Chin. Phys. B, 2014, Vol. 23(12): 124502    DOI: 10.1088/1674-1056/23/12/124502
 ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS Prev   Next

Noether's theorems of a fractional Birkhoffian system within Riemann–Liouville derivatives

Zhou Yana, Zhang Yib
a College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China;
b College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China
Abstract

The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann–Liouville fractional derivatives. Firstly, the fractional Birkhoff's equations and the corresponding transversality conditions are given. Secondly, from special to general forms, Noether's theorems of a standard Birhoffian system are given, which provide an approach and theoretical basis for the further research on the Noether symmetry of the fractional Birkhoffian system. Thirdly, the invariances of the fractional Pfaffian action under a special one-parameter group of infinitesimal transformations without transforming the time and a general one-parameter group of infinitesimal transformations with transforming the time are studied, respectively, and the corresponding Noether's theorems are established. Finally, an example is given to illustrate the application of the results.

Received:  13 May 2014      Published:  15 December 2014
 PACS: 45.10.Hj (Perturbation and fractional calculus methods) 45.20.Jj (Lagrangian and Hamiltonian mechanics) 11.30.Na (Nonlinear and dynamical symmetries (spectrum-generating symmetries)) 02.30.Xx (Calculus of variations)
Fund:

Project supported by the National Natural Science Foundation of China (Grant Nos. 10972151 and 11272227) and the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province, China (Grant No. CXZZ110949).

Corresponding Authors:  Zhang Yi     E-mail:  weidiezh@gmail.com