Narrow Results

This paper is devoted to studying symmetries of k-symplectic Hamiltonian and Lagrangian first-order classical field theories. In particular, we define symmetries and Cartan symmetries and study the problem of associating conservation laws to these symmetries, stating and proving Noether's theorem in different situations for the Hamiltonian and Lagrangian cases. We also characterize equivalent Lagrangians, which lead to an introduction of Lagrangian gauge symmetries, as well as analyzing their relation with Cartan symmetries.

In terms of the quantitative causal principle, this paper obtains a general variational principle, gives unified expressions of the general, Hamilton, Voss, Hölder, Maupertuis–Lagrange variational principles of integral style, the invariant quantities of the general, Voss, Hölder, Maupertuis–Lagrange variational principles are given, finally the Noether conservation charges of the general, Voss, Hölder, Maupertuis–Lagrange variational principles are deduced, and the intrinsic relations among the invariant quantities and the Noether conservation charges of all the integral variational principles are achieved.

Given the $\beta$ functions of the closed string sigma model up to one loop in ${\alpha }^{\prime }$, the effective action implements the condition $\beta =0$ to preserve conformal symmetry at quantum level. One of the more powerful and striking results of string theory is that this effective action contains Einstein gravity as an emergent dynamics in space–time. We show from the $\beta$ functions and its relation with the equations of motion of the effective action that the differential identities are the Noether identities associated with the effective action and its gauge symmetries. From here, we reconstruct the gauge and space–time symmetries of the effective action. In turn, we can show that the differential identities are the contracted Bianchi identities of the field strength $H$ and Riemann tensor $R$. Next, we apply the same ideas to DFT. Taking as starting point that the generalized $\beta$ functions in DFT are proportional to the equations of motion, we construct the generalized differential identities in DFT. Relating the Noether identities with the contracted Bianchi identities of DFT, we were able to reconstruct the generalized gauge and space–time symmetries. Finally, we recover the original $\beta$ functions, effective action, differential identities, and symmetries when we turn off the $\tilde{x}$ space–time coordinates from DFT.

In this letter, we briefly recall the quantization of CP4f¹ nonlinear σ model coupling with non-Abelian Chern–Simons fields in Faddeev–Senjanovic (FS) path-integral quantization formalism, and discuss the quantum symmetries in this system through the quantum-extended Noether theorem. We obtain interesting and important results, that is the fractional spins of the constrained system are from the orbitally rotational motion of magneton of the non-Abelian Chern–Simons fields, that is, from the interaction between magnetons coupling to the propagator and magnetons coupling to the fixed lattice, with the former orbitally rotating around the latter. This means that anomalous spins are from the interactions among magnetons of Chern–Simons fields.

We study various definitions of the gravitational field energy based on the usage of isometric embeddings in the Regge–Teitelboim approach. For the embedding theory, we consider the coordinate translations on the surface as well as the coordinate translations in the flat bulk. In the latter case, the independent definition of gravitational energy–momentum tensor appears as a Noether current corresponding to global inner symmetry. In the field-theoretic form of this approach (splitting theory), we consider Noether procedure and the alternative method of energy–momentum tensor defining by varying the action of the theory with respect to flat bulk metric. As a result, we obtain energy definition in field-theoretic form of embedding theory which, among the other features, gives a nontrivial result for the solutions of embedding theory which are also solutions of Einstein equations. The question of energy localization is also discussed.

These notes provide a detailed catalog of surface charge formulas for different classes of gravity theories. The present catalog reviews and extends the existing literature on the topic. Part of the focus is on reviewing the method to compute quasi-local surface charges for gauge theories in order to clarify conceptual issues and their range of applicability. Many surface charge formulas for gravity theories are expressed in metric, tetrads-connection, Chern–Simons connection, and even BF variables. For most of them, the language of differential forms is exploited and contrasted with the more popular metric components language. The gravity theory is coupled with matter fields as scalar, Maxwell, Skyrme, Yang–Mills, and spinors. Furthermore, three examples with ready-to-download notebook codes, show the method in full action. Several new results are highlighted through the notes.

In contrast with QFT, classical field theory can be formulated in strict mathematical terms of fibre bundles, graded manifolds and jet manifolds. Second Noether theorems provide BRST extension of this classical field theory by means of ghosts and antifields for the purpose of its quantization.

We calculate the relative conserved currents, superpotentials and conserved quantities between two homogeneous and isotropic universes. In particular, we prove that their relative "energy" (defined as the conserved quantity associated to cosmic time coordinate translations for a comoving observer) is vanishing and so are the other conserved quantities related to a Lie subalgebra of vector fields isomorphic to the Poincaré algebra. These quantities are also conserved in time. We also find a relative conserved quantity for such a kind of solution which is conserved in time though non-vanishing. This example provides at least two insights in the theory of conserved quantities in General Relativity. First, the contribution of the cosmological matter fluid to the conserved quantities is carefully studied and proved to be vanishing. Second, we explicitly show that our superpotential (that happens to coincide with the so-called KBL potential although it is generated differently) provides strong conservation laws under much weaker hypotheses than the ones usually required. In particular, the symmetry generator is not needed to be Killing (nor Killing of the background, nor asymptotically Killing), the prescription is quasi-local and it works fine in a finite region too and no matching condition on the boundary is required.

