Lepage equivalents of Lagrangians are a higher order, field-theoretical generalization of the notion of Poincare-Cartan form from mechanics and play a similar role: they give rise to a geometric formulation (and to a geometric understanding) of the calculus of variations. A long-standing open problem is the determination, for field-theoretical Lagrangians of order greater than one, of a Lepage equivalent with the so-called closure property: the Lepage equivalent is a closed differential form if and only if the Lagrangian has vanishing Euler-Lagrange expressions. The present paper proposes a solution to this problem, for general Lagrangians, of any order. The construction is a local one; yet, we show that in most of the cases of interest for physical applications, the obtained Lepage equivalent is actually globally defined. A variant of this construction, which is advantageous for reducible Lagrangians, is also introduced. In particular, for reducible Lagrangians of order two, the obtained Lepage equivalents are of order one.
The integrable systems known as "AKS systems" admit a natural formulation in terms of a Hamiltonian picture. The Lagrangian side of these systems are far less known; a version in these terms can be found in a work of Feher et al. The purpose of these notes in to provide a novel description of AKS systems in terms of a variational problem different from the usual in mechanics. Additionally, and using techniques borrowed from an article of M. Gotay, it was possible to build the Hamiltonian side of this variational problem, allowing us to establish the equivalence with the usual approach to these…
The Carathéodory form of the calculus of variations belongs to the class of Lepage equivalents of first-order Lagrangians in field theory. Here, this equivalent is generalized for second- and higher-order Lagrangians by means of intrisic geometric operations applied to the well-known Poincaré--Cartan form and principal component of Lepage forms, respectively. For second-order theory, our definition coincides with the previous result obtained by Crampin and Saunders in a different way. The Carathéodory equivalent of the Hilbert Lagrangian in general relativity is discussed.
A setting for global variational geometry on Grassmann fibrations is presented. The integral variational functionals for finite dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert form in the classical mechanics. Prolongations of immersions, diffeomorphisms and vector fields to the Grassmann fibrations are introduced as geometric tools for the variations of immersions. The first infinitesimal variation formula together with its consequences, the Euler-Lagrang…
The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) de Donder-Weyl (dDW) one. One of the main points is the fact that the Legendre transform in the dDW approach is replaced by a Legendre correspondence in the LP theory (This correspondence behaves differently: ignoring the singularities whenever the Lagrangian is dege…
Hamilton equations based not only upon the Poincare--Cartan equivalent of a first-order Lagrangian, but rather upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton--De Donder theory, but regularizable in this generalized sense are studied. Legendre transformation for regularizable Lagrangians is proposed, and Hamilton equations, equivalent with the Euler--Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field …
By considering the closure property of a Lagrangian multiform as a conservation law, we use Noether's theorem to show that every variational symmetry of a Lagrangian leads to a Lagrangian multiform. In doing so, we provide a systematic method for constructing Lagrangian multiforms for which the closure property and the multiform Euler-Lagrange (EL) equations both hold. We present three examples, including the first known example of a Lagrangian 3-form: a multiform for the Kadomtsev-Petviashvili equation. We also present a new proof of the multiform EL equations for a Lagrangian k-form for arbi…
A systematic procedure is proposed for inclusion of Stueckelberg fields. The procedure begins with the involutive closure when the original Lagrangian equations are complemented by all the lower order consequences. The involutive closure can be viewed as Lagrangian analogue of complementing constrained Hamiltonian system with secondary constraints. The involutively closed form of the field equations allows for explicitly covariant degree of freedom number count, which is stable with respect to deformations. If the original Lagrangian equations are not involutive, the involutive closure will be…
The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) de Donder-Weyl (dDW) one. For instance the Legendre transform in the dDW approach is replaced by a Legendre correspondence in the LP theory, while ignoring singularities whenever the Lagrangian is degenerate. Moreover we show that there exist two different definition…
In this work, we present another example of the Lagrangian 1-form structure for the hy- perbolic Calogero-Moser system both in discrete-time level and continuous-time level. The discrete-time hyperbolic Calogero-Moser system is obtained by considering pole-reduction of the semi-discrete Kadomtsev-Petviashvili (KP) equation. The key relation called the discrete-time closure relation is directly obtained from the compatibility between the temporal Lax matrices. The continuous-time hierarchy of the hyperbolic Calogero-Moser system is obtained through two successive continuum limits. The continuou…
