The objective of this work is twofold: First, we analyze the relation between the k-cosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms in classical field theories. In particular, we prove the equivalence between k-symplectic field theories and the so-called autonomous k-cosymplectic field theories, extending in this way the description of the symplectic formalism of autonomous systems as a particular case of the cosymplectic formalism in non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric character of the solutions to the Hamilton-de Donder-Weyl and the Euler-Lagrange equations in these formalisms. Second, we study the equivalence between k-cosymplectic and a particular kind of multisymplectic Hamiltonian and Lagrangian field theories (those where the configuration bundle of the theory is trivial).
This book is devoted to review two of the most relevant approaches to the study of classical field theories of first order, say k-symplectic and k-cosymplectic. In the last part, we relate the k-symplectic and k-cosymplectic manifolds with the multisymplectic theory.
We establish a link between the multisymplectic and the covariant phase space approach to geometric field theory by showing how to derive the symplectic form on the latter, as introduced by Crnkovic-Witten and Zuckerman, from the multisymplectic form. The main result is that the Poisson bracket associated with this symplectic structure, according to the standard rules, is precisely the covariant bracket due to Peierls and DeWitt.
This paper presents a generalization of symplectic geometry to a principal bundle over the configuration space of a classical field. This bundle, the vertically adapted linear frame bundle, is obtained by breaking the symmetry of the full linear frame bundle of the field configuration space, and it inherits a generalized symplectic structure from the full frame bundle. The geometric structure of the vertically adapted frame bundle admits vector-valued field observables and produces vector-valued Hamiltonian vector fields, from which we can define a Poisson bracket on the field observables. We …
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 …
This paper presents generalized momentum mappings for covariant Hamiltonian field theories. The new momentum mappings arise from a generalization of symplectic geometry to $L_VY$, the bundle of vertically adapted linear frames over the bundle of field configurations $Y$. Specifically, the generalized field momentum observables are vector-valued momentum mappings on the vertically adapted frame bundle generated from automorphisms of $Y$. The generalized symplectic geometry on $L_VY$ is a covering theory for multisymplectic geometry on the multiphase space $Z$, and it follows that the field mome…
Many physically important mechanical systems may be described with a Lie group $G$ as configuration space. According to the well-known Noether's theorem, underlying symmetries of the Lie group may be used to considerably reduce the complexity of the problems. However, these reduction techniques, used without care for general problems (waves, field theory), may lead to uncomfortable infinite dimensional spaces. As an alternative, the \emph{covariant} formulation allows to consider a finite dimensional configuration space by increasing the number of independent variables. But the geometric eleme…
Given a vector field on a manifold M, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well-behaved under transgression to spaces of maps into M. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. We show that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family o…
We discuss the interplay between lagrangian distributions and connections in symplectic geometry, beginning with the traditional case of symplectic manifolds and then passing to the more general context of poly- and multisymplectic structures on fiber bundles, which is relevant for the covariant hamiltonian formulation of classical field theory. In particular, we generalize Weinstein's tubular neighborhood theorem for symplectic manifolds carrying a (simple) lagrangian foliation to this situation. In all cases, the Bott connection, or an appropriately extended version thereof, plays a central …
A Lie 2-algebra is a "categorified" version of a Lie algebra: that is, a category equipped with structures analogous those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an "n-plectic manifold": a finite-dimensional manifold equipped with a closed nondegenerate (n+1)-form. Here we consider the case n = 2…
From 2-covariant tensor fields to Poisson and Jacobi structures
Instituto de Ciencias Matemáticas (ICMAT–CSIC)（数学科学研究所）；Real Academia de Ciencias Exactas, Físicas y Naturales de España（西班牙皇家精确、物理和自然科学院）；Great Bay University（大湾区大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；Universitat Rovira i Virgili（罗维拉-伊维尔基里大学）
This paper presents a "historical" formalism for dynamical systems, in its Hamiltonian version (Lagrangian version was presented in a previous paper). It is universal, in the sense that it applies equally well to time dynamics and to field theories on space-time. It is based on the notion of (Hamiltonian) histories, which are sections of the (extended) phase space bundle. It is developed in the space of sections, in contradistinction with the usual formalism which works in the bundle manifold. In field theories, the formalism remains covariant and does not require a spitting of space-time. It …
This paper presents (in its Lagrangian version) a very general "historical" formalism for dynamical systems, including time-dynamics and field theories. It is based on the universal notion of history. Its condensed and universal formulation provides a synthesis and a generalization different approaches of dynamics. It is in our sense closer to its real essence. The formalism is by construction explicitely covariant and does not require the introduction of time, or of a time function in relativistic theories. It considers space-time (in field theories) exactly in the same manner than time in us…
A novel reduction procedure for covariant classical field theories, reflecting the generalized symplectic reduction theory of Hamiltonian systems, is presented. The departure point of this reduction procedure consists in the choice of a submanifold of the manifold of solutions of the equations describing a field theory. Then, the covariance of the geometrical objects involved, will allow to define equations of motion on a reduced space. The computation of the canonical geometrical structure is performed neatly by using the geometrical framework provided by the multisymplectic description of co…
This is the first paper of a five part work in which we study the Lagrangian and Hamiltonian structure of classical field theories with constraints. Our goal is to explore some of the connections between initial value constraints and gauge transformations in such theories (either relativistic or not). To do this, in the course of these four papers, we develop and use a number of tools from symplectic and multisymplectic geometry. Of central importance in our analysis is the notion of the ``energy-momentum map'' associated to the gauge group of a given classical field theory. We hope to demonst…
The multisymplectic formalism of field theories developed by many mathematicians over the last fifty years is extended in this work to deal with manifolds that have boundaries. In particular, we develop a multisymplectic framework for first order covariant Hamiltonian field theories on manifolds with boundaries. This work is a geometric fulfillment of Fock's characterization of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin [Ca14]. This framework leads to a true geometric understanding of conventional choices for boundary conditions. For example, the boundary con…
With the covariant formulation in hand from the first paper of this series (physics/9801019), we begin in this second paper to study the canonical (or ``instantaneous'') formulation of classical field theories. The canonical formluation works with fields defined as time-evolving cross sections of bundles over a Cauchy surface, rather than as sections of bundles over spacetime as in the covariant formulation. In Chapter 5 we begin to relate these approaches to classical field theory; in particular, we show how covariant multisymplectic geometry induces the instantaneous symplectic geometry of c…
Observable currents are spacetime local objects that induce physical observables when integrated on an auxiliary codimension one surface. Since the resulting observables are independent of local deformations of the integration surface, the currents themselves carry most of the information about the induced physical observables. I introduce observable currents in a multisymplectic framework for Lagrangian field theory over discrete spacetime. One family of examples is composed by Noether currents. A much larger family of examples is composed by currents, spacetime local objects, that encode the…
