Noether-Type Conserved Quantities on Time Scales for Birkhoffian Systems with Delayed Arguments

摘要

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.