56 research outputs found
Reduction of polysymplectic manifolds
The aim of this paper is to generalize the classical Marsden-Weinstein
reduction procedure for symplectic manifolds to polysymplectic manifolds in
order to obtain quotient manifolds which in- herit the polysymplectic
structure. This generalization allows us to reduce polysymplectic Hamiltonian
systems with symmetries, such as those appearing in certain kinds of classical
field theories. As an application of this technique, an analogous to the
Kirillov-Kostant-Souriau theorem for polysymplectic manifolds is obtained and
some other mathematical examples are also analyzed. Our procedure corrects some
mistakes and inaccuracies in previous papers [29, 50] on this subject.Comment: Latex file. 33 pages. New examples, comments and references are adde
Higher-order Cartan symmetries in k-symplectic field theory
For k-symplectic Hamiltonian field theories, we study infinitesimal
transformations generated by certain kinds of vector fields which are not
Noether symmetries, but which allow us to obtain conservation laws by means of
a suitable generalization of the Noether theorem.Comment: 11 page
Hamilton-Jacobi theory in multisymplectic classical field theories
The geometric framework for the Hamilton-Jacobi theory developed in previous
works is extended for multisymplectic first-order classical field theories. The
Hamilton-Jacobi problem is stated for the Lagrangian and the Hamiltonian
formalisms of these theories as a particular case of a more general problem,
and the classical Hamilton-Jacobi equation for field theories is recovered from
this geometrical setting. Particular and complete solutions to these problems
are defined and characterized in several equivalent ways in both formalisms,
and the equivalence between them is proved. The use of distributions in jet
bundles that represent the solutions to the field equations is the fundamental
tool in this formulation. Some examples are analyzed and, in particular, the
Hamilton-Jacobi equation for non-autonomous mechanical systems is obtained as a
special case of our results.Comment: 44 p
On a kind of Noether symmetries and conservation laws in k-cosymplectic field theory
This paper is devoted to studying symmetries of certain kinds of k-cosymplectic Hamiltonian systems in first-order classical field theories. Thus, we introduce a particular class of symmetries and study the problem of associating conservation laws to them by means of a suitable generalization of Noether’s theorem.Preprin
Hamilton-Jacobi theory in multisymplectic classical field theories
The geometric framework for the Hamilton-Jacobi theory developed in [14, 17, 39] is extended for multisymplectic first-order classical field theories. The Hamilton-Jacobi problema is stated for the Lagrangian and the Hamiltonian formalisms of these theories as a particular case of a more general problem, and the classical Hamilton-Jacobi equation for field theories is recovered from this geometrical setting. Particular and complete solutions to these problems are defined and characterized in several equivalent ways in both formalisms, and the equivalence between them is proved. The use of distributions in jet bundles that represent
the solutions to the field equations is the fundamental tool in this formulation. Some examples are analyzed and, in particular, the Hamilton-Jacobi equation for non-autonomous mechanical systems is obtained as a special case of our results.Preprin
Reduction of polysymplectic manifolds
The aim of this paper is to generalize the classical Marsden-
Weinstein reduction procedure
for symplectic manifolds to polysymplectic manifolds in or
der to obtain quotient manifolds which in-
herit the polysymplectic structure. This generalization a
llows us to reduce polysymplectic Hamiltonian
systems with symmetries, suuch as those appearing in certai
n kinds of classical field theories. As an
application of this technique, an analogous to the Kirillov
-Kostant-Souriau theorem for polysymplectic
manifolds is obtained and some other mathematical examples
are also analyzed.
Our procedure corrects some mistakes and inaccuracies in pr
evious papers [28, 48] on this subject.Preprin
Hamilton-Jacobi theory in multisymplectic classical field theories
The geometric framework for the Hamilton-Jacobi theory developed in the studies of Carinena et al. [Int. J. Geom. Methods Mod. Phys. 3(7), 1417-1458 (2006)], Carinena et al. [Int. J. Geom. Methods Mod. Phys. 13(2), 1650017 (2015)], and de Léon et al. [Variations, Geometry and Physics (Nova Science Publishers, New York, 2009)] is extended for multisymplectic first-order classical field theories. The Hamilton-Jacobi problem is stated for the Lagrangian and the Hamiltonian formalisms of these theories as a particular case of a more general problem, and the classical Hamilton-Jacobi equation for field theories is recovered from this geometrical setting. Particular and complete solutions to these problems are defined and characterized in several equivalent ways in both formalisms, and the equivalence between them is proved. The use of distributions in jet bundles that represent the solutions to the field equations is the fundamental tool in this formulation. Some examples are analyzed and, in particular, the Hamilton-Jacobi equation for non-autonomous mechanical systems is obtained as a special case of our results.Peer ReviewedPostprint (author's final draft
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