A variational principle of Herglotz type with a Lagrangian that depends on fractional derivatives of both real and complex orders is formulated, and the invariance of this principle under the action of a local group of symmetries is determined. By the Noether theorem the conservation law for the corresponding fractional Euler–Lagrange equation is obtained. A sequence of approximations of a fractional Euler–Lagrange equation by systems of integer order equations is used for the construction of a sequence of conservation laws which, with certain assumptions, weakly converge to the one for the basic Herglotz variational principle. Results are illustrated by two examples.

提出了具有拉格朗日的Herglotz型变分原理，该变分原理取决于实数阶和复数阶的分数导数，并且确定了该原理在局部对称性组的作用下的不变性。通过Noether定理，得到了相应的分数阶Euler-Lagrange方程的守恒律。分数阶欧拉-拉格朗日方程与整数阶方程组的近似序列用于构造一系列守恒律，在一定的假设下，这些守恒律弱收敛于基本的Herglotz变分原理。结果用两个例子说明。