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.

中文翻译：

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时间尺度上的 Noether 型定理

我们在位移和非位移 Δ 变分演算的框架内证明了与对称群和守恒定律相关的诺特定理的时间尺度版本。我们的结果扩展了 Noether 定理的连续版本以及离散版本，并纠正了 Bartosiewicz 和 Torres 的先前陈述 [“时间尺度上的 Noether 定理”，J. Math。肛门。应用程序 342(2), 1220–1226 (2008)]。这个结果还意味着时间尺度上的第二个欧拉-拉格朗日方程是由 Bartosiewicz、Martins 和 Torres 推导出来的[“时间尺度上变分微积分的第二个欧拉-拉格朗日方程，”Eur。J. 控制 17(1), 9–18 (2011)]。使用 Caputo 引入的 Caputo 对偶原理，[“时间尺度：从 Nabla 演算到 delta 演算，反之亦然，通过对偶，Int。J. 不同。方程 5, 25–40, (2010)], 我们在移位和非移位 ∇ 变分演算的框架中提供了时间尺度上的相应诺特定理。我们所有的结果都用数值模拟支持的许多例子来说明。