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New Lagrangian and Hamiltonian methods in field theory. (English) Zbl 0913.58001
Singapore: World Scientific. viii, 456 p. (1997).
The present book is intended to develop a formulation of classical field theory based on the notion of jets.
First the authors recall the main notions they use for the geometric formulation of classical field theory, such as fibred manifolds, jet manifolds, and connections (Chapters 1 and 2). The notion of fibred manifold includes, for instance, fibre bundles, vector bundles, principal, and associated bundles. The jet formalism introduced in the second chapter is of first and second order, which is sufficient for physical applications.
The purpose of Chapter 3 is to present the first order Lagrangian formalism. The authors derive the first variational formula for a general Lagrangian density of first order, describe the Euler-Lagrange operators associated with a Lagrangian density, and lead to Lagrangian polysymplectic structures. Finally, Lagrangian conservation laws and conservation laws in gauge theory as well as in gravitational theory are discussed.
Chapter 4 gives a detailed composition of the polysymplectic Hamiltonian formalism defined on the Legendre bundle given by a Lagrangian density. The notion of Legendre bundle is a generalization of the usual phase space in a purely symplectic context.
Chapter 5 is devoted to special topics including higher order and infinite order jets, applications of the variational techniques, formal integrability concepts for differential equations, and their applications to the Yang-Mills and Einstein-Yang-Mills equations of field theory.
The book contains all necessary mathematical material, which makes it very convenient for the reader to follow the ideas.

70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
58A20 Jets in global analysis
58E30 Variational principles in infinite-dimensional spaces
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics