The Geometry of Lagrange Spaces: Theory and ApplicationsDifferential-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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Page 197
R. Miron, Mihai Anastasiei. §6 . Electromagnetic Fields In a Lagrange space the electromagnetic field was introduced by extending an idea suggested by a Lagrangian from electrodynamics ( see ( 5.1 ) in Ch . IX ) . This idea may also be ...
R. Miron, Mihai Anastasiei. §6 . Electromagnetic Fields In a Lagrange space the electromagnetic field was introduced by extending an idea suggested by a Lagrangian from electrodynamics ( see ( 5.1 ) in Ch . IX ) . This idea may also be ...
Page 236
... Electromagnetic Tensors For the nondispersive media there exists an electromagnetic tensor field given by the skew - symmetric part of the h - covariant deflection tensor . In general , for a dispersive medium we may consider two ...
... Electromagnetic Tensors For the nondispersive media there exists an electromagnetic tensor field given by the skew - symmetric part of the h - covariant deflection tensor . In general , for a dispersive medium we may consider two ...
Page 237
... electromagnetic tensorsF , and f of a space GL " satisfy the Maxwell equations ( 6.2 ) , Ch.X. In our case , it is easy to prove Theorem 5.2 . The electromagnetic tensor fields F , and f1 of the space GL " with the metric tensor ( 1.4 ) ...
... electromagnetic tensorsF , and f of a space GL " satisfy the Maxwell equations ( 6.2 ) , Ch.X. In our case , it is easy to prove Theorem 5.2 . The electromagnetic tensor fields F , and f1 of the space GL " with the metric tensor ( 1.4 ) ...
Contents
Morphisms of Vector Bundles | 11 |
Connections in Fibre Bundles | 19 |
Vertical and Horizontal Lifts | 27 |
Copyright | |
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The Geometry of Lagrange Spaces: Theory and Applications R. Miron,Mihai Anastasiei Limited preview - 2012 |
The Geometry of Lagrange Spaces: Theory and Applications R. Miron,Mihai Anastasiei No preview available - 2012 |
The Geometry of Lagrange Spaces: Theory and Applications R. Miron,Mihai Anastasiei No preview available - 2014 |
Common terms and phrases
1-forms absolute energy AFL-space associated autoparallel curves Bianchi identities canonical metrical d-connection canonical non-linear connection Cartan connection Ch.III coefficients covariant derivatives d-tensor field d-vector defined Definition deflection tensors denote dependent Lagrangian direct calculation dt dt Einstein equations electromagnetic tensor endowed exists an unique əxi əyi F₁ field of type field on TM Finsler space fundamental function gauge d-connection gauge transformation geometrical given h(hh Hermitian homogeneous horizontal distribution Lagrange space Lagrangian linear connection Liouville vector field local coefficients mapping Maxwell equations metric structure metric tensor metrical d-connection CT(N morphism non-linear connection normal d-connection obtains principal bundle Proof properties Proposition regular Lagrangian rheonomic Lagrange space Ricci identities Riemannian satisfy semispray space F subbundle tangent bundle Theorem 2.1 theory torsions total space vanishes vector field vertical distribution x(TM ду дус дх