Lagrangian Field Theory Research Articles

Covariantization of nonlinear elasticity

In this paper we make a connection between covariant elasticity based on covariance of energy balance and Lagrangian field theory of elasticity with two background metrics. We use Kuchař’s idea of reparametrization of field theories and make elasticity generally covariant by introducing a "covariance field", which is a time-independent spatial diffeomorphism. We define a modified action for parameterized elasticity and show that the Doyle-Ericksen formula and spatial homogeneity of the Lagrangian density are among its Euler–Lagrange equations.

ON HIGHER DERIVATIVES AS CONSTRAINTS IN FIELD THEORY: A GEOMETRIC PERSPECTIVE

We formalize geometrically the idea that the (de Donder) Hamiltonian formulation of a higher derivative Lagrangian field theory can be constructed understanding the latter as a first derivative theory subjected to constraints.

Regular hyperbolicity, dominant energy condition and causality for Lagrangian theories of maps

The goal of this paper is threefold. First is to clarify the connection between the dominant energy condition and hyperbolicity properties of Lagrangian field theories. Second is to provide further analysis on the breakdown of hyperbolicity for the Skyrme model, sharpening the results of Crutchfield and Bell and comparing against a result of Gibbons, and provide a local well-posedness result for the dynamical problem in the Skyrme model. Third is to provide a short summary of the framework of regular hyperbolicity of Christodoulou for the relativity community. In the process, a general theorem about dominant energy conditions for Lagrangian theories of maps is proved, as well as several results concerning hyperbolicity of those maps.

Local BRST cohomologyin(non-)Lagrangian field theory

Some general theorems are established on the local BRST cohomology for not necessarily Lagrangian gauge theories. Particular attention is given to the BRST groups with direct physical interpretation. Among other things, the groups of rigid symmetries and conservation laws are shown to be still connected, though less tightly than in the Lagrangian theory. The connection is provided by the elements of another local BRST cohomology group whose elements are identified with Lagrange structures. This extends the cohomological formulation of the Noether theorem beyond the scope of Lagrangian dynamics. We show that each integrable Lagrange structure gives rise to a Lie bracket in the space of conservation laws, which generalizes the Dickey bracket of conserved currents known in Lagrangian field theory. We study the issues of existence and uniqueness of the local BRST complex associated with a given set of field equations endowed with a compatible Lagrange structure. Contrary to the usual BV formalism, such a complex does not always exist for non-Lagrangian dynamics, and when exists it is by no means unique. The ambiguity and obstructions are controlled by certain cohomology classes, which are all explicitly identified.

Forms of Lepage type and the balance systems

In this work we apply infinitesimal variational calculus to the systems of balance equations. We determine a class of the exterior n + ( n + 1 ) -forms Θ on the jet bundle of infinite order over a configurational bundle π : Y n + m → X n similar to the class of Lepage n-forms. Systems of differential equations obtained in the way similar to one used in the Lagrangian field theory, include the Euler–Lagrange equations corresponding to a Lagrangian functions as well as arbitrary regular systems of balance equations. For a balance system with a symmetry group G we present the Noether balance laws corresponding to the generators of the Lie algebra of the group G.

VARIATIONAL THEORY OF BALANCE SYSTEMS

In this work, we apply the Poincaré–Cartan formalism of Classical Field Theory to study the systems of balance equations (balance systems). We introduce the partial k-jet bundles [Formula: see text] of the configurational bundle π : Y → X and study their basic properties: partial Cartan structure, prolongation of vector fields, etc. A constitutive relation C of a balance system [Formula: see text] is realized as the mapping between a (partial) k-jet bundle [Formula: see text] and the extended dual bundle [Formula: see text] similar to the Legendre mapping of the Lagrangian Field Theory. The invariant (variational) form of the balance system [Formula: see text] corresponding to a constitutive relation [Formula: see text] is studied. Special cases of balance systems — Lagrangian systems of order 1 with arbitrary sources and RET (Rational Extended Thermodynamics) systems are characterized in geometrical terms. The action of automorphisms of the bundle π on the constitutive mappings [Formula: see text] is studied and it is shown that the symmetry group [Formula: see text] of [Formula: see text] acts on the sheaf of solutions [Formula: see text] of balance system [Formula: see text]. A suitable version of Noether theorem for an action of a symmetry group is presented together with the special forms for semi-Lagrangian and RET balance systems. Examples of energy momentum and gauge symmetries balance laws are provided. At the end, we introduce the secondary balance laws for a balance system and classify these laws for the Cattaneo heat propagation system.

A Geometric Theory of Growth Mechanics

In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and "shape". We show that time dependency of material metric will affect the energy balance and the entropy production inequality; both the energy balance and the entropy production inequality have to be modified. We then obtain the governing equations covariantly by postulating invariance of energy balance under time-dependent spatial diffeomorphisms. We use the principle of maximum entropy production in deriving an evolution equation for the material metric. In the case of isotropic growth, we find those growth distributions that do not result in residual stresses. We then look at Lagrangian field theory of growing elastic solids. We will use the Lagrange-d'Alembert's principle with Rayleigh's dissipation functions to derive all the governing equations. We make an explicit connection between our geometric theory and the conventional multiplicative decomposition of deformation gradient $\mathbf{F}=\mathbf{F}_e\mathbf{F}_g$ into growth and elastic parts. We linearize the nonlinear theory and derive a linearized theory of growth mechanics. Finally, we obtain the stress-free growth distributions in the linearized theory.

