中国物理B ›› 2007, Vol. 16 ›› Issue (11): 3176-3181.doi: 10.1088/1009-1963/16/11/005

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Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for Lagrange systems

郭永新1, 陈向炜2, 罗绍凯3   

  1. (1)Department of Physics, Liaoning University, Shenyang 110036, China; (2)Department of Physics, Shangqiu Teachers College, Shangqiu 476000, China; (3)Institute of Mathematical Mechanics and Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China;Key Laboratory of Advanced Textile Materials and Manufacturing Technology (Zhejiang Sci-Tech University), Ministry of Education, Hangzhou
  • 出版日期:2007-11-20 发布日期:2007-11-20
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos 10472040 and 10372053), the Natural Science Foundation of Hunan Province, China (Grant No 03JJY3005), the Natural Science Foundation of Henan Province, China (Grant No 031101

Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for Lagrange systems

Luo Shao-Kai(罗绍凯)a)b)†, Chen Xiang-Wei(陈向炜)c), and Guo Yong-Xin(郭永新)d)   

  1. a Institute of Mathematical Mechanics and Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China; b Key Laboratory of Advanced Textile Materials and Manufacturing Technology (Zhejiang Sci-Tech University), Ministry of Education, Hangzhou 310018, China; c Department of Physics, Shangqiu Teachers College, Shangqiu 476000, China; d Department of Physics, Liaoning University, Shenyang 110036, China
  • Online:2007-11-20 Published:2007-11-20
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos 10472040 and 10372053), the Natural Science Foundation of Hunan Province, China (Grant No 03JJY3005), the Natural Science Foundation of Henan Province, China (Grant No 031101

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摘要： Based on the invariance of differential equations under infinitesimal transformations of group, Lie symmetries, exact invariants, perturbation to the symmetries and adiabatic invariants in form of non-Noether for a Lagrange system are presented. Firstly, the exact invariants of generalized Hojman type led directly by Lie symmetries for a Lagrange system without perturbations are given. Then, on the basis of the concepts of Lie symmetries and higher order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for the system with the action of small disturbance is investigated, the adiabatic invariants of generalized Hojman type for the system are directly obtained, the conditions for existence of the adiabatic invariants and their forms are proved. Finally an example is presented to illustrate these results.

关键词: Lagrange system, Lie symmetrical perturbation, exact invariant, adiabatic invariant of generalized Hojman type

Abstract: Based on the invariance of differential equations under infinitesimal transformations of group, Lie symmetries, exact invariants, perturbation to the symmetries and adiabatic invariants in form of non-Noether for a Lagrange system are presented. Firstly, the exact invariants of generalized Hojman type led directly by Lie symmetries for a Lagrange system without perturbations are given. Then, on the basis of the concepts of Lie symmetries and higher order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for the system with the action of small disturbance is investigated, the adiabatic invariants of generalized Hojman type for the system are directly obtained, the conditions for existence of the adiabatic invariants and their forms are proved. Finally an example is presented to illustrate these results.

Key words: Lagrange system, Lie symmetrical perturbation, exact invariant, adiabatic invariant of generalized Hojman type

中图分类号:  (Lagrangian and Hamiltonian mechanics)

  • 45.20.Jj
45.10.Hj (Perturbation and fractional calculus methods) 45.50.Pk (Celestial mechanics)