Geometry of Hamiltonian n-vector fields in multisymplectic field theory

Paufler, Cornelius; Römer, Hartmann

doi:10.1016/s0393-0440(02)00031-1

Cited by 39 publications

(48 citation statements)

References 16 publications

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“…Finally, notice that Eqs. (7) cover Euler-Lagrange equations (4) via leg −1 in the following sense. If σ : M −→ J † is a solution of (7), then leg −1 • σ = j l+1 s : M −→ J l+1 for a solution s : M −→ E of (4) (see [11] for a detailed proof).…”

Section: Proposition 3 Diagrammentioning

confidence: 99%

The Hamilton–jacobi Formalism for Higher-Order Field Theories

Vitagliano

2010

Int. J. Geom. Methods Mod. Phys.

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We extend the geometric Hamilton-Jacobi formalism for Hamiltonian mechanics to higher-order field theories with regular Lagrangian density. We also investigate the dependence of the formalism on the Lagrangian density in the class of those yielding the same Euler-Lagrange equations.The Hamilton-Jacobi (HJ) formalism is a cornerstone of the calculus of variations (see for instance [1]) and the theory of Hamiltonian systems. Moreover, it is a first important step through the quantization of a mechanical system (see, for instance [2], see also [3]). HJ formalism can be readily extended to first-order Lagrangian (and Hamiltonian) field theories [1,4]. Moreover, both its original version and its first-order field theoretic extension have an effective geometric formulation in terms of symplectic [5] and multisymplectic [7-9] geometry respectively. Finally, in [6] the authors formulate in geometric terms a generalized HJ problem depending on the sole equations of motion (and not directly on the Lagrangian, nor the Hamiltonian function itself). In particular, such generalized problem can be stated for any SODE on the tangent bundle and any vector field on the cotangent bundle of a configuration manifold, and thus it has a wide range of applicability. The aim of the present paper is to formulate in geometric terms a (generalized) HJ problem for higher-order Lagrangian field theories, in view of its application to both variational calculus and theoretical physics. Recall that higher-order Lagrangian field theory has got a very elegant geometric formulation (see, for instance, [10]). Moreover, the Hamiltonian formulation of Lagrangian mechanics has been recently extended to

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Section: Proposition 3 Diagrammentioning

confidence: 99%

The Hamilton–jacobi Formalism for Higher-Order Field Theories

Vitagliano

2010

Int. J. Geom. Methods Mod. Phys.

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“…(A further local analysis of these multivector fields solution and other additional details can be found in [11] and [41]). …”

Section: Remarkmentioning

confidence: 99%

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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“…As in the case of odes, a Hamiltonian formulation is also possible in this case [13], [23], [42], [68], [79], [85], after defining the corresponding Legendre map. In all of them, the so-called multimomentum phase space is a fiber bundle over Y endowed with a multisymplectic structure, which is canonical in some cases, although in others it is constructed using additional elements (connections or sections in the bundle).…”

Section: Name Equation Lagrangianmentioning

confidence: 99%

On Some Aspects of the Geometry of Differential Equations in Physics

Gràcia¹,

Muñoz-Lecanda²,

Román‐Roy³

2004

Int. J. Geom. Methods Mod. Phys.

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In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular differential equations, and partial differential equations in field theories. The geometric structures underlying these systems are presented and commented. The main results concerning these structures are stated and discussed, as well as their influence on the study of the differential equations with which they are related. Furthermore, research to be developed in these areas is also commented.

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Geometry of Hamiltonian n-vector fields in multisymplectic field theory

Cited by 39 publications

References 16 publications

The Hamilton–jacobi Formalism for Higher-Order Field Theories

The Hamilton–jacobi Formalism for Higher-Order Field Theories

Extended Hamiltonian systems in multisymplectic field theories

On Some Aspects of the Geometry of Differential Equations in Physics

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