A geometrical approach to Classical Field Theories: a constraint algorithm for singular theories

León, Manuel de; Marín-Solano, Jesús; Marrero, Juan Carlos

doi:10.1007/978-94-009-0149-0_22

Cited by 48 publications

(103 citation statements)

References 31 publications

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“…Then, the associated connection ∇ h P , which is a connection along the submanifold P (see [30], [31] and [34]), is called a Hamilton-De Donder-Weyl connection for (J 1 π * , P, h P ), and satisfies the equation…”

Section: Restricted Almost-regular Hamiltonian Systemsmentioning

confidence: 99%

“…The most frequently used choice is to take the so-called restricted multimomentum bundle, denoted by J 1 π * ; that is analogous to T * Q×R in the mechanical case. The Hamiltonian formalism in J 1 π * has been extensively studied [6], [11], [31], [37]. Nevertheless, this bundle does not have a canonical multisymplectic form and the physical information, given by a Hamiltonian section, is used to obtain the geometric structure.…”

Section: Introductionmentioning

confidence: 99%

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Extended Hamiltonian systems in multisymplectic field theories

Echeverría-Enríquez¹,

León

Muñoz-Lecanda³

et al. 2007

Journal of Mathematical Physics

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We consider Hamiltonian systems in first-order multisymplectic field theories. We review the properties of Hamiltonian systems in the so-called restricted multimomentum bundle, including the variational principle which leads to the Hamiltonian field equations. In an analogous way to how these systems are defined in the so-called extended (symplectic) formulation of non-autonomous mechanics, we introduce Hamiltonian systems in the extended multimomentum bundle. The geometric properties of these systems are studied, the Hamiltonian equations are analyzed using integrable multivector fields, the corresponding variational principle is also stated, and the relation between the extended and the restricted Hamiltonian systems is established. All these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories.

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Section: Restricted Almost-regular Hamiltonian Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Extended Hamiltonian systems in multisymplectic field theories

Echeverría-Enríquez¹,

León

Muñoz-Lecanda³

et al. 2007

Journal of Mathematical Physics

Self Cite

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“…More precisely, if n ≥ 2, we have that Leg L is a local multisymplectomorphism between the multisymplectic manifolds (Z, Ω L ) and (Z * , Ω h ). For singular lagrangians, a constraint algorithm was developed in [23] (see Section 6).…”

Section: And It Is Locally Expressed Asmentioning

confidence: 99%

“…This problem is solved by constructing a submanifold of Z f where such a solution exists (see [23,24] and below for more details).…”

Section: A New Geometric Setting Consider the Fibered Productmentioning

confidence: 99%

A new geometric setting for classical field theories

León

Marrero

Diego

2003

Classical and Quantum Integrability

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Abstract.A new geometrical setting for classical field theories is introduced. This description is strongly inspired by the one due to Skinner and Rusk for singular lagrangian systems. For a singular field theory a constraint algorithm is developed that gives a final constraint submanifold where a well-defined dynamics exists. The main advantage of this algorithm is that the second order condition is automatically included.

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“…Here we present one of the most standard ways of defining Hamiltonian systems, which is based on using Hamiltonian sections [16]; although this construction can also be done taking Hamiltonian densities [16,39,79,91]. In particular, the construction of Hamiltonian systems which are the Hamiltonian counterpart of Lagrangian systems is carried out by using the Legendre map associated with the Lagrangian system, and this problem has been studied by different authors in the (hyper) regular case [16,92], and in the singular (almost-regular) case [39,69,91]. In Section 3 we review some of these constructions.…”

Section: Introductionmentioning

confidence: 99%

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

Román‐Roy¹

2009

SIGMA

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Abstract. This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been developed in the last years), following the original ideas of Rusk and Skinner for mechanical systems.

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A geometrical approach to Classical Field Theories: a constraint algorithm for singular theories

Cited by 48 publications

References 31 publications

Extended Hamiltonian systems in multisymplectic field theories

Extended Hamiltonian systems in multisymplectic field theories

A new geometric setting for classical field theories

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

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