On the k-Symplectic, k-Cosymplectic and Multisymplectic Formalisms of Classical Field Theories

Abstract

The objective of this work is twofold: First, we analyze the relation between the k-cosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms in classical field theories. In particular, we prove the equivalence between k-symplectic field theories and the so-called autonomous k-cosymplectic field theories, extending in this way the description of the symplectic formalism of autonomous systems as a particular case of the cosymplectic formalism in non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric character of the solutions to the Hamilton-de Donder-Weyl and the Euler-Lagrange equations in these formalisms. Second, we study the equivalence between k-cosymplectic and a particular kind of multisymplectic Hamiltonian and Lagrangian field theories (those where the configuration bundle of the theory is trivial).

25 pages

References in corpus (2)

• Extended Hamiltonian systems in multisymplectic field theories
• Gunther's formalism (k-symplectic formalism) in classical field theory: Skinner-Rusk approach and the evolution operator

Cited by in corpus (20)

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• The Tulczyjew triple for classical fields
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• k-cosymplectic classical field theories: Tulczyjew, Skinner--Rusk and Lie-algebroid formulations
• Nonautonomous k-contact field theories
• Remarks on multisymplectic reduction
• The Herglotz variational principle for dissipative field theories
• Symmetries, Newtonoids vector fields and conservation laws in the Lagrangian $k$-symplectic formalism
• Quantization of Polysymplectic Manifolds
• Hamilton-Jacobi theory for gauge field theories
• Symmetries in Lagrangian Field Theory
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• Conditions for symmetry reduction of polysymplectic and polycosymplectic structures
• Higher-order Cartan symmetries in k-symplectic field theory
• Regularized polysymplectic geometry and first steps towards Floer theory for covariant field theories
• Cotangent bundle reduction and Routh reduction for polysymplectic manifolds