Fractional Noether Research Articles

Fractional calculus of variations of several independent variables

We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians depending on generalized partial integrals and derivatives. A generalized fractional Noether's theorem, a formulation of Dirichlet's principle and an uniqueness result are given.

Noether’s theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space

This paper focuses on studying Noether’s theorem in phase space for fractional variational problems from extended exponentially fractional integral introduced by El-Nabulsi. Both holonomic and nonholonomic systems are studied. First, the fractional variational problem from extended exponentially fractional integral, as well as El-Nabulsi–Hamilton’s canonical equations are established; second, the definitions and criteria of fractional Noether symmetric transformations and fractional Noether quasi-symmetric transformations are presented which are based on the invariance of El-Nabulsi–Hamilton action under the infinitesimal group transformations; finally, the fractional Noether’s theorem is established, which reveals the inner relationship between a fractional Noether symmetry and a fractional conserved quantity.

Symmetries and conserved quantities for fractional action-like Pfaffian variational problems

The fractional Pfaffian variational problems and the fractional Noether theory are studied under a fractional model presented by El-Nabulsi. Firstly, the fractional action-like Pfaffian variational problem is presented, the El-Nabulsi–Pfaff–Birkhoff–d’Alembert fractional principle is established, then the El-Nabulsi–Birkhoff fractional equations are derived; secondly, the definitions and criteria of the fractional Noether symmetric transformations are given, which are based on the invariance of El-Nabulsi–Pfaffian action under the infinitesimal transformations of group, then the inner relationship between a fractional Noether symmetry and a fractional conserved quantity is established; finally, two examples are given to illustrate the application of the results.

Hamilton formalism and noether symmetry for mechanico—electrical systems with fractional derivatives

This paper presents extensions to the traditional calculus of variations for mechanico—electrical systems containing fractional derivatives. The Euler—Lagrange equations and the Hamilton formalism of the mechanico—electrical systems with fractional derivatives are established. The definition and the criteria for the fractional generalized Noether quasi-symmetry are presented. Furthermore, the fractional Noether theorem and conseved quantities of the systems are obtained by virtue of the invariance of the Hamiltonian action under the infinitesimal transformations. An example is presented to illustrate the application of the results.

A continuous/discrete fractional Noether’s theorem

We prove a fractional Noether’s theorem for fractional Lagrangian systems invariant under a symmetry group both in the continuous and discrete cases. This provides an explicit conservation law (first integral) given by a closed formula which can be algorithmically implemented. In the discrete case, the conservation law is moreover computable in a finite number of steps.

Maxwell's equations and electromagnetic Lagrangian density in fractional form

The fractional form of the electromagnetic Lagrangian density is presented using the Riemann-Liouville fractional derivative. Agrawal procedure is employed to obtain Maxwell's equations in fractional form. The Hamilton equations of motion resulting from the electromagnetic Lagrangian density are obtained. Conserved quantities, such as energy density, momentum, and Poynting's vector, are also derived using fractional Noether's theorem.

Symmetry theories of Hamiltonian systems with fractional derivatives

The Noether and Lie symmetries as well as the conserved quantities of Hamiltonian system with fractional derivatives are established. The definitions and criteria for the fractional symmetrical transformations and quasi-symmetrical transformations in the Noether sense of Hamiltonian system are first discussed. Then, using the invariance of Hamiltonian action under the infinitesimal transformations with respect to time, generalized coordinates and generalized momentums, the fractional Noether theorem of the system is obtained. Further, the Lie symmetry and conserved quantity of the system are acquired. Two examples are presented to illustrate the application of the results.

Fractional Noether’s theorem in the Riesz–Caputo sense

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.

Fractional embedding of differential operators and Lagrangian systems

This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the stochastic embedding theory developed with Darses [C. R. Acad. Sci. Ser. I: Math 342, 333 (2006); (preprint IHES 06/27, p. 87, 2006)], we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivatives. For Lagrangian systems, our method provides a fractional Euler-Lagrange equation. We prove, developing the corresponding fractional calculus of variations, that such equation can be derived via a fractional least-action principle. We then obtain naturally a fractional Noether theorem and a fractional Hamiltonian formulation of fractional Lagrangian systems. All these constructions are coherents, i.e., the embedding procedure is compatible with the fractional calculus of variations. We then extend our results to cover the Ostrogradski formalism. Using the fractional embedding and following a previous work of Riewe [Phys. Rev. E 53, 1890 (1996); Phys. Rev. E 55, 3581 (1997)], we obtain a fractional Ostrogradski formalism which allows us to derive nonconservative dynamical systems via a fractional generalized least-action principle. We also discuss the Whittaker equations and obtain a fractional Lagrangian formulation. Last, we discuss the fractional embedding of continuous Lagrangian systems. In particular, we obtain a fractional Lagrangian formulation of the classical fractional wave equation introduced by Schneider and Wyss [J. Math. Phys. 30, 134 (1989)] as well as the fractional diffusion equation.