A Multisymplectic Framework for Classical Field Theory and the Calculus of Variations

Gotay, Mark J.

doi:10.1016/b978-0-444-88958-4.50012-4

Cited by 108 publications

(133 citation statements)

References 29 publications

Supporting

Mentioning

132

Contrasting

Order By: Relevance

“…The analogue of the Poisson bracket in the DW formulation is deduced from the object, called the polysymplectic form, which in local coordinates can be written in the form 1 Ω = −dy a ∧ dp µ a ∧ ω µ and is viewed as a field theoretic generalization of the symplectic form within the DW formulation. Note that if Σ, Σ : (y a = y a (x), t = t), denotes the Cauchy data surface in the covariant configuration space (y a , x µ ) the standard symplectic form in field theory, ω S , can be expressed as the integral over Σ of the restriction of Ω to Σ, Ω| Σ [16,17], i.e.…”

Section: Classical Theorymentioning

confidence: 99%

“…Besides, there are intimate relations, not fully studied as yet, between the structures of the canonical Hamiltonian formalism and the structures of the Lepagean formulations [16][17][18] which point to the latter as a natural intermediate step when formulating the field theories canonically proceeding from the Lagrangian level. For this reason, henceforth we refer to the finite dimensional covariant Hamiltonian-like formulations based on different Lepagean theories as "precanonical ".…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, the already understood character of connections between the De Donder-Weyl theory and the standard Hamiltonian formalism [16,22] seems to provide us with a clue to a possible approach to this problem. In fact, the standard symplectic form and the standard equaltime canonical brackets in field theory can be obtained by integrating the polysymplectic form Ω and the canonical brackets (2.3) over the Cauchy data surface Σ : (y a = y a (x), t = const) in the covariant configuration space (y a , x µ ) [27].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Untitled

Kanatchikov

2001

International Journal of Theoretical Physics

View full text Add to dashboard Cite

A nonpertubative approach to quantum gravity using precanonical field quantization originating from the covariant De Donder-Weyl Hamiltonian formulation which treats space and time variables on equal footing is presented. A generally covariant "multi-temporal" generalized Schrödinger equation on the finite dimensional space of metric and space-time variables is obtained. An important ingredient of the formulation is the "bootstrap condition" which introduces a classical space-time geometry as an approximate concept emerging as the quantum average in a self-consistent with the underlying quantum dynamics manner. An independence of the theory from an arbitrarily fixed background is ensured in this way. The prospects and unsolved problems of precanonical quantization of gravity are outlined. *

show abstract

Section: Classical Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Untitled

Kanatchikov

2001

International Journal of Theoretical Physics

View full text Add to dashboard Cite

show abstract

“…This subject has a long and distinguished history that we shall not review in this article; rather, we follow the framework established in Gotay [1991], Gotay et al [1992], and Gotay and Marsden [1992], wherein relativistic field theories with Dirac-Bergmann type constraints are considered in a Lagrangian formalism, while the Hamiltonian formalism relies on a "space + time" (or 3+1) split. These references contain citations to much of the important literature and history of the subject.…”

Section: Introductionmentioning

confidence: 99%

Multisymplectic geometry, covariant Hamiltonians, and water waves

Marsden

Shkoller

1999

Math. Proc. Camb. Phil. Soc.

115

102

View full text Add to dashboard Cite

Abstract. This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. This theory generalizes and unifies the classical Hamiltonian formalism of particle mechanics as well as the many pre-symplectic 2-forms used by Bridges [1997]. In this theory, solutions of a PDE are sections of a fiber bundle Y over a base manifold X of dimension n+1, typically taken to be spacetime. Given a connection on Y , a covariant Hamiltonian density H is then intrinsically defined on the primary constraint manifold P L , the image of the multisymplectic version of the Legendre transformation. One views P L as a subbundle of J 1 (Y ) ⋆ , the affine dual of J 1 (Y ), the first jet bundle of Y . A canonical multisymplectic (n+2)-form Ω H is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original PDE as well as the Euler-Lagrange equations of the corresponding Lagrangian. Furthermore, we show that the n+1 2-forms ω (µ) defined by Bridges [1997] are a particular coordinate representation for a single multisymplectic (n+2)-form, and in the presence of symmetries, can be assembled into Ω H . A generalized Hamiltonian Noether theory is then constructed which relates the action of the symmetry groups lifted to P L with the conservation laws of the system. These conservation laws are defined by our generalized Noether's theorem which recovers the vanishing of the divergence of the vector of n+1 distinct momentum mappings defined in Bridges [1997] and, when applied to water waves, recovers Whitham's conservation of wave action. In our view, the multisymplectic structure provides the natural setting for studying dispersive wave propagation problems, particularly the instability of water waves, as discovered by Bridges. After developing the theory, we show its utility in the study of periodic pattern formation and wave instability.

show abstract

“…First, several Hamiltonian models can be stated, and the equivalence among them is not always clear (see, for instance, [2], [12], [19], [21], [22], [23], [40], [43]). Furthermore, there are equivalent Lagrangian models with non-equivalent Hamiltonian descriptions [26], [27], [28].…”

Section: Introductionmentioning

confidence: 99%

Extended Hamiltonian systems in multisymplectic field theories

Echeverría-Enríquez¹,

León

Muñoz-Lecanda³

et al. 2007

Journal of Mathematical Physics

View full text Add to dashboard Cite

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.

show abstract

A Multisymplectic Framework for Classical Field Theory and the Calculus of Variations

Cited by 108 publications

References 29 publications

Untitled

Untitled

Multisymplectic geometry, covariant Hamiltonians, and water waves

Extended Hamiltonian systems in multisymplectic field theories

Contact Info

Product

Resources

About