Abstract

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

Highlights

  • The objective of this work is twofold: First, we analyze the relation between the kcosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms in classical field theories

  • In the k-cosymplectic formalism, every solution to the Hamilton-de Donder-Weyl equations is, an integral section of an integrable k-vector field that is a solution to the geometrical field equations in the Hamiltonian formalism

  • The relation between the k-cosymplectic and the k-symplectic Hamiltonian formalism is studied here, proving the equivalence between ksymplectic Hamiltonian systems and a class of k-cosymplectic Hamiltonian systems: the so-called autonomous k-cosymplectic Hamiltonian systems. This generalizes the situation in classical mechanics, where the symplectic formalism for describing autonomous Hamiltonian systems can be recovered as a particular case of the cosymplectic Hamiltonian formalism when systems described by time-independent Hamiltonian functions are considered

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Summary

Autonomous k-cosymplectic Hamiltonian systems

Following a terminology analogous to that in mechanics, we define: Definition 6 A k-cosymplectic Hamiltonian system (Rk × (Tk1)∗Q, H) is said to be autonomous if. (Proof ): If X ∈ XkH(Rk × (Tk1)∗Q) is an integrable k-vector field, denote by Sthe set of its integral sections (i.e., solutions to the the HDW-equations (12)). By proposition 2 we can construct an integrable k-vector field X ∈ XkH ((Tk1)∗Q) for which S is its set of integral sections (which are admissible solutions to the HDW-equations (7)). If the Lagrangian is regular, the above equations are equivalent to (ΓA)i = vAi. The last group of these equations is the local expression of the condition that Γ is a sopde (see [25]), and if it is integrable, its integral sections are first prolongations φ(1) : Rk → Tk1Q of maps φ : Rk → Q, and using the first group of equations, we deduce that φ(1) are solutions to the Euler-Lagrange equations (25). The k-symplectic and k-cosymplectic Lagrangian and Hamiltonian systems are related by means of the Legendre maps F L and F L

Multisymplectic Hamiltonian formalism
Relation with the k-cosymplectic Hamiltonian formalism
It is a straighforward consequence of the above item because
Multisymplectic Lagrangian formalism
Relation between multisymplectic and k-cosymplectic Lagrangian systems
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