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Chin. Phys. B, 2017, Vol. 26(8): 084501    DOI: 10.1088/1674-1056/26/8/084501

Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales

Jing Song(宋静)1, Yi Zhang(张毅)2
1 College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China;
2 College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China

This paper focuses on studying the Noether symmetry and the conserved quantity with non-standard Lagrangians, namely exponential Lagrangians and power-law Lagrangians on time scales. Firstly, for each case, the Hamilton principle based on the action with non-standard Lagrangians on time scales is established, with which the corresponding Euler-Lagrange equation is given. Secondly, according to the invariance of the Hamilton action under the infinitesimal transformation, the Noether theorem for the dynamical system with non-standard Lagrangians on time scales is established. The proof of the theorem consists of two steps. First, it is proved under the infinitesimal transformations of a special one-parameter group without transforming time. Second, utilizing the technique of time-re-parameterization, the Noether theorem in a general form is obtained. The Noether-type conserved quantities with non-standard Lagrangians in both classical and discrete cases are given. Finally, an example in Friedmann-Robertson-Walker spacetime and an example about second order Duffing equation are given to illustrate the application of the results.

Keywords:  time scale      non-standard Lagrangian      Noether symmetry      conserved quantity     
Received:  28 February 2017      Published:  05 August 2017
PACS:  45.20.Jj (Lagrangian and Hamiltonian mechanics)  
  11.30.Na (Nonlinear and dynamical symmetries (spectrum-generating symmetries))  
  45.10.Db (Variational and optimization methods)  

Project supported by the National Natural Science Foundation of China (Grant Nos. 11572212 and 11272227) and the Innovation Program of Suzhou University of Science and Technology, China (Grant No. SKYCX16_012).

Corresponding Authors:  Yi Zhang     E-mail:
About author:  0.1088/1674-1056/26/8/

Cite this article: 

Jing Song(宋静), Yi Zhang(张毅) Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales 2017 Chin. Phys. B 26 084501

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