Some remarks on Lagrangian and Poisson reduction for field theories

Abstract

Given a Hamiltonian system on a fiber bundle, the Poisson covariant formulation of the Hamilton equations is described. When the fiber bundle is a G-principal bundle and the Hamiltonian density is G-invariant, the reduction of this formulation is studied thus obtaining the analog of the Lie-Poisson reduction for field theories. The relation of this reduction with the Lagrangian reduction and the Lagrangian and Poisson reduction for electromagnetism are also analyzed.

On a kind of Noether symmetries and conservation laws in k-cosymplectic field theory

“…These problems have been treated for k-symplectic field theories in [23,28], generalizing the results obtained for non-autonomous mechanical systems (see, in particular, [17], and references quoted therein). We further remark that the problem of symmetries in field theory has also been analyzed using other geometric frameworks, such as the multisymplectic models (see, for instance, [5,7,9,10,13,14,18]).…”

Extended Hamiltonian systems in multisymplectic field theories

“…In particular, if we take i(Xh)(τ * ω) = 1 we are choosing a representative of the class ofτ -transverse multivector fields solution to (7). (This is equivalent to putting f = 1 in the local expression (6)).…”

“…Furthermore, there are equivalent Lagrangian models with non-equivalent Hamiltonian descriptions [26], [27], [28]. Among the different geometrical descriptions to be considered for describing field theories, we focus our attention on the multisymplectic models [7], [20], [24], [25], [38]; where the geometric background is in the realm of multisymplectic manifolds, which are manifolds endowed with a closed and 1-nondegenerate k-form, with k ≥ 2. In these models, this form plays a similar role to the symplectic form in mechanics.…”

Cartan forms for first order constrained variational problems

“…The interest of the subject has recently increased even more due to the new approach to the problem of Lagrangian reduction, according to which a certain kind of variational problems, called reducible, can be reduced to constrained variational problems of a lower order, which serves as motivation to take these last problems together with the associated structures as objects of a possible variational category which includes the Lagrangian reduction procedure as one of its fundamental operations (see, for example [6][7][8]10,11,31,35]). …”