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

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.