This paper deals with the Noether symmetry and conserved quantity for Birkhoffian system with delayed arguments in version of time scales. Firstly, the Pfaff–Birkhoff principle with delayed arguments in which the variables are defined on time scales is established, through which the corresponding time-delayed Birkhoff’s equations on time scales are derived. Secondly, the invariance of the Pfaff action with delayed arguments on time scales under the infinitesimal transformations of one-parameter group is analyzed. Further, the Noether theorem for the system which reveals the relationship between the symmetry and conserved quantity is proved by applying a technique of time reparameterization and performing the linear change of the variables with delayed arguments. Finally, the results incorporate the classical Noether-type conserved quantity, the discrete Noether-type conserved quantity and the quantum Noether-type conserved quantity. Also, an example illustrates the application of the results.

本文讨论了时标版本中具有延迟参数的伯克霍夫系统的Noether对称性和守恒量。首先，建立了带有延迟参数的Pfaff-Birkhoff原理，其中在时标上定义了变量，由此推导了在时标上相应的时滞Birkhoff方程。其次，分析了在一参数组无穷小变换下具有时标的时滞参数的Pfaff动作的不变性。此外，通过应用时间重新参数化技术并执行带有延迟参数的变量的线性变化，证明了揭示对称性与守恒量之间关系的系统Noether定理。最后，结果结合了经典的Noether型守恒量，离散的Noether型守恒量和量子Noether型守恒量。另外，一个示例说明了结果的应用。