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A new Lie symmetrical method of finding conserved quantity for Birkhoffian systems

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Abstract

For a Birkhoffian system, a new Lie symmetrical method to find a conserved quantity is given. Based on the invariance of the equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. Then, several important relationships which reveal the interior properties of the Birkhoffian system are given. By using these relationships, a new Lie symmetrical conservation law for the Birkhoffian system is presented. The new conserved quantity is constructed in terms of infinitesimal generators of the Lie symmetry and the system itself without solving the structural equation which may be very difficult to solve. Furthermore, several deductions are given in the special infinitesimal transformations and the results are reduced to a Hamiltonian system. Finally, one example is given to illustrate the method and results of the application.

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Acknowledgements

This work is partly supported by National Natural Science Foundation of China (grant Nos. 10972127 and 10372053).

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Authors and Affiliations

  1. Institute of Mathematical Mechanics and Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou, 310018, P.R. China

    Zhuangjun Li & Shaokai Luo

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  1. Zhuangjun Li
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  2. Shaokai Luo
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Correspondence to Shaokai Luo.

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Li, Z., Luo, S. A new Lie symmetrical method of finding conserved quantity for Birkhoffian systems. Nonlinear Dyn 70, 1117–1124 (2012). https://doi.org/10.1007/s11071-012-0517-9

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