Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory

Abstract

This is the first paper of a five part work in which we study the Lagrangian and Hamiltonian structure of classical field theories with constraints. Our goal is to explore some of the connections between initial value constraints and gauge transformations in such theories (either relativistic or not). To do this, in the course of these four papers, we develop and use a number of tools from symplectic and multisymplectic geometry. Of central importance in our analysis is the notion of the ``energy-momentum map'' associated to the gauge group of a given classical field theory. We hope to demonstrate that many different and apparently unrelated facets of field theories can be thereby tied together and understood in an essentially new way. In Part I we develop some of the basic theory of classical fields from a spacetime covariant viewpoint. We begin with a study of the covariant Lagrangian and Hamiltonian formalisms, on jet bundles and multisymplectic manifolds, respectively. Then we discuss symmetries, conservation laws, and Noether's theorem in terms of ``covariant momentum maps.''

LaTeX2e, 68 pages, 1 figure, GIMMsy 1; Updated, with minor revisions and corrections

Cited by in corpus (33)

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• Covariant Poisson Brackets in Geometric Field Theory
• Geometric formulation of the Covariant Phase Space methods with boundaries
• Symmetries in Classical Field Theory
• Multi-moment maps
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• On the Multimomentum Bundles and the Legendre Maps in Field Theories
• The Hamilton-Pontryagin Principle and Multi-Dirac Structures for Classical Field Theories
• Covariant canonical formulations of classical field theories
• On higher Dirac structures
• On a kind of Noether symmetries and conservation laws in k-cosymplectic field theory
• Proof of the equivalence of the symplectic forms derived from the canonical and the covariant phase space formalisms
• Generating Functionals and Lagrangian PDEs
• Curved space-times by crystallization of liquid fiber bundles
• R-adaptive multisymplectic and variational integrators
• Tulczyjew triples: from statics to field theory
• Momentum sections in Hamiltonian mechanics and sigma models
• Covariant Momentum Map Thermodynamics for Parametrized Field Theories
• The homotopy momentum map of general relativity
• Linking Covariant and Canonical General Relativity via Local Observers
• Hamilton-Jacobi theory for gauge field theories
• Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings
• A review on geometric formulations for classical field theory: the Bonzom-Livine model for gravity
• Covariant constructive gravity: A step-by-step guide towards alternative theories of gravity
• A global version of Gunther's polysymplectic formalism using vertical projections
• Principal Symbol of Euler-Lagrange Operators
• Graded Poisson and Graded Dirac structures
• Clifford Algebraic approach to the De Donder-Weyl Hamiltonian theory
• Geometry of Classical Nambu-Goldstone Fields
• Multisymplectic Formalism for Cubic Horndeski Theories
• Reduction of $L_{\infty }$-Algebras of Observables on Multisymplectic Manifolds
• Routh Reduction of Palatini Gravity in Vacuum
• Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds