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Lie symmetries, symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems

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Abstract

For a nonlinear nonholonomic constrained mechanical system with the action of small forces of perturbation, Lie symmetries, symmetrical perturbation, and a new type of non-Noether adiabatic invariants are presented in general infinitesimal transformation of Lie groups. Based on the invariance of the equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, constraints restriction equations, additional restriction equations, and exact invariants of the system are given. Then, under the action of small forces of perturbation, the determining equations, constraints restriction equations, and additional restriction equations of the Lie symmetrical perturbation are obtained, and adiabatic invariants of the Lie symmetrical perturbation, the weakly Lie symmetrical perturbation, and the strongly Lie symmetrical perturbation for the disturbed nonholonomic system are obtained, respectively. Furthermore, a set of non-Noether exact invariants and adiabatic invariants are given in the special infinitesimal transformations. Finally, one example is given to illustrate the application of the method and results.

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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, Wenan Jiang & Shaokai Luo

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  1. Zhuangjun Li
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  2. Wenan Jiang
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  3. Shaokai Luo
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Correspondence to Shaokai Luo.

Additional information

This work is partly supported by National Natural Science Foundation of China (Grant No. 10372053).

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Li, Z., Jiang, W. & Luo, S. Lie symmetries, symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems. Nonlinear Dyn 67, 445–455 (2012). https://doi.org/10.1007/s11071-011-9993-6

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