Noether’s theorem for fractional variational problems of variable order

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Tatiana Odzijewicz ◽  
Agnieszka Malinowska ◽  
Delfim Torres
Keyword(s):  
Optimality Condition ◽  
Variational Problems ◽  
Time Variable ◽  
Variable Order ◽  
Necessary Optimality Condition ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Fractional Variational Problems ◽  
Derivatives Of

AbstractWe 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.

Noether’s Symmetries and Its Inverse for Fractional Logarithmic Lagrangian Systems

2019 ◽  
Vol 7 (1) ◽  
pp. 90-98 ◽  
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shuli Xu
Keyword(s):  
Variational Problems ◽  
Conserved Quantity ◽  
Inverse Theorem ◽  
Lagrangian Systems ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Hamilton Principle ◽  
Fractional Variational Problems

Abstract In this paper, Noether’s theorem and its inverse theorem are proved for the fractional variational problems based on logarithmic Lagrangian systems. The Hamilton principle of the systems is derived. And the definitions and the criterions of Noether’s symmetry and Noether’s quasi-symmetry of the systems based on logarithmic Lagrangians are given. The intrinsic relation between Noether’s symmetry and the conserved quantity is established. At last an example is given to illustrate the application of the results.

scholarly journals Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether’s Theorem

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shougui Li
Keyword(s):  
Fractional Derivative ◽  
Sufficient Conditions ◽  
Variational Problems ◽  
Necessary And Sufficient Conditions ◽  
Optimal Conditions ◽  
Lagrange Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Necessary And Sufficient

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented.

scholarly journals An Euler-Lagrange Equation only Depending on Derivatives of Caputo for Fractional Variational Problems with Classical Derivatives

2020 ◽  
Vol 8 (2) ◽  
pp. 590-601
Author(s):  
Melani Barrios ◽  
Gabriela Reyero
Keyword(s):  
Exact Solution ◽  
Variational Problem ◽  
Lagrange Equation ◽  
Integral Form ◽  
Variational Problems ◽  
The Other ◽  
Isoperimetric Problems ◽  
Caputo Derivatives ◽  
Fractional Variational Problems ◽  
Derivatives Of

In this paper we present advances in fractional variational problems with a Lagrangian depending on Caputofractional and classical derivatives. New formulations of the fractional Euler-Lagrange equation are shown for the basic and isoperimetric problems, one in an integral form, and the other that depends only on the Caputo derivatives. The advantage is that Caputo derivatives are more appropriate for modeling problems than the Riemann-Liouville derivatives and makes the calculations easier to solve because, in some cases, its behavior is similar to the behavior of classical derivatives. Finally, anew exact solution for a particular variational problem is obtained.

scholarly journals Noether’s theorem for non-smooth extremals of variational problems with time delay

2013 ◽  
Vol 93 (1) ◽  
pp. 153-170 ◽  
Author(s):  
Gastão S.F. Frederico ◽  
Tatiana Odzijewicz ◽  
Delfim F.M. Torres
Keyword(s):  
Time Delay ◽  
Variational Problems ◽  
Noether’S Theorem ◽  
Noether's Theorem

scholarly journals Noether’s Theorem and Symmetry

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 744 ◽  
Author(s):  
Amlan Halder ◽  
Andronikos Paliathanasis ◽  
Peter Leach
Keyword(s):  
Boundary Function ◽  
Original Form ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Independent Variables ◽  
Point Transformations ◽  
Original Presentation ◽  
Parametric Functions ◽  
Nonlocal Terms ◽  
Derivatives Of

In Noether’s original presentation of her celebrated theorem of 1918, allowance was made for the dependence of the coefficient functions of the differential operator, which generated the infinitesimal transformation of the action integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades, this diminution of the power of Noether’s theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether’s theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence on the independent variables.

scholarly journals Fractional Variational Problems Depending on Fractional Derivatives of Differentiable Functions with Application to Nonlinear Chaotic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Matheus Jatkoske Lazo
Keyword(s):  
Dynamical Systems ◽  
Fractional Derivatives ◽  
Chaotic Systems ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Necessary Condition ◽  
Chaotic Dynamical Systems ◽  
Differentiable Functions ◽  
Fractional Variational Problems ◽  
Derivatives Of

We formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously known results in the literature and enables us to construct simple Lagrangians for nonlinear systems. As examples of application, we obtain Lagrangians for some chaotic dynamical systems.

LAGRANGIAN AND HAMILTONIAN FORMALISM ON THE h-DEFORMED QUANTUM PLANE

2000 ◽  
Vol 12 (05) ◽  
pp. 711-724
Author(s):  
SUNGGOO CHO ◽  
SANG-JUN KANG ◽  
KWANG SUNG PARK
Keyword(s):  
Hamiltonian Formalism ◽  
Quantum Plane ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Commutation Relations ◽  
Quantum Deformations ◽  
Initial Choice ◽  
Derivatives Of ◽  
Noncommuting Coordinates

It is known that there are only two quantum planes which are covariant under the quantum deformations of GL(2) admitting a central determinant. Contrary to the q-deformed quantum plane, the h-deformed quantum plane has a structure suitable for defining time derivatives and variations as closely as in the ordinary plane. From these we derive differential calculi including the skew-derivatives of Wess–Zumino as well as variational calculi on the quantum plane. These calculi enable us to generalize the Lagrangian and Hamiltonian formalism on the ordinary plane to the quantum plane. In particular, we construct commutation relations between noncommuting coordinates and momenta which do not depend on the initial choice of Lagrangian. We also discuss the symmetry of a Lagrangian and Noether's theorem.

scholarly journals A non-standard class of variational problems of Herglotz type

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Natália Martins
Keyword(s):  
Variational Problem ◽  
Optimality Condition ◽  
Unknown Function ◽  
Variational Problems ◽  
Standard Theory ◽  
Necessary Optimality Condition ◽  
Natural Boundary ◽  
Special Cases ◽  
Independent Variable ◽  
Natural Boundary Conditions

<p style='text-indent:20px;'>In this paper, we extend the variational problem of Herglotz considering the case where the Lagrangian depends not only on the independent variable, an unknown function <inline-formula><tex-math id="M1">\begin{document}$ x $\end{document}</tex-math></inline-formula> and its derivative and an unknown functional <inline-formula><tex-math id="M2">\begin{document}$ z $\end{document}</tex-math></inline-formula>, but also on the end points conditions and a real parameter. Herglotz's problems of calculus of variations of this type cannot be solved using the standard theory. Main results of this paper are necessary optimality condition of Euler-Lagrange type, natural boundary conditions and the Dubois-Reymond condition for our non-standard variational problem of Herglotz type. We also prove a necessary optimality condition that arises as a consequence of the Lagrangian dependence of the parameter. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases. In addition, two examples are given in order to illustrate our results.</p>

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