A Multisymplectic Unified Formalism for Second Order Classical Field Theories (original) (raw)

A new multisymplectic unified formalism for second order classical field theories

Journal of Geometric Mechanics, 2015

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.

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.

Multisymplectic Lagrangian and Hamiltonian formalisms of First-order Classical Field theories

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.

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.

On the multisymplectic formalism for first order field theories

Differential Geometry and its Applications, 1991

Cariiiena, J.F., M. Crampin and L.A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl. 1 (1991) 345-374. Abstract: The general purpose of this paper is to attempt to clarify the geometrical foundations of first order Lagrangian and Hamiltonian field theories by introducing in a systematic way multisymplectic manifolds, the field theoretical analogues of the symplectic structures used in geometrical mechanics. Much of the confusion surrounding such terms as gauge transformation and symmetry transformation as they are used in the context of Lagrangian theory is thereby eliminated, as we show. We discuss Noether's theorem for general symmetries of Lagrangian and Hamiltonian field theories. The cohomology associated to a group of symmetries of Hamiltonian or Lagrangian field theories is constructed and its relation with the structure of the current algebra is made apparent. Zt'eywords: Multisymplectic structure, jet bundle, first order field theory, multimomentum map, gauge transformation. MS classification: 53C80, 58A20. 0926-2245/91/$03.50 01991 -Elsevier Science Publishers B.V. All rights reserved J. F. Cariiiena, M. Crampin, L.A. Ibort

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.

VARIATIONAL PRINCIPLES FOR MULTISYMPLECTIC SECOND-ORDER CLASSICAL FIELD THEORIES

International Journal of Geometric Methods in Modern Physics, 2015

We state a unified geometrical version of the variational principles for second-order classical field theories. The standard Lagrangian and Hamiltonian formulations of these principles and the corresponding field equations are recovered from this unified framework.

A new Hamiltonian formalism for singular Lagrangian theories

2009

We introduce a "nonlinear" version of the Hamiltonian formalism which allows a self-consistent description of theories with degenerate Lagrangian. A generalization of the Legendre transform to the case when the Hessian is zero is done using the mixed (envelope/general) solutions of the multidimensional Clairaut equation. The corresponding system of equations of motion is equivalent to the Lagrange equations, but contains "nondynamical" momenta and unresolved velocities. This system is reduced to the physical phase space and presented in the Hamiltonian form by introducing a new (non-Lie) bracket.