In this paper, we obtain the Euler-Lagrange equations for different kind of variational problems with the Lagrangian function containing the Riesz-Hilfer fractional derivative. Since the Riesz-Hilfer fractional derivative is a generalization for the Riesz-Riemann-Liouville and the Riesz-Caputo derivative, then our results generalize many recent works in which the Lagrangian function involving the Riesz-Riemann-Liouville or the Riesz-Caputo derivative. We also study the problem in the presence of delay derivatives and establish a version for Noether theorem in the Riesz-Hilfer sense. In order to achieve our aims we derive some formulas to integration by parts for the Riesz-Hilfer fractional derivative. In the last section, examples are given to clarify the possibility of applicability of our results. In order to clarify the significant conclusions of the paper, we refer to our techniques enable to study many different variational problems containing the Riesz-Hilfer derivative.

在本文中，我们使用包含Riesz-Hilfer分数导数的Lagrangian函数，获得了针对不同类型变分问题的Euler-Lagrange方程。由于Riesz-Hilfer分数导数是Riesz-Riemann-Liouville和Riesz-Caputo导数的推广，因此我们的结果推广了许多最近的工作，其中涉及Riesz-Riemann-Liouville或Riesz-Caputo导数的拉格朗日函数。我们还研究了存在延迟导数的问题，并建立了Riesz-Hilfer意义上的Noether定理的版本。为了实现我们的目标，我们为Riesz-Hilfer分数导数推导了部分积分公式。在最后一部分中，给出了一些例子以阐明我们的结果适用的可能性。