Multisymplectic Lagrangian and Hamiltonian formalisms of First-order Classical Field theories (original) (raw)

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

2009

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.

A Multisymplectic Unified Formalism for Second Order Classical Field Theories

2014

We present a new multisymplectic framework for second-order classical field theories which is based on an extension of the unified Lagrangian-Hamiltonian formalism to these kinds of systems. Our model allows us to overcome all the ambiguities inherent to these theories. It therefore provides a straightforward and simple way to define the Poincaré-Cartan form and clarifies the construction of the Legendre map (univocally obtained as a consequence of the constraint algorithm). Likewise, it removes the undesirable arbitrariness in the solutions to the field equations, which are analyzed in-depth, and written in terms of holonomic sections and multivector fields. Our treatment therefore completes previous attempt to achieve this aim. The formulation is applied to describing some physical examples; in particular, to giving another alternative multisymplectic description of the Korteweg-de Vries equation.

A new multisymplectic unified formalism for second order classical field theories

Journal of Geometric Mechanics, 2015

Lagrangian-Hamiltonian unified formalism for field theory

2004

The Lagrangian-Hamiltonian unified formalism of R. Skinner and R. Rusk was originally stated for autonomous dynamical systems in classical mechanics. It has been generalized for non-autonomous first-order mechanical systems, as well as for first-order and higher-order field theories. However, a complete generalization to higher-order mechanical systems has yet to be described. In this work, after reviewing the natural geometrical setting and the Lagrangian and Hamiltonian formalisms for higher-order autonomous mechanical systems, we develop a complete generalization of the Lagrangian-Hamiltonian unified formalism for these kinds of systems, and we use it to analyze some physical models from this new point of view.

Geometry of multisymplectic Hamiltonian first-order field theories

Journal of Mathematical Physics, 2000

In the jet bundle description of field theories ͑multisymplectic models, in particu-lar͒, there are several choices for the multimomentum bundle where the covariant Hamiltonian formalism takes place. As a consequence, several proposals for this formalism can be stated, and, on each one of them, the differentiable structures needed for setting the formalism are obtained in different ways. In this work we make an accurate study of some of these Hamiltonian formalisms, showing their equivalence. In particular, the geometrical structures ͑canonical or not͒ needed for the Hamiltonian formalism, are introduced and compared, and the derivation of Hamiltonian field equations from the corresponding variational principle is shown in detail. Furthermore, the Hamiltonian formalism of systems described by Lagrangians is performed, both for the hyper-regular and almost-regular cases. Finally, the role of connections in the construction of Hamiltonian field theories is clarified.

Fe b 20 04 LAGRANGIAN-HAMILTONIAN UNIFIED FORMALISM FOR FIELD THEORY

2008

The Rusk-Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems (including dynamical equations and solutions, constraints, Legendre map, evolution operators, equivalence, etc.). In this work we extend this unified framework to first-order classical field theories, and show how this description comprises the main features of the Lagrangian and Hamiltonian formalisms, both for the regular and singular cases. This formulation is a first step toward further applications in optimal control theory for PDE’s.

Multivector field formulation of Hamiltonian field theories: equations and symmetries

Journal of Physics A: Mathematical and General, 1999

We state the intrinsic form of the Hamiltonian equations of first-order Classical Field theories in three equivalent geometrical ways: using multivector fields, jet fields and connections. Thus, these equations are given in a form similar to that in which the Hamiltonian equations of mechanics are usually given.

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.

2 Lagrangian formalism 2 . 1 The setting for classical field theories

2007

The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.

Hamiltonian Systems in Multisymplectic Field Theories

We consider Hamiltonian systems in first-order multisymplectic field theories. First we review the construction and properties of Hamiltonian systems in the so-called restricted multimomentum bundle using Hamiltonian sections, including the variational principle which leads to the Hamiltonian field equations. Then, we introduce Hamiltonian systems in the extended multimomentum bundle, in an analogous way to how these systems are defined in non-autonomous (symplectic) mechanics or in the so-called extended (symplectic) formulation of autonomous mechanics. The corresponding variational principle is also stated for these extended Hamiltonian systems and, after studying the geometric properties of these systems, we establish the relation between the extended and the restricted ones. These definitions and properties are also adapted to submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories.

Multisymplectic Lagrangian and Hamiltonian formalisms of First-order Classical Field theories (original) (raw)

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