Acta Scientiarum Naturalium Universitatis Pekinensis ›› 2016, Vol. 52 ›› Issue (4): 658-668.DOI: 10.13209/j.0479-8023.2016.068

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Noether Symmetry and Conserved Quantity for Fractional Birkhoffian Systems in Terms of Riesz Derivatives

ZHANG Yi1, ZHOU Yan2   

  1. 1. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011
    2. Suzhou Industrial Park Loufeng School, Suzhou 215021
  • Received:2015-10-07 Revised:2016-02-10 Online:2016-07-20 Published:2016-07-20
  • Contact: ZHANG Yi, E-mail: zhy(at)


张毅1, 周燕2   

  1. 1. 苏州科技大学土木工程学院, 苏州 215011
    2. 苏州市工业园区娄葑学校, 苏州 215021
  • 通讯作者: 张毅, E-mail: zhy(at)
  • 基金资助:
    国家自然科学基金(10972151, 11272227, 11572212)和江苏省普通高校研究生科研创新计划(CXZZ11_0949)资助


The Noether symmetry and the conserved quantity for a fractional Birkhoffian system in terms of Riesz fractional derivatives are studied. The fractional Pfaff variational problems are presented and the fractional Birkhoff’s equations are established within Riesz-Riemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives, respectively. Based on the invariance of the Pfaff action under the infinitesimal transformations, the Noether theorems for the fractional Birkhoffian system are given. The proof of the Noether theorem is done in two steps: first, the Noether theorem under a special one-parameter group of infinitesimal transformations without transforming the time is proved; second, by using a technique of time-reparameterization, the Noether theorem in its general form is obtained. Two examples are given to illustrate the application of the results.

Key words: fractional Birkhoffian system, Noether symmetry, fractional conserved quantity, Riesz fractional derivative


提出并研究Riesz分数阶导数下分数阶Birkhoff系统的Noether对称性与守恒量。分别在Riesz-Riemann-Liouville分数阶导数和Riesz-Caputo分数阶导数下, 建立分数阶Pfaff变分问题, 给出分数阶Birkhoff方程。基于分数阶Pfaff作用量在无限小变换下的不变性, 建立分数阶Birkhoff系统的Noether定理。定理的证明分成两步: 一是在时间不变的无限小变换下给出证明; 二是利用时间重新参数化技术得到一般情况下的分数阶Noether定理。最后举例说明结果的应用。

关键词: 分数阶Birkhoff系统, Noether对称性, 分数阶守恒量, Riesz分数阶导数

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