Chin. Phys. B, 2017, Vol. 26(8): 084501    DOI: 10.1088/1674-1056/26/8/084501
 ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS Prev   Next

# 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
Abstract

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.

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)
Fund:

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:  zhy@mail.usts.edu.cn