Noether’s-type theorems on time scales

We prove a time scales version of the Noether theorem relating group of symmetries and conservation laws in the framework of the shifted and nonshifted Δ calculus of variations. Our result extends the continuous version of the Noether theorem as well as the discrete one and corrects a previous statement of Bartosiewicz and Torres [“Noether’s theorem on time scales,” J. Math. Anal. Appl. 342(2), 1220–1226 (2008)]. This result implies also that the second Euler–Lagrange equation on time scales is derived by Bartosiewicz, Martins, and Torres [“The second Euler–Lagrange equation of variational calculus on time scales,” Eur. J. Control 17(1), 9–18 (2011)]. Using the Caputo duality principle introduced by Caputo, [“Time scales: From Nabla calculus to delta calculus and vice versa via duality, Int. J. Differ. Equations 5, 25–40, (2010)], we provide the corresponding Noether theorem on time scales in the framework of the shifted and nonshifted ∇ calculus of variations. All our results are illustrated with numerous examples supported by numerical simulations.