Multisymplectic Field Theory Research Articles

More Insights into Symmetries in Multisymplectic Field Theories

This work provides a general overview for the treatment of symmetries in classical field theories and (pre)multisymplectic geometry. The geometric characteristics of the relation between how symmetries are interpreted in theoretical physics and in the geometric formulation of these theories are clarified. Finally, a general discussion is given on the structure of symmetries in the presence of constraints appearing in singular field theories. Symmetries of some typical theories in theoretical physics are analyzed through the construction of the relevant multimomentum maps which are the conserved quantities (by Noether’s theorem) on the (pre)multisymplectic phase spaces.

Multisymplectic Hamiltonian Variational Integrators

Variational integrators have traditionally been constructed from the perspective of Lagrangian mechanics, but there have been recent efforts to adopt discrete variational approaches to the symplectic discretization of Hamiltonian mechanics using Hamiltonian variational integrators. In this paper, we will extend these results to the setting of Hamiltonian multisymplectic field theories. We demonstrate that one can use the notion of Type II generating functionals for Hamiltonian partial differential equations as the basis for systematically constructing Galerkin Hamiltonian variational integrators that automatically satisfy a discrete multisymplectic conservation law, and establish a discrete Noether's theorem for discretizations that are invariant under a Lie group action on the discrete dual jet bundle. In addition, we demonstrate that for spacetime tensor product discretizations, one can recover the multisymplectic integrators of Bridges and Reich, and show that a variational multisymplectic discretization of a Hamiltonian multisymplectic field theory using spacetime tensor product Runge–Kutta discretizations is well-defined if and only if the partitioned Runge-Kutta methods are symplectic in space and time.

Hamilton–Jacobi theory for gauge field theories

Recently, M. de León et al. (Campos et al., 2015) have developed a geometrical description of Hamilton–Jacobi theory for multisymplectic field theory. In our paper we analyze in the same spirit a special kind of field theories which are gauge field theories. The Hamilton–Jacobi theory for this kind of fields is shown.

Hamilton-Jacobi theory in multisymplectic classical field theories

The geometric framework for the Hamilton-Jacobi theory developed in the studies of Cariñena et al. [Int. J. Geom. Methods Mod. Phys. 3(7), 1417–1458 (2006)], Cariñena et al. [Int. J. Geom. Methods Mod. Phys. 13(2), 1650017 (2015)], and de León et al. [Variations, Geometry and Physics (Nova Science Publishers, New York, 2009)] 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.

Multisymplectic effective general boundary field theory

The transfer matrix in lattice field theory connects the covariant and the initial data frameworks; in spin foam models, it can be written as a composition of elementary cellular amplitudes/propagators. We present a framework for discrete spacetime classical field theory in which solutions to the field equations over elementary spacetime cells may be amalgamated if they satisfy simple gluing conditions matching the composition rules of cellular amplitudes in spin foam models. Furthermore, the formalism is endowed with a multisymplectic structure responsible for local conservation laws. Some models within our framework are effective theories modeling a system at a given scale. Our framework allows us to study coarse graining and the continuum limit.

Extended Hamiltonian systems in multisymplectic field theories

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 nonautonomous 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.

Variational integrators and the finite element method

Based on the finite element method, discretizations of Lagrangians in multisymplectic field theory are presented in this paper. In particular, we obtain discrete Lagrangians of high order. The corresponding variational integrators are constructed. The convergence of the variational integrators is discussed. Numerical experiments are also reported.

