Geometry of Classical Fields book cover

Geometry of Classical Fields

This volume is an introduction to differential methods in physics. Part I contains a comprehensive presentation of the geometry of manifolds and Lie groups, including infinite dimensional settings. The differential geometric notions introduced in Part I are used in Part II to develop selected topics in field theory, from the basic principles up to the present state of the art. This second part is a systematic development of a covariant Hamiltonian formulation of field theory starting from the principle of stationary action.

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Published: November 1988

Imprint: North-holland

ISBN: 978-0-444-70544-0

Contents

  • Part I: Differential Geometric Preliminaries. Manifolds and Lie Groups. Manifolds. Tangent Manifolds. Flows. The Theorem of Frobenius. Lie Groups. Immersed Lie Groups. Examples. Aut G for a Connected G. The Semidirect Product. Vector Bundles. Fibre Bundles. Vector Bundles. Construction of Vector Bundles. The Pull Back. Homotopy. &Lgr;mE. Section Modules of E. Orientation in E. The Jet Bundle. The Canonical 1-Form on JkN. Vertical and Horizontal Bundles. Connections. Riemmanian Structures on Vector Bundles. Elementary Differential Geometry. The Lemma of Poincaré. Induced Riemannian Metrics, Covariant Derivatives and Second Fundamental Tensors on Submanifolds of Euclidean Spaces. Linear Connections, Sprays, Geodesics and the Exponential Map. The Canonical One- and Two-Form on T*M, Riemannian Spray and the Levi-Cività Connection. Curvature Tensors and the Bianchi Identity. Embeddings, the Weingarten Map and the Second Fundamental Form, the Equations of Gauss and Codazzi, the Mean and the Gaussian Curvature. Geodesic Spray of a Right resp. Left Invariant Metric on a Lie Group. Principal Bundles and Connections. Preliminaries. Principal Bundles. Examples. Associated Bundles. Connections. The Special Case G → G/H. Invariant Connections on Principal Bundles. Linear Connections in Vector Bundles. Connection Forms and Linear Connections. Function Space. Space of Functions and Distributions. Globally Defined Function Spaces. Remarks on Calculus. Ck(M,N) as a Manifold. Examples of Manifolds of Maps and Some Tangent Mappings. Gauge Groups. On the Deformation of Differentials of Immersions.

    Part II: Covariant Hamiltonian Dynamics. Non-Relativistic Dynamics. Action Principle. Canonical Hamiltonian Formalism. Symplectic Manifolds and Poisson Algebras. Degenerate Lagrangians and Constraints. Cartan Equations and Symmetries. Generalized Hamiltonian Dynamics. Constraints and Reduction. Dynamics of Classical Fields. Action Principle and Field Equations. Boundary Conditions, Symmetries and Conservation Laws. Cartan Formalism in the Space of Cauchy Data. Hamiltonian Formulation. Constraints and Reduction. Yang-Mills Theory. Field Equations. Gauge Transformations and Conservation Laws. Hamiltonian Formulation. Minkowski Space. Maxwell-Dirac Theory. Yang-Mills Charges. General Relativity. Field Equations. Conservation Laws and Constraints. Hamiltonian Formulation. Asymptotically Flat Space Times. References. Index. Glossary.

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