, relativistic geometrical optics, and the geometry of time-dependent Lagrangians. Prerequisites for using the book are a good foundation in general manifold theory and a general background in geometrical models in physics.
For mathematical physicists and applied mathematicians interested in the theory and applications of differential-geometric methods.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Engineering & Technology Technology in General Maths for Engineers Science Science in General Maths for Scientists Applied Mathematics Geometry

Differential & Riemannian Geometry Physics Mathematical Physics

Preface
Fibre Bundles. General Theory	p. 1
Fibre Bundles	p. 1
Principal Fibre Bundles	p. 3
Vector Bundles	p. 7
Morphisms of Vector Bundles	p. 11
Vector Subbundles	p. 13
Operations with Vector Bundles	p. 14
Principal Bundle Associated with a Vector Bundle	p. 15
Sections in Vector Bundles	p. 16
Connections in Fibre Bundles	p. 19
Non-linear Connections in Vector Bundles	p. 19
Local Representations of a Non-linear Connection	p. 21
Other Characterisations of a Non-linear Connection	p. 23
Vertical and Horizontal Lifts	p. 27
Curvature of a Non-linear Connection	p. 30
Affine Morphisms of Vector Bundles	p. 33
Geometry of the Total Space of a Vector Bundle	p. 35
d-Connections on the Total Space of a Vector Bundle	p. 35
Local Representation of d-Connections	p. 40
Torsion and Curvature of d-Connections	p. 44
Structure Equations of a d-Connection	p. 51
Metric Structures on the Total Space of a Vector Bundle	p. 55
Geometrical Theory of Embeddings of Vector Bundles	p. 66
Embeddings of Vector Bundles	p. 66
Moving Frame on E' in E	p. 68
Induced Non-linear Connections. Relative Covariant Derivative	p. 70
The Gauss-Weingarten Formulae	p. 75
The Gauss-Codazzi Equations	p. 78
Einstein Equations	p. 80
Einstein Equations	p. 80
Einstein Equations in the Case m = 1	p. 84
Another Form of the Einstein Equations	p. 87
Einstein Equations for some particular metrics on E	p. 90
Generalized Einstein-Yang Mills Equations	p. 94
Gauge Transformations	p. 94
Gauge Covariant Derivatives	p. 98
Metrical Gauge d-Connections	p. 101
Generalized Einstein-Yang Mills Equations	p. 104
Geometry of the Total Space of a Tangent Bundle	p. 106
Non-linear Connections in Tangent Bundle	p. 106
Semisprays, Sprays and Non-linear Connections	p. 111
Torsions and Curvature of a Non-linear Connections	p. 115
Transformations of Non-linear Connections	p. 118
Normal d-Connections on TM	p. 121
Metrical Structures on TM	p. 123
Some Remarkable Metrics on TM	p. 126
Finsler Spaces	p. 129
The Notion of Finsler Space	p. 129
Non-linear Cartan Connection	p. 132
Geodesics	p. 136
Metrical Cartan Connection	p. 138
Structure Equations. Bianchi Identities	p. 142
Remarkable Finslerian Connections	p. 147
Almost Kahlerian Model of a Finsler Space	p. 150
Subspaces in a Finsler Space	p. 153
Lagrange Spaces	p. 157
The Notion of Lagrange Space	p. 158
Euler-Lagrange Equations. Canonical Non-linear Connection	p. 160
Canonical Metrical d-Connection	p. 163
Gravitational and Electromagnetic Fields	p. 166
Lagrange Space of Electrodynamics	p. 170
Almost Finslerian Lagrange Spaces	p. 173
Almost Kahlerian Model of a Lagrange Space	p. 178
Generalized Lagrange Space	p. 180
Notion of Generalized Lagrange Space	p. 180
Metrical d-Connections in a GL[superscript n] Space	p. 183
Structure Equations. Parallelism	p. 187
On h-Covariant Constant d-Tensor Fields	p. 191
Gravitational Field	p. 194
Electromagnetic Field	p. 197
Almost Hermitian Model of a GL[superscript n] Space	p. 199
Applications of the GL[superscript n] Spaces with the Metric Tensor e[superscript 2[delta](x,y)][gamma][subscript ij](x,y)	p. 204
EPS conditions and the Metric e[superscript 2[delta](x,y)][gamma][subscript ij](x,y)	p. 204
Canonical Metrical d-Connection	p. 206
Electromagnetic and Gravitational Fields	p. 210
Two Particular Cases	p. 213
GL[superscript n] Spaces with the Metric e[superscript 2[delta](x,y)][gamma][subscript ij](x,y)	p. 214
Antonelli's Metrics	p. 217
General Case	p. 220
Relativistic Geometrical Optics	p. 223
Synge Metric in Dispersive Media	p. 224
A Post-Newtonian Estimation	p. 225
A Non-linear Connection	p. 229
Canonical Metrical d-Connection	p. 233
Electromagnetic Tensors	p. 236
Einstein Equations	p. 238
Locally Minkowski GL[superscript n] Spaces	p. 241
Almost Hermitian Model	p. 244
A Finslerian Approach to the Relativistic Optics	p. 246
Geometry of Time Dependent Lagrangians	p. 250
Non-linear Connections in [xi] = (R x TM, [pi], R x M)	p. 250
Time Dependent Lagrangians	p. 255
Non-linear Connection and Semisprays	p. 258
Normal d-Connections on R x TM	p. 260
Metrical Normal d-Connections on R x TM	p. 262
Rheonomic Finsler Spaces	p. 265
Remarkable Time Dependent Lagrangians	p. 268
Metrical Almost Contact Model of a Rheonomic Lagrange Space	p. 271
Generalized Rheonomic Lagrange Spaces	p. 273
Bibliography	p. 276
Index	p. 284
Table of Contents provided by Blackwell. All Rights Reserved.

The Geometry of Lagrange Spaces : Theory and Applications

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