Reduced classical field theories: k-cosymplectic formalism on Lie algebroids

In this paper we introduce a geometric description of Lagrangian and Hamiltonian classical field theories on Lie algebroids in the framework of k-cosymplectic geometry. We discuss the relation between Lagrangian and Hamiltonian descriptions through a convenient notion of Legendre transformation. The theory is a natural generalization of the standard one; in addition, other interesting examples are studied, mainly on reduction of classical field theories.

Modeling Steady Acoustic Fields Bounded in Cavities with Geometrical Imperfections

A mathematical method is derived within the framework of classical Lagrangian field theory, which is suitable for the determination of the eigenstates of acoustic resonators of nearly spherical shape. The method is based on the expansion of the Helmholtz differential operator and the boundary condition in a power series of a small geometrical perturbation parameter \({\epsilon}\) . The method extends to orders higher than \({\epsilon^2}\) the calculation of the perturbed acoustic eigenvalues, which was previously limited by the use of variational formalism and the methods of Morse and Ingard. A specific example is worked out for radial modes of a prolate spheroid, with the frequency perturbation calculated to order \({\epsilon^3}\) . A possible strategy to tackle the problem of calculating the acoustic eigenvalues for cavities presenting non-smooth geometrical imperfections is also described.

Field-theoretic parametrization of low-energy nucleon form factors

A field-theoretic parametrization is proposed for nucleon electromagnetic form factors at momentum transfer less than 600 MeV. The parametrization is part of a larger effective field theory Lagrangian that is Lorentz covariant and chirally symmetric and that has been used to successfully describe bulk and single-particle properties of medium- to heavy-mass nuclei. The parametrization is based on vector meson dominance and a derivative expansion of nucleon couplings to the electromagnetic fields. At lowest order in the expansion, it is possible to fit all four parameters to modern data on the rms radii of the nucleon form factors. At next-to-leading order it is possible to fit the form factors to within a few percent up to momentum transfers of 600 MeV. The vector meson dominance contributions are crucial in this fit, because a simple expansion in powers of momentum transfer would require many terms to achieve comparable accuracy. The ability to fit single-nucleon form factors up to 600 MeV momentum transfer makes possible the study of two-body electromagnetic exchange currents within this effective field theory framework.

The Lagrangian-Hamiltonian formalism for higher order field theories

We generalize the Lagrangian-Hamiltonian formalism of Skinner and Rusk to higher order field theories on fiber bundles. As a byproduct we solve the long standing problem of defining, in a coordinate free manner, a Hamiltonian formalism for higher order Lagrangian field theories. Namely, our formalism does only depend on the action functional and, therefore, unlike previously proposed ones, is free from any relevant ambiguity.

Unambiguous formalism for higher order Lagrangian field theories

The aim of this paper is to propose an unambiguous intrinsic formalism for higher order field theories which avoids the arbitrariness in the generalization of the conventional description of field theories, and implies the existence of different Cartan forms and Legendre transformations. We propose a differential-geometric setting for the dynamics of a higher order field theory, based on the Skinner and Rusk formalism for mechanics. This approach incorporates aspects of both the Lagrangian and the Hamiltonian description, since the field equations are formulated using the Lagrangian on a higher order jet bundle and the canonical multisymplectic form on its affine dual. As both of these objects are uniquely defined, the Skinner–Rusk approach has the advantage that it does not suffer from the arbitrariness in conventional descriptions. The result is that we obtain a unique and global intrinsic version of the Euler–Lagrange equations for higher order field theories. Several examples illustrate our construction.

Lagrangian field theory of reaction-diffusion

It is commonly accepted that reaction-diffusion equations cannot be obtained by a Lagrangian field theory. Guided by the well known connection between quantum and diffusion equations, we implement here a Lagrangian approach valid for totally general nonlinear reacting-diffusing systems which allows the definition of global conserved observables derived using Nöther's theorem.

K-symplectic formalism on Lie algebroids

In this paper we introduce a geometric description of Lagrangian and Hamiltonian classical field theories on Lie algebroids in the framework of k-symplectic geometry. We discuss the relation between the Lagrangian and Hamiltonian descriptions through a convenient notion of Legendre transformation. The theory is a natural generalization of the standard one; in addition, other interesting examples are studied, in particular, systems with symmetry and Poisson-sigma models.

Janus field theories from non-linear BF theories for multiple M2-branes

We integrate the nonpropagating B_{\mu} gauge field for the non-linear BF Lagrangian describing N M2-branes which includes terms with even number of the totally antisymmetric tensor M^{IJK} in arXiv:0808.2473 and for the two-types of non-linear BF Lagrangians which include terms with odd number of M^{IJK} as well in arXiv:0809:0985. For the former Lagrangian we derive directly the DBI-type Lagrangian expressed by the SU(N) dynamical A_{\mu} gauge field with a spacetime dependent coupling constant, while for the low-energy expansions of the latter Lagrangians the B_{\mu} integration is iteratively performed. The derived Janus field theory Lagrangians are compared.

