In this paper, we research Noether’s theorems of fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivatives. First, the generalized Pfaff–Birkhoff principle and Birkhoff’s equations with classical and combined Caputo derivatives are given. Then, in the case of without and with transforming time, respectively, we obtain two kinds of Noether symmetry and their conserved quantities by the method of time re-parameterization. Finally, by taking case of generalized Birkhoffian systems, we study the relationship between Noether symmetry and Mei symmetry. It is concluded that the generalized Birkhoff equations, Noether identity and Noether conserved quantity obtained by using the generalized Pfaff–Birkhoff principle based on the action functional with dynamical functions after infinitesimal transformation are completely consistent with the criterion equation, structural equation and Mei conserved quantity of the fractional generalized Birkhoffian system, respectively.

中文翻译：

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根据经典和组合 Caputo 导数的分数广义 Birkhoffian 系统的 Noether 对称性

在本文中，我们根据经典和组合 Caputo 导数研究分数广义 Birkhoffian 系统的 Noether 定理。首先，给出了广义 Pfaff-Birkhoff 原理和具有经典和组合 Caputo 导数的 Birkhoff 方程。然后，分别在不变换时间和有变换时间的情况下，通过时间重参数化的方法得到两种Noether对称及其守恒量。最后，以广义Birkhoffian系统为例，研究了Noether对称与Mei对称之间的关系。得出的结论是，广义 Birkhoff 方程，