Geometrical theory of balance systems and the entropy principle

In this work we present the theory of balance equations of Continuum Thermodynamics (balance systems) in a geometrical form using the Poincare-Cartan formalism of Multisymplectic Field Theory. A constitutive relation C of a balance system BC is realized as a mapping between a (partial) 1-jet bundle of the configurational bundle π : Y → X and the dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system BC is presented in three different forms and the space of admissible variations is defined and studied. Action of automorphisms of the bundle π on the constitutive mappings C is studied and it is shown that the symmetry group Sym(C) of the constitutive relation C acts on the space of solutions of balance system BC. A version of the Noether Theorem for an action of a symmetry group of vertical automorphisms is presented with the usage of conventional multimomentum mapping. The "entropy principle" of Irreducible Thermodynamics (the requirement that the entropy balance should be the consequence of the balance system) is studied for general balance systems. The structure of corresponding (secondary) balance laws of a balance system BC is studied for the cases of different (partial) 1-jet bundles as domains of constitutive laws. Corresponding results may be considered as a generalization of the transition to the dual, Lagrange-Liu picture of Rational Extended Thermodynamics (RET).

Discrete Lagrangian field theories on Lie groupoids

We present a geometric framework for discrete classical field theories, where fields are modeled as “morphisms” defined on a discrete grid in the base space, and take values in a Lie groupoid. We describe the basic geometric set-up and derive the field equations from a variational principle. We also show that the solutions of these equations are multisymplectic in the sense of Bridges and Marsden. The groupoid framework employed here allows us to recover not only some previously known results on discrete multisymplectic field theories, but also to derive a number of new results, most notably a notion of discrete Lie–Poisson equations and discrete reduction. In a final section, we establish the connection with discrete differential geometry and gauge theories on a lattice.

Hamiltonian multivector fields and Poisson forms in multisymplectic field theory

We present a general classification of Hamiltonian multivector fields and of Poisson forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories. This is a prerequisite for computing explicit expressions for the Poisson bracket between two Poisson forms.

A multisymplectic integrator for the periodic nonlinear Schrödinger equation

A multisymplectic integrator for the periodic nonlinear Schrodinger equation is presented in this paper. Its accuracy is proved. By introducing a norm, we investigate its nonlinear stability. We also discuss the relationship between this multisymplectic integrator and two variational integrators which are derived by using the discrete multisymplectic field theory and the finite element method.

Geometry of Hamiltonian n-vector fields in multisymplectic field theory

Multisymplectic geometry—which originates from the well known De Donder–Weyl (DW) theory—is a natural framework for the study of classical field theories. Recently, two algebraic structures have been put forward to encode a given theory algebraically. Those structures are formulated on finite dimensional spaces, which seems to be surprising at first. In this paper, we investigate the correspondence of Hamiltonian functions and certain antisymmetric tensor products of vector fields. The latter turn out to be the proper generalisation of the Hamiltonian vector fields of classical mechanics. Thus we clarify the algebraic description of solutions of the field equations.

Total Variation in Discrete Multisymplectic Field Theory and Multisymplectic-Energy-Momentum Integrators

A total variation calculus in discrete multisymplectic field theory is developed in this Letter. Using this discrete total variation calculus, we obtain multisymplectic-energy-momentum integrators. The multisymplectic discretization for the nonlinear Schrodinger equation is also presented.

Variational methods, multisymplectic geometry and continuum mechanics

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constraint. The results obtained in this paper set the stage for multisymplectic reduction and for the further development of Veselov-type multisymplectic discretizations and numerical algorithms. The latter will be the subject of a companion paper.

A variational approach to second-order multisymplectic field theory

This paper presents a geometric-variational approach to continuous and discrete second-order field theories following the methodology of [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351–395]. Staying entirely in the Lagrangian framework and letting Y denote the configuration fiber bundle, we show that both the multisymplectic structure on J 3 Y as well as the Noether theorem arise from the first variation of the action function. We generalize the multisymplectic form formula derived for first-order field theories in [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351–395], to the case of second-order field theories, and we apply our theory to the Camassa–Holm (CH) equation in both the continuous and discrete settings. Our discretization produces a multisymplectic-momentum integrator, a generalization of the Moser–Veselov rigid body algorithm to the setting of nonlinear PDEs with second-order Lagrangians.