中国物理B ›› 2009, Vol. 18 ›› Issue (7): 2634-2641.doi: 10.1088/1674-1056/18/7/003

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Noether conserved quantities and Lie point symmetries of difference Lagrange--Maxwell equations and lattices

Jimé nez Salvador1, Vá zquez Luis2, 黄健飞3, 赵维加3, 傅景礼4, 唐贻发5, 聂宁明6   

  1. (1)Departamento de Matem'atica Aplicada TTII, E. T. S. I. Telecomunicaci'on, Universidad Polit'ecnica de Madrid, 28040-Madrid, Spain; (2)Departamento de Matem'atica Aplicada, Facultad de Inform'atica, Instituto de Matem'atica Interdisciplinar (IMI), Universidad Complutense de Madrid, 28040-Madrid, Spai; (3)Department of Mathematics, Qingdao University, Qingdao 266071, China; (4)Institute of Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China; (5)State Key Laboratory of Scientific and Engineering Computing, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;Departamento de Matem'atica Aplicada, Facultad de Matem'aticas, Instituto de Matem'atica Interdisciplinar (IMI), Universidad Complutense de Madrid, 28040-Madrid, Spain; (6)State Key Laboratory of Scientific and Engineering Computing, ICMSEC, Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing 100190, China
  • 收稿日期:2008-09-26 修回日期:2008-12-04 出版日期:2009-07-20 发布日期:2009-07-20
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grants Nos 10672143 and 60575055) and State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences. Tang Yi-Fa acknowledges the support under Sabbatical Program (SAB2006-0070) of the Spanish Ministry of Education and Science. Jim\'enez S and V\'azquez L acknowledge support of the Spanish Ministry of Education and Science (Grant No MTM2005-05573).

Noether conserved quantities and Lie point symmetries of difference Lagrange--Maxwell equations and lattices

Fu Jing-Li(傅景礼)a)†, Nie Ning-Ming(聂宁明)b), Huang Jian-Fei(黄健飞)c) Jimenez Salvadord), Tang Yi-Fa(唐贻发)b)e), Vazquez Luisf), and Zhao Wei-Jia(赵维加)c)   

  1. a Institute of Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China; b State Key Laboratory of Scientific and Engineering Computing, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; c Department of Mathematics, Qingdao University, Qingdao 266071, China; d Departamento de Matemática Aplicada TTII, E. T. S. I. Telecomunicación, Universidad Politécnica de Madrid, 28040-Madrid, Spaine Departamento de Matemática Aplicada, Facultad de Matemáticas, Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, 28040-Madrid, Spainf Departamento de Matemática Aplicada, Facultad de Informática, Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, 28040-Madrid, Spain
  • Received:2008-09-26 Revised:2008-12-04 Online:2009-07-20 Published:2009-07-20
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grants Nos 10672143 and 60575055) and State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences. Tang Yi-Fa acknowledges the support under Sabbatical Program (SAB2006-0070) of the Spanish Ministry of Education and Science. Jim\'enez S and V\'azquez L acknowledge support of the Spanish Ministry of Education and Science (Grant No MTM2005-05573).

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摘要： This paper presents a method to find Noether-type conserved quantities and Lie point symmetries for discrete mechanico-electrical dynamical systems, which leave invariant the set of solutions of the corresponding difference scheme. This approach makes it possible to devise techniques for solving the Lagrange--Maxwell equations in differences which correspond to mechanico-electrical systems, by adapting existing differential equations. In particular, it obtains a new systematic method to determine both the one-parameter Lie groups and the discrete Noether conserved quantities of Lie point symmetries for mechanico-electrical systems. As an application, it obtains the Lie point symmetries and the conserved quantities for the difference equation of a model that represents a capacitor microphone.

Abstract: This paper presents a method to find Noether-type conserved quantities and Lie point symmetries for discrete mechanico-electrical dynamical systems, which leave invariant the set of solutions of the corresponding difference scheme. This approach makes it possible to devise techniques for solving the Lagrange--Maxwell equations in differences which correspond to mechanico-electrical systems, by adapting existing differential equations. In particular, it obtains a new systematic method to determine both the one-parameter Lie groups and the discrete Noether conserved quantities of Lie point symmetries for mechanico-electrical systems. As an application, it obtains the Lie point symmetries and the conserved quantities for the difference equation of a model that represents a capacitor microphone.

Key words: Lagrange--Maxwell equation, Lie point symmetry, discrete mechanico-electrical system, conserved quantity

中图分类号:  (Lagrangian and Hamiltonian mechanics)

  • 45.20.Jj
03.50.De (Classical electromagnetism, Maxwell equations) 02.20.Qs (General properties, structure, and representation of Lie groups) 45.05.+x (General theory of classical mechanics of discrete systems) 02.30.Hq (Ordinary differential equations) 02.30.Jr (Partial differential equations)