The Geometry of Lagrange Spaces: Theory and Applications: (59 Fundamental Theories of Physics)

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.

Table of Contents:
I. Fibre Bundles. General Theory.- 1. Fibre Bundles.- 2. Principal Fibre Bundles.- 3. Vector Bundles.- 4. Morphisms of Vector Bundles.- 5. Vector Subbundles.- 6. Operations with Vector Bundles.- 7. Principal Bundle Associated with a Vector Bundle.- 8. Sections in Vector Bundles.- II. Connections in Fibre Bundles.- 1. Non-linear Connections in Vector Bundles.- 2. Local Representations of a Non-linear Connection.- 3. Other Characterisations of a Non-linear Connection.- 4. Vertical and Horizontal Lifts.- 5. Curvature of a Non-linear Connection.- 6. Affine Morphisms of Vector Bundles.- III. Geometry of the Total Space of a Vector Bundle.- 1. d-Connections on the Total Space of a Vector Bundle.- 2. Local Representation of d-Connections.- 3. Torsion and Curvature of d-Connections.- 4. Structure Equations of a d-Connection.- 5. Metric Structures on the Total Space of a Vector Bundle.- IV. Geometrical Theory of Embeddings of Vector Bundles.- 1. Embeddings of Vector Bundles.- 2. Moving Frame on E? in E.- 3. Induced Non-linear Connections. Relative Covariant Derivative.- 4. The Gauss-Weingarten Formulae.- 5. The Gauss-Codazzi Equations.- V. Einstein Equations.- 1. Einstein Equations.- 2. Einstein Equations in the Case m = 1.- 3. Another Form of the Einstein Equations.- 4. Einstein Equations for some particular metrics on E.- VI. Generalized Einstein-Yang Mills Equations.- 1. Gauge Transformations.- 2. Gauge Covariant Derivatives.- 3. Metrical Gauge d-Connections.- 4. Generalized Einstein-Yang Mills Equations.- VII. Geometry of the Total Space of a Tangent Bundle.- 1. Non-linear Connections in Tangent Bundle.- 2. Semisprays, Sprays and Non-linear Connections.- 3. Torsions and Curvature of a Non-linear Connections.- 4. Transformations of Non-linear Connections.- 5. Normald-Connections on TM.- 6. Metrical Structures on TM.- 7. Some Remarkable Metrics on TM.- VIII. Finsler Spaces.- 1. The Notion of Finsler Space.- 2. Non-linear Cartan Connection.- 3. Geodesics.- 4. Metrical Cartan Connection.- 5. Structure Equations. Bianchi Identities.- 6. Remarkable Finslerian Connections.- 7. Almost Kählerian Model of a Finsler Space.- 8. Subspaces in a Finsler Space.- IX. Lagrange Spaces.- 1. The Notion of Lagrange Space.- 2. Euler-Lagrange Equations. Canonical Non-linear Connection.- 3. Canonical Metrical d-Connection.- 4. Gravitational and Electromagnetic Fields.- 5. Lagrange Space of Electrodynamics.- 6. Almost Finslerian Lagrange Spaces.- 7. Almost Kählerian Model of a Lagrange Space.- X. Generalized Lagrange Space.- 1. Notion of Generalized Lagrange Space.- 2. Metrical d-Connections in a GLn Space.- 3. Structure Equations. Parallelism.- 4. On h-Covariant Constant d-Tensor Fields.- 5. Gravitational Field.- 6. Electromagnetic Field.- 7. Almost Hermitian Model of a GLn Space.- XI. Applications of the GLn Spaces with the Metric Tensor e2?(x,y)?ji(x,y).- 1. EPS conditions and the Metric e2?(x,y)?ij(x).- 2. Canonical Metrical d-Connection.- 3. Electromagnetic and Gravitational Fields.- 4. Two Particular Cases.- 5. GLn Spaces with the Metric e2?(x,y)?ij(y).- 6. Antonelli’s Metrics.- 7. General Case.- XII. Relativistic Geometrical Optics.- 1. Synge Metric in Dispersive Media.- 2. A Post-Newtonian Estimation.- 3. A Non-linear Connection.- 4. Canonical Metrical d-Connection.- 5. Electromagnetic Tensors.- 6. Einstein Equations.- 7. Locally Minkowski GLn Spaces.- 8. Almost Hermitian Model.- 9. A Finslerian Approach to the Relativistic Optics.- XIII. Geometry of Time Dependent Lagrangians.- 1. Non-linear Connections in ? = (R x TM,?,R x M).- 2.Time Dependent Lagrangians.- 3. Non-linear Connections and Semisprays.- 4. Normal d-Connections on R x TM.- 5. Metrical Normal d-Connections on RxTM.- 6. Rheonomic Finsler Spaces.- 7. Remarkable Time Dependent Lagrangians.- 8. Metrical Almost Contact Model of a Rheonomic Lagrange Space.- 9. Generalized Rheonomic Lagrange Spaces.