NOETHER'S THEOREM ON TIME SCALES

We prove a time scales version of the Noether's theorem relating group of symmetries and conservation laws in the framework of the shifted and nonshifted ∆ calculus of variations. Our result extends the continuous version of the Noether's theorem as well as the discrete one and corrects a previous statement of Bartosiewicz and Torres in [3]. This result implies also that the second Euler-Lagrange equation on time scales as derived by Bartosiewicz, Martins and Torres is false. Using the Caputo duality principle, we provide the corresponding Noether's 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.