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Noether’s-type theorems on time scales

Abstract : 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 ∆ 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 in [Z. Bartosiewicz and D.F.M. 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 as derived by Bartosiewicz, Martins and Torres is in [Z. Bartosiewicz, N. Martins and D.F.M. Torres. The second Euler-Lagrange equation of variational calculus on time scales. Eur. J. Control 17 (2011), no. 1, 9-18]. Using the Caputo duality principle introduced in [C. Caputo, Time Scales: From Nabla Calculus to Delta Calculus and Vice Versa via Duality, International Journal of Difference Equations 2540, 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.
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Contributor : Khaled Hariz-Belgacem <>
Submitted on : Tuesday, July 13, 2021 - 1:07:16 AM
Last modification on : Thursday, July 15, 2021 - 3:35:15 AM

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Baptiste Anerot, Jacky Cresson, Khaled Hariz Belgacem, Frederic Pierret. Noether’s-type theorems on time scales. Journal of Mathematical Physics, American Institute of Physics (AIP), 2020, 61 (11), pp.113502. ⟨10.1063/1.5140201⟩. ⟨hal-03284877⟩

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