On a kind of Noether symmetries and conservation laws in <i>k</i>-cosymplectic field theory

Marrero, Juan Carlos; Román‐Roy, Narciso; Salgado, Modesto; Vilariño, Silvia

doi:10.1063/1.3545969

Cited by 9 publications

(11 citation statements)

References 27 publications

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“…This result extends the work developed by Marmo and Mukunda in [36]. The study of symmetries in field theory, using various geometric frameworks, has been done in [5,11,19,30,37].…”

Section: Introductionsupporting

confidence: 75%

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SYMMETRIES, NEWTONOID VECTOR FIELDS AND CONSERVATION LAWS IN THE LAGRANGIAN k-SYMPLECTIC FORMALISM

2012

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In this paper we study symmetries, Newtonoid vector fields, conservation laws, Noether's Theorem and its converse, in the framework of the k-symplectic formalism, using the Frölicher-Nijenhuis formalism on the space of k 1 -velocities of the configuration manifold.For the case k = 1, it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether's Theorem and its converse. For the case k > 1, we provide a new proof for Noether's Theorem, which shows that, in the ksymplectic formalism, each Cartan symmetry induces a conservation law. We prove that, under some assumptions, the converse of Noether's Theorem is also true and we provide examples when this is not the case. We also study the relations between dynamical symmetries, Newtonoid vector fields, Cartan symmetries and conservation laws, showing when one of them will imply the others. We use several examples of partial differential equations to illustrate when these concepts are related and when they are not.

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“…This result extends the work developed by Marmo and Mukunda in [36]. The study of symmetries in field theory, using various geometric frameworks, has been done in [5,11,19,30,37].…”

Section: Introductionsupporting

confidence: 75%

“…One of the advantages of using these formalisms is that only the tangent and cotangent bundles of the configuration manifold are required to develop them. Others papers related with the k-symplectic and k-cosymplectic formalism are [20,27,28,37,40,46,47].…”

Section: Introductionmentioning

confidence: 99%

SYMMETRIES, NEWTONOID VECTOR FIELDS AND CONSERVATION LAWS IN THE LAGRANGIAN k-SYMPLECTIC FORMALISM

2012

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“…This machinery is later used to discuss symmetries in this context, extending some previous results (see [1,26]). …”

Section: Introductionmentioning

confidence: 88%

Symmetries in Lagrangian Field Theory

Búa¹,

Bucataru

León

et al. 2015

Reports on Mathematical Physics

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By generalising the cosymplectic setting for time-dependent Lagrangian mechanics, we propose a geometric framework for the Lagrangian formulation of classical field theories with a Lagrangian depending on the independent variables. For that purpose we consider the first order jet bundles J 1 π of a fiber bundle π : E → R k where R k is the space of independent variables. Generalized symmetries of the Lagrangian are introduced and the corresponding Noether Theorem is proved.Keywords: Symmetries, Cartan theorem, Noether Theorem, conservation laws, jet bundles. IntroductionAs it is well-know, the natural arena for studying mechanics is symplectic geometry. One interesting problem is to extend this geometric framework for the case of classical field theories. Several different geometric approaches are well known: the polysymplectic formalisms developed by Sardanashvily et al. [11,12,35], and by Kanatchikov [17], as 2 well as the n-symplectic formalism of Norris [27,29], and the k-cosymplectic of de León et al. [22,23].Let us remark that the multisymplectic formalism is the most ambitious program for developing the Classical Field Theory (see for example [2,9,10,13,14,18], and references quoted therein).The aims of this paper are• To give a new Lagrangian description of first order Classical Field Theory, by considering a fibration E → R k , which has as particular cases the cosymplectic setting for time-dependent Lagrangian mechanics, and it is related with the kcosymplectic [23] and multisymplectic formalisms. Let us observe that although every fiber bundle over R k is a trivial bundle since R k is contractible (Steenrod 1951 [36]), we do not use this fact to develop our Lagrangian description.• To introduce and to study the generalized symmetries in first order Lagrangian Field Theory. For time-dependent Lagrangian Mechanics this was done by J.F. Cariñena et al. [5].In the present paper we present a new approach for Lagrangian Field Theory, working with the first order jet bundle J 1 π of a fiber bundle π :The crucial point is that each 1-form on R k defines a tensor field of type (1, 1) on J 1 π, see Saunders [38]. The paper is organized as follows. The main tools to be used are those of vector fields, k-vector fields and forms along maps, the general definitions of which are given in Section 2..In Section 3. we introduce the geometric elements on J 1 π necessary to develop the geometric formulation of the Euler-Lagrange field equations in Section 4. and to the study of symmetries and conservations laws. The principal tools, here described, are the canonical vector fields, the k vertical endomorphisms and a kind of k-vector fields, known as sopdes, which describe systems of second order partial differential equations.This machinery is later used to discuss symmetries in this context, extending some previous results (see [1,26]).The geometric formulation of the Euler-Lagrange field equations is given in Section 4., see Theorem 1. For this purpose we introduce the k Poincaré-Cartan 1-forms using the Lagrangian and...

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“…The study of symmetries and conservation laws for k-symplectic Hamiltonian systems is, like in the classical case, a topic of great interest and was developped recently by M. Salgado, N. Roman-Roy, S. Vilarino in [27], [28] and L. Bua, I. Bucȃtaru, M. Salgado in [5]. Further more, in the paper [16] J.C. Marrero, N. Roman-Roy, M. Salgado, S. Vilarino begin the study of symmetries and conservation laws for k-cosymplectic Hamiltonian systems, like an extension to field theories of the standard cosymplectic formalism for nonautonomous mechanics ( [13], [14]). In [27] the Noether's theorem, obtained for a k-symplectic Hamiltonian system, associates conservation laws to so-called Cartan symmetries.…”

Section: Introductionmentioning

confidence: 99%

Symmetries, pseudosymmetries and conservation laws in Lagrangian and Hamiltonian k-symplectic formalisms

Munteanu

2016

Int. J. Geom. Methods Mod. Phys.

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In this paper we will present Lagrangian and Hamiltonian k-symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will obtain new kinds of conservation laws for k-symplectic Hamiltonian systems and k-symplectic Lagrangian systems.AMS Subject Classification (2010): 70S05, 70S10, 53D05.

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

Cited by 9 publications

References 27 publications

SYMMETRIES, NEWTONOID VECTOR FIELDS AND CONSERVATION LAWS IN THE LAGRANGIAN k-SYMPLECTIC FORMALISM

SYMMETRIES, NEWTONOID VECTOR FIELDS AND CONSERVATION LAWS IN THE LAGRANGIAN k-SYMPLECTIC FORMALISM

Symmetries in Lagrangian Field Theory

Symmetries, pseudosymmetries and conservation laws in Lagrangian and Hamiltonian k-symplectic formalisms

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