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Multisymplectic Lagrangian and Hamiltonian formalisms of First-order Classical Field theories
Román Roy, Narciso
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada IV; Universitat Politècnica de Catalunya. Geometria Diferencial, Sistemes Dinàmics i Aplicacions
This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically first-order classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been recently developed), following the original ideas of Rusk and Skinner for mechanical systems.
Differential geometry
First-order field theories
Lagrangian and Hamiltonian formalisms
Fiber bundles
Multisymplectic manifolds
Geometria diferencial
Classificació AMS::70 Mechanics of particles and systems {For relativistic mechanics, see 83A05 and 83C10; for statistical mechanics, see 82-xx}::70S Classical field theories [See also 37Kxx, 37Lxx, 78-xx, 81Txx, 83-xx]
Classificació AMS::55 Algebraic topology::55R Fibspaces and bundles [See also 18F15, 32Lxx, 46M20, 57R20, 57R22, 57R25]er
Classificació AMS::53 Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx}::53C Global differential geometry [See also 51H25, 58-xx; for related bundle theory, see 55Rxx, 57Rxx]
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