Wave--current interaction (WCI) dynamics energizes and mixes the ocean thermocline by producing a combination of Langmuir circulation, internal waves and turbulent shear flows, which interact over a wide range of time scales. Two complementary approaches exist for approximating different aspects of WCI dynamics. These are the Generalized Lagrangian Mean (GLM) approach and the Gent--McWilliams (GM) approach. Their complementarity is evident in their Kelvin circulation theorems. GLM introduces a wave pseudomomentum per unit mass into its Kelvin circulation integrand, while GM introduces a an add…
We present further developments on the Lagrangian 1-form description for one-dimensional integrable systems in both discrete and continuous levels. A key feature of integrability in this context called a closure relation will be derived from the local variation of the action on the space of independent variables. The generalised Euler-Lagrange equations and constraint equations are derived directly from the variation of the action on the space of dependent variables. This set of Lagrangian equations gives rise to a crucial property of integrable systems known as the multidimensional consistenc…
General Lagrangian theory of discrete one-dimensional integrable systems is illustrated by a detailed study of Bäcklund transformations for Toda-type systems. Commutativity of Bäcklund transformations is shown to be equivalent to consistency of the system of discrete multi-time Euler-Lagrange equations. The precise meaning of the commutativity in the periodic case, when all maps are double-valued, is established. It is shown that gluing of different branches is governed by the so called superposition formulas. The closure relation for the multi-time Lagrangian 1-form on solutions of the variat…
Multidimensional consistency has emerged as a key integrability property for partial difference equations (P$Δ$Es) defined on the "space-time" lattice. It has led, among other major insights, to a classification of scalar affine-linear quadrilateral P$Δ$Es possessing this property, leading to the so-called ABS list. Recently, a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete Lagrangian multi-forms. This description is based on a fundamental and highly nontrivial property of Lagrangians for those integrable lattice equations, name…
We study the Lagrange formalism of the (rational) Ruijsenaars-Schneider (RS) system, both in discrete time as well as in continuous time, as a further example of a Lagrange 1-form structure in the sense of the recent paper [24]. The discrete-time model of the RS system was established some time ago arising via an Ansatz of a Lax pair, and was shown to lead to an exactly integrable correspondence (multivalued map)[15]. In this paper we consider an extended system representing a family of commuting flows of this type, and establish a connection with the lattice KP system. In the Lagrangian 1-for…
Building towards a more covariant approach to canonical classical and quantum gravity we outline an approach to constrained dynamics that de-emphasizes the role of the Hamiltonian phase space and highlights the role of the Lagrangian phase space. We identify a "Lagrangian one-form" to replace the standard symplectic one-form, which we use to construct the canonical constraints and an associated constraint algebra. The method is particularly useful for generally covariant systems and systems with a degenerate canonical symplectic form, such as Einstein Cartan gravity, to which we apply the meth…
Exact, degenerate two-forms on time-extended space R X M which are invariant under the unsteady, incompressible fluid motion on 3D region M are introduced. The equivalence class up to exact one-forms of each potential one-form is splitted by the velocity field. The components of this splitting corresponds to Lagrangian and Eulerian conservation laws for helicity densities. These are expressed as the closure of three-forms which depend on two discrete and a continuous parameter. Each two-form is extended to a symplectic form on R X M. The subclasses of potential one-forms giving rise to Euleria…
We consider the variational principle for the Lagrangian 1-form structure for long-range models of Calogero-Moser (CM) type. The multiform variational principle involves variations with respect to both the field variables as well as the independent variables corresponding to deformations of the time-curves in a multi-time space. The ensuing generalised Euler-Lagrange (gEL) equations comprise a system of multi-time EL equations, as well as constraints from so-called `alien derivatives' and `corner equations' arising from how variations on different coordinate curves match up. The closure relati…
We discuss a geometrical formulation for gauge field theories, called Gauge-Natural, which has been mostly developed at the University of Turin (Italy) and at the Masaryk University in Brno (Czech Republic). In this framework, the group of automorphisms of some suitable principal bundle P, called the structure bundle, acts on fields by means of gauge transformations. After showing that the classical Wess-Zumino model can be consistently described within this formalism, we consider minimal supergravity (mSugra). We find that to embed mSugra in a Gauge-Natural framework, further investigations a…