Noether theorem plays a central role in linking symmetries and first integrals in Lagrangian mechanics. The situation is different in the nonholonomic context, but in the last decades there have been several extensions of Noether theorem to the nonholonomic setting. We provide an overview of this subject which is as elementary as possible.

In this paper, we discuss possible extensions of the concept of the Cartan form of classical mechanics to higher-order mechanics on manifolds, higher-order field theory on jet bundles and to parametric variational problems on slit tangent bundles and on bundles of nondegenerate velocities. We present a generalization of the Cartan form, known as a Lepage form, and basic properties of the Lepage forms. Both earlier and recent examples of differential forms generalizing the Cartan form are reviewed.

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 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 is realized as the mapping between a (partial) k-jet bundle and the extended dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. The invariant (variational) form of the balance system corresponding to a constitutive relation 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 is studied and it is shown that the symmetry group of acts on the sheaf of solutions of balance system . 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.

This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals.

In this paper, we will present Lagrangian and Hamiltonian $k$-symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will obtain new kinds of conservation laws for $k$-symplectic Hamiltonian systems and $k$-symplectic Lagrangian systems.

In this paper, Noether symmetry and Killing symmetry analyses of the curved traversable wormholes of $(3+1)$-dimensional spacetime metric in a Riemannian space are discussed. Moreover, a Lie algebra analysis is shown. Using the first and second Cartan’s structure equations, we find connection forms and then the curvature 2-forms are obtained. Finally, the Ricci scalar tensor and the components of Einstein curvature are computed.

Coherently with the principle of analogy suggested by Dirac, we describe a general setting for reducing a classical dynamics, and the role of the Noether theorem — connecting symmetries with constants of the motion — within a reduction. This is the first of two papers, and it focuses on the reduction within the Poisson and the symplectic formalism.

Teleparallel gravity has been formulated in the symplectic-geometrical fashion. For that purpose, the symplectic potential and the pre-symplectic structure on the manifold of all solutions of the field equation of teleparallel gravity have been firstly constructed from the Lagrangian density of the theory. That the obtained pre-symplectic structure is a symplectic structure also has been proved. It has been shown that the symplectic potential obtained from the Lagrangian density up to a factor is equal to the superpotential of teleparallel gravity. The Hamiltonian formulation of teleparallel gravity has been derived. Furthermore, the Poisson bracket between arbitrary two classical observables (i.e. real-valued functional) defined on the symplectic manifold of all solutions of the field equation of teleparallel gravity has constructed. The Hamiltonian vector field associated to an observable has been obtained. The formulation of Noether theorem in the symplectic formulation of teleparallel gravity has been investigated.

We review an algorithm, in the context of gauge and gravity theories described by differential forms, to read off the symmetries of a physical system out of its action, which was originally proposed in [C. Corral and Y. Bonder, Symmetry algebra in gauge theories of gravity, Class. Quantum Grav. 36 (2019) 045002]. In particular, we study the interplay between gauge symmetries and a gauge-covariant version of diffeomorphisms dubbed local translations. We focus on the role of boundary terms and conserved charges that were not discussed in the original proposal. We provide a review of the formalism and analyze several examples of classical theories, including gauge theories coupled to matter fields in flat space and gravitational theories in different dimensions.

In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries [1], mixed heavenly equation and Husain equation appear as closely related canonical equations admitting partner symmetries. Here for the mixed heavenly equation and Husain equation, formulated in a two-component form, we present recursion operators, Lax pairs of Olver–Ibragimov–Shabat type and discover their Lagrangians, symplectic and bi-Hamiltonian structure. We obtain all point and second-order symmetries, integrals and bi-Hamiltonian representations of these systems and their symmetry flows together with infinite hierarchies of nonlocal higher symmetries.

We introduce exterior differential systems associated with Euler-Lagrange equations on fibred manifolds and investigate their symmetries and conservation laws.

In this talk we are presenting the systems of balance equations of Continuum Thermodynamics (balance systems) in a variational form using Poincare-Cartan formalism. A constitutive relation of a balance system is realized as a mapping between a k-jet bundle J^k(π) and the extended dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system is studied, the class of admissible variations is defined. A case of semi-Lagrangian balance systems is characterized in geometrical terms. Action of automorphisms of the bundle π on the constitutive mappings is studied and it is shown that the symmetry group of acts on the space of solutions of balance system . Suitable version of Noether Theorem for an action of a symmetry group is presented together with the special form for semi-Lagrangian balance systems and examples of energy momentum and gauge symmetries balance laws.