This paper presents a formulation of Noether's theorem for fractional classical fields.
We extend the variational formulations for fractional discrete systems to fractional field systems. By applying the variational principle to a fractional action $S$, we obtain the fractional Euler-Lagrange equations of motion. Considerations of the Noether's variational problem for discrete systems whose action is invariant under gauge transformations will be extended to fractional variational problems for classical fields. The conservation laws associated with fractional classical fields are derived. As an example we present the conservation laws for the fractional Dirac fields.
We prove a Noether's theorem for fractional variational problems with Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and optimal control are given.
Since the seminal work of Emmy Noether it is well know that all conservations laws in physics, \textrm{e.g.}, conservation of energy or conservation of momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether's theorem can not yields informations about constants of motion for non-conservative systems since it is not possible to formulate physically meaningful Lagragians for this kind of systems in classical calculus of variation. On the other hand, in recent years the fractional calculus of variation within Lagrangians dep…
We prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether's theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.
In the present work, by taking advantage of a so-called practical limitation of fractional derivatives, namely, the absence of a simple chain and Leibniz's rules, we proposed a generalized fractional calculus of variation where the Lagrangian function depends on fractional derivatives of differentiable functions. The Euler-Lagrange equation we obtained generalizes previously results and enables us to construct simple Lagrangians for nonlinear systems. Furthermore, in our main result, we formulate a Noether-type theorem for these problems that provides us with a means to obtain conservative qua…
A Lagrangian formulation with nonlocality is investigated in this paper. The nonlocality of the Lagrangian is introduced by a new nonlocal argument that is defined as a nonlocal residual satisfying the zero mean condition. The nonlocal Euler-Lagrangian equation is derived from the Hamilton's principle. The Noether's theorem is extended to this Lagrangian formulation with nonlocality. With the help of the extended Noether's theorem, the conservation laws relevant to energy, linear momentum, angular momentum and the Eshelby tensor are determined in the nonlocal elasticity associated with the mec…
Quasi-Noether differential systems are more general than variational systems and are quite common in mathematical physics. They include practically all differential systems of interest, at least those that have conservation laws. In this paper, we discuss quasi-Noether systems that possess infinite-dimensional (infinite) symmetries involving arbitrary functions of independent variables. For quasi-Noether systems admitting infinite symmetries with arbitrary functions of all independent variables, we state and prove an extension of the Second Noether Theorem. In addition, we prove that infinite …
We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, transversality conditions for free boundary value problems, and a generalized Noether type theorem.
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.
We consider Noether symmetries of the equations defined by the sections of characteristic line bundles of nondegenerate 1-forms and of the associated perturbed systems. It appears that this framework can be used for time-dependent systems with constraints and nonconservative forces, allowing a quite simple and transparent formulation of the momentum equation and the Noether theorem in their general forms.
Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined via the (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive the local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton-Jacobi equation applicable in the field theory then follows readily. In addition, we discuss th…
Symmetries and, in particular, Cartan (Noether) symmetries and conserved quantities (conservation laws) are studied for the multisymplectic formulation of first and second order Lagrangian classical field theories. Noether-type theorems are stated in this geometric framework. The concept of gauge symmetry and its geometrical meaning are also discussed in this formulation. The results are applied to study Noether and gauge symmetries for the multisymplectic description of the electromagnetic and the gravitational theory; in particular, the Einstein--Hilbert and the Einstein--Palatini approaches…
The C*-algebraic formulation of generic interacting quantum field theories, recently presented by Detlev Buchholz and one of the authors (KF), is enriched by a unitary version of the Master Ward Identity, which was postulated some time ago by Franz Marc Boas, Ferdinand Brennecke and two of us (MD,KF). It is shown that the corresponding axiom implies the validity of the time slice axiom. Moreover, it opens the way for a new approach to Noether's Theorem where it yields directly the unitaries implementing the symmetries. It also unravels interesting aspects of the role of anomalies in quantum fi…
We obtain Euler-Lagrange equations, transversality conditions and a Noether-like theorem for Herglotz-type variational problems with Lagrangians depending on generalized fractional derivatives. As an application, we consider a damped harmonic oscillator with time-depending mass and elasticity, and arbitrary memory effects.
