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Noether symmetries and conserved quantities for Birkhoffian systems with time delay

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Abstract

The Noether symmetries and conserved quantities for Birkhoffian systems with time delay are proposed and studied. First, the Pfaff–Birkhoff principle with time delay is proposed, and Birkhoff’s equations with time delay are obtained. Second, based on the invariance of the Pfaff action with time delay under a group of infinitesimal transformations, the Noether symmetric transformations and the Noether quasisymmetric transformations of the system are defined, and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theorems for Birkhoffian systems with time delay are established. Some examples are given to illustrate the application of the results.

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Acknowledgments

This study is supported by the National Natural Science Foundation of China (Grant Nos.10972151 and 11272227), the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province (No. CXZZ13_0853), and the Innovation Program for Postgraduate of Suzhou University of Science and Technology (No. SKCX13S_051).

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  1. College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou , 215009, People’s Republic of China

    Xiang-Hua Zhai

  2. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou , 215011, People’s Republic of China

    Yi Zhang

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  1. Xiang-Hua Zhai
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Correspondence to Yi Zhang.

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Zhai, XH., Zhang, Y. Noether symmetries and conserved quantities for Birkhoffian systems with time delay. Nonlinear Dyn 77, 73–86 (2014). https://doi.org/10.1007/s11071-014-1274-8

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