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Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations

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

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

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

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  • 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.

    Mathematics Subject Classification: Primary: 26A33, 70-XX, 70G65, 37M15, 49-XX.

    Citation:

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  • Figure 1.  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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