We consider gravity in four dimensions in the vielbein formulation, where the fundamental variables are a tetrad $e$ and a SO(3,1) connection $ω$. We start with the most general action principle compatible with diffeomorphism invariance which includes, besides the standard Palatini term, other terms that either do not change the equations of motion, or are topological in nature. For our analysis we employ the covariant Hamiltonian formalism where the phase space $Γ$ is given by solutions to the equations of motion. We consider spacetimes that include a boundary at infinity, satisfying asymptot…
In a series of previous articles by the author, it was shown that one could effectively give a variational formulation to non-conservative mechanical systems, as well as ones that subject to non-holonomic constraints by starting with the virtual work functional instead of an action functional. In this article, it is shown that when one starts with a virtual work functional that is not exact, so it does not admit an action functional, instead of conservation laws the extension of Noether's theorem gives balance principles for non-conserved currents. Examples of this generalized Noether identity…
We consider the variational formulation for vertical slice models introduced in Cotter and Holm (Proc Roy Soc, 2013). These models have a Kelvin circulation theorem that holds on all materially-transported closed loops, not just those loops on isosurfaces of potential temperature. Potential vorticity conservation can be derived directly from this circulation theorem. In this paper, we show that this property is due to these models having a relabelling symmetry for every single diffeomorphism of the vertical slice that preserves the density, not just those diffeomorphisms that preserve the pote…
Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Prin…
In the paper the generalisation of classical rate independent plasticity using fractional calculus is presented. This new formulation is non-local due to properties of applied fractional differential operator during definition of kinematics. In the description small fractional strains assumption is hold together with additive decomposition of total fractional strains into elastic and plastic parts. Classical local rate independent plasticity is recovered as a special case.
We propose a new fractional derivative, the Hilfer-Katugampola fractional derivative. Motivated by the Hilfer derivative this formulation interpolates the well-known fractional derivatives of Hilfer, Hilfer-Hadamard, Riemann-Liouville, Hadamard, Caputo, Caputo-Hadamard, Liouville, Weyl, generalized and Caputo-type. As an application, we consider a nonlinear fractional differential equation with an initial condition using this new formulation. We show that this equation is equivalent to a Volterra integral equation and demonstrate the existence and uniqueness of solution to the nonlinear initia…
Fractional derivatives and integrations of non-integers orders was introduced more than three centuries ago but only recently gained more attention due to its application on nonlocal phenomenas. In this context, several formulations of fractional electromagnetic fields was proposed, but all these theories suffer from the absence of an effective fractional vector calculus, and in general are non-causal or spatially asymmetric. In order to deal with these difficulties, we propose a spatially symmetric and causal gauge invariant fractional electromagnetic field from a Lagrangian formulation. From…
In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then, the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractiona…
We consider some possible approaches to the fractional-order generalization of definition of variation (functional) derivative. Some problems of formulation of a fractional-order variational derivative are discussed. To give a consistent definition of the fractional-order variations, we use a fractional generalization of Gateaux differential.
Despite the fact that conserved currents have dimensions that are determined solely by dimensional analysis (and hence no anomalous dimensions), Nature abounds in examples of anomalous diffusion in which $L\propto t^γ$, with $γ\ne 1/2$, and heat transport in which the thermal conductivity diverges as $L^α$. Aside from breaking of Lorentz invariance, the true common link in such problems is an anomalous dimension for the underlying conserved current, thereby violating the basic tenet of field theory. We show here that the phenomenological non-local equations of motion that are used to describe …
A mathematical model for a one-phase change problem (particularly a Stefan problem) with a memory flux, is obtained. The hypothesis that the weighted sum of fluxes back in time is proportional to the gradient of temperature is considered. The model obtained involves fractional derivatives with respect on time in the sense of Caputo and in the sense of Riemann--Liouville. An integral relationship for the free boundary which is equivalent to the `fractional Stefan condition' is also obtained.
We present the ${\mathscr P}(φ)_2$ model on de Sitter space in the canonical formulation. We discuss the role of the Noether theorem and we provide explicit expressions for the energy-stress tensor of the interacting model.
