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A Lie symmetrical basic integral variable relation and a new conservation law for generalized Hamiltonian systems

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Abstract

For a generalized Hamiltonian system, a new Lie symmetrical method to find a conserved quantity is presented in a general infinitesimal transformation of Lie groups. Based on the invariance of the differential equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. And a number of important relationships of the Lie symmetrical method for a generalized Hamiltonian system are investigated, which reveal the interior properties of the system. By using the relationships, a Lie symmetrical basic integral variable relation and a new Lie symmetrical conservation law for generalized Hamiltonian systems is presented. The basic integral variable relation not only can be used for a linear dynamic system but can also be used for a nonlinear dynamic system. The new conserved quantity is constructed in terms of infinitesimal generators of Lie symmetry and the interior properties of the system itself without solving the structural equation that may be very difficult to solve. Then, the method is applied in the generalized Hamiltonian system of even dimensions and the Hamiltonian system, respectively. Furthermore, the relationship between the generalized Hamiltonian system and a Birkhoffian system is studied, and a Lie symmetrical conservation law for a semiautonomous (or autonomous) Birkhoffian system is obtained. Finally, one example is given to illustrate the method and results of the application.

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Authors and Affiliations

  1. Institute of Mathematical Mechanics and Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou, 310018, People’s Republic of China

    Shao-Kai Luo, Zhuang-Jun Li, Wang Peng & Lin Li

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  1. Shao-Kai Luo
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  2. Zhuang-Jun Li
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  3. Wang Peng
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  4. Lin Li
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Correspondence to Shao-Kai Luo.

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Luo, SK., Li, ZJ., Peng, W. et al. A Lie symmetrical basic integral variable relation and a new conservation law for generalized Hamiltonian systems. Acta Mech 224, 71–84 (2013). https://doi.org/10.1007/s00707-012-0733-x

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  • DOI: https://doi.org/10.1007/s00707-012-0733-x

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