# American Institute of Mathematical Sciences

doi: 10.3934/jgm.2021012
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## Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations

 1 Laboratoire de Mathématiques et de leurs Applications, Université de Pau et des Pays de l'Adour, Pau, CNRS 5142, France 2 Departamento de Matemáticas Aplicadas a la Ingeniería Industrial, Universidad Politécnica de Madrid, José Gutiérrez Abascal 2, 28006 Madrid, Spain 3 Department of Mathematics, University of Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany

* Corresponding author: fernando.jimenez.alburquerque@upm.es

Received  February 2021 Early access July 2021

Fund Project: This work has been funded by the EPSRC project: "Fractional Variational Integration and Optimal Control"; ref: EP/P020402/1

We prove a Noether's theorem of the first kind for the so-called restricted fractional Euler-Lagrange equations and their discrete counterpart, introduced in [26,27], based in previous results [11,35]. Prior, we compare the restricted fractional calculus of variations to the asymmetric fractional calculus of variations, introduced in [14], and formulate the restricted calculus of variations using the discrete embedding approach [12,18]. The two theories are designed to provide a variational formulation of dissipative systems, and are based on modeling irreversbility by means of fractional derivatives. We explicit the role of time-reversed solutions and causality in the restricted fractional calculus of variations and we propose an alternative formulation. Finally, we implement our results for a particular example and provide simulations, actually showing the constant behaviour in time of the discrete conserved quantities outcoming the Noether's theorems.

Citation: Jacky Cresson, Fernando Jiménez, Sina Ober-Blöbaum. Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021012
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##### References:
Top-Left: $\rho = 0$; this is the classical dynamics of the non-fractional harmonic oscillator (without dissipation). Top-Right: $\rho = 0.2$, $\alpha = 0.2$. Bottom-Left: $\rho = 0.2$, $\alpha = 0.4$. Bottom-Right: $\rho = 0.2$, $\alpha = 0.5$; this is the case of the usual linearly damped harmonic oscillator. Naturally, we observe that the dissipation increases as $\alpha$ increases for equivalent $\rho$
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