Quantum liquid signatures in dwarf stars

We develop further the proposal of arXiv:0806.3692 that a new state of matter — charged condensate of spin-0 nuclei — may exist in helium-core dwarf stars. The charged condensate and its fluctuations are described by an effective field theory Lagrangian. The spectrum of bosonic fluctuations is gapped, while electrons, at temperatures of interest, give rise to gapless excitations near the Fermi surface. These properties determine the evolution of the dwarfs with condensed cores. In particular, we show that such dwarf stars would cool significantly faster than their crystallized counterparts. As a result, the luminosity function for the helium-core dwarfs will have a sharp drop-off after the condensation. It is tempting to interpret the recently discovered abrupt termination of a sequence of 24 helium-core dwarf candidates in NGC 6397 as a signature of the charged condensation.

On the notion of gauge symmetries of generic Lagrangian field theory

General Lagrangian theory of even and odd fields on an arbitrary smooth manifold is considered. Its nontrivial reducible gauge symmetries and their algebra are defined in this very general setting by means of the inverse second Noether theorem. In contrast with gauge symmetries, nontrivial Noether and higher-stage Noether identities of Lagrangian theory can be intrinsically defined by constructing the exact Koszul–Tate complex. The inverse second Noether theorem that we prove associates with this complex the cochain sequence with the ascent operator whose components define nontrivial gauge and higher-stage gauge symmetries. These gauge symmetries are said to be algebraically closed if the ascent operator can be extended to a nilpotent operator. The necessary conditions for this extension are stated. The characteristic examples of Yang–Mills supergauge theory, topological Chern–Simons theory, gauge gravitation theory, and topological background field (BF) theory are presented.

Secondary calculus and the covariant phase space

The covariant phase space of a lagrangian field theory is the solution space of the associated Euler–Lagrange equations. It is, in principle, a nice environment for covariant quantization of a lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) “presymplectic structure” ω (as first noticed by Zuckerman in 1986) whose degeneracy (functional) distribution is naturally interpreted as the Lie algebra of gauge transformations. We propose a fully rigorous approach to the covariant phase space in the framework of jet spaces and (A. M. Vinogradov’s) secondary calculus. In particular, we describe the degeneracy distribution of ω . As a byproduct we rederive the existence of a Lie bracket among gauge invariant functions on the covariant phase space.

A study on relativistic lagrangian field theories with non-topological soliton solutions

We perform a general analysis of the dynamic structure of two classes of relativistic lagrangian field theories exhibiting static spherically symmetric non-topological soliton solutions. The analysis is concerned with (multi-) scalar fields and generalized gauge fields of compact semi-simple Lie groups. The lagrangian densities governing the dynamics of the (multi-) scalar fields are assumed to be general functions of the kinetic terms, whereas the gauge-invariant lagrangians are general functions of the field invariants. These functions are constrained by requirements of regularity, positivity of the energy and vanishing of the vacuum energy, defining what we call “admissible” models. In the scalar case we establish the general conditions which determine exhaustively the families of admissible lagrangian models supporting this kind of finite-energy solutions. We analyze some explicit examples of these different families, which are defined by the asymptotic and central behaviour of the fields of the corresponding particle-like solutions. From the variational analysis of the energy functional, we show that the admissibility constraints and the finiteness of the energy of the scalar solitons are necessary and sufficient conditions for their linear static stability against small charge-preserving perturbations. Furthermore, we perform a general spectral analysis of the dynamic evolution of the small perturbations around the statically stable solitons, establishing their dynamic stability. Next, we consider the case of many-components scalar fields, showing that the resolution of the particle-like field problem in this case reduces to that of the one-component case. The study of these scalar models is a necessary step in the analysis of the gauge fields. In this latter case, we add the requirement of parity invariance to the admissibility constraints. We determine the general conditions defining the families of admissible gauge-invariant models exhibiting finite-energy electrostatic spherically symmetric solutions which, unlike the (multi-) scalar case, are not always stable. The variational analysis of the energy functional leads now to supplementary restrictions to be imposed on the lagrangian densities in order to ensure the linear stability of the solitons. We establish a correspondence between any admissible soliton-supporting (multi-) scalar model and a family of admissible generalized gauge models supporting finite-energy electrostatic point-like solutions. Conversely, for each admissible soliton-supporting gauge-invariant model there is an associated unique admissible (multi-) scalar model with soliton solutions. This shows the exhaustive character of the admissibility and stability conditions in determining the class of soliton-supporting generalized gauge models. The usual Born–Infeld electrodynamic theory and its non-abelian extensions are shown to be (very particular) examples of one of these families.

Constraining maximally supersymmetric membrane actions

We study the recent construction of maximally supersymmetric field theory Lagrangians in three spacetime dimensions that are based on algebras with a triple product. Assuming that the algebra has a positive definite metric compatible with the triple product, we prove that the only non-trivial examples are either the well known case based on a four dimensional algebra or direct sums thereof.