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The Geometry of Lagrange Spaces: Theory and Applications

Author: Radu Miron; Mihai Anastasiei
Publisher: Dordrecht : Springer Netherlands : Imprint : Springer, 1994.
Series: Fundamental Theories of Physics, An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application ;, 59.
Edition/Format:   eBook : Bibliographic data : EnglishView all editions and formats
Summary:
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.  Read more...
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Details

Genre/Form: Electronic books
Additional Physical Format: Printed edition:
Material Type: Bibliographic data, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Radu Miron; Mihai Anastasiei
ISBN: 9789401107884 9401107882 9789401043380 9401043388
OCLC Number: 840308655
Description: 1 online resource (304 pages).
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. Normal d-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.
Series Title: Fundamental Theories of Physics, An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application ;, 59.
Responsibility: by Radu Miron, Mihai Anastasiei.

Abstract:

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,  Read more...

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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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