Abstract This paper examines a new Noether theorem for Hamiltonian systems with Caputo Δ derivatives based on fractional time-scales calculus, which overcomes the difficulties unified to study the Noether theorems of fractional continuous systems and fractional discrete systems. To begin with, the fractional time-scales definitions and properties with Caputo Δ derivatives are introduced. Next, the fractional time-scales Hamilton canonical equations with Caputo Δ derivatives are formulated. For the fractional time-scales Hamiltonian system, the definitions and criteria of Noether symmetries without transforming time and with transforming time are given, respectively. Furthermore, the corresponding Noether theorem without transforming time and its Noether theorem with transforming time are obtained. The latter one can reduce to the time-scales Noether theorem with Δ derivatives or the fractional Noether theorem with Caputo derivatives for Hamiltonian systems. Finally, the fractional time-scales damped oscillator and Kepler problem are taken as examples to verify the correctness of the results.

中文翻译：

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分数时间尺度 Noether 定理与哈密顿系统的 Caputo Δ 导数

摘要 本文研究了一种新的基于分数阶时间尺度微积分的具有 Caputo Δ 导数的哈密顿系统的诺特定理，它克服了统一研究分数连续系统和分数离散系统的诺特定理的困难。首先，介绍了 Caputo Δ 导数的分数时间尺度定义和属性。接下来，制定了具有 Caputo Δ 导数的分数时间尺度 Hamilton 正则方程。对于分数时间尺度的哈密顿系统，分别给出了不带变换时间和带变换时间的Noether对称性的定义和判据。进而得到相应的不带变换时间的Noether定理及其带变换时间的Noether定理。对于哈密顿系统，后者可以简化为具有 Δ 导数的时间尺度 Noether 定理或具有 Caputo 导数的分数 Noether 定理。最后，以分数时间尺度阻尼振荡器和开普勒问题为例，验证了结果的正确性。