Two formulations and solutions of the inverse problems for Lie symmetries in dynamics of a Birkhoffian system

Journal of East China Normal University(Natural Sc ›› 2012, Vol. 2012 ›› Issue (3): 49-55.

• Article • Previous Articles Next Articles

Two formulations and solutions of the inverse problems for Lie symmetries in dynamics of a Birkhoffian system

LONG Zi-xuan ¹, ZHANG Yi ²

1. College of Mathematics and Physics, Suzhou University of Science and Technology, Jiangsu 215009, China; 2. College of Civil Engineering, Suzhou University of
Science and Technology, Jiangsu 215011, China

Received:2011-06-01 Revised:2011-09-01 Online:2012-05-25 Published:2012-05-22

Abstract

Abstract: First, the determining equations, the structure equations and the conserved quantities of Lie symmetries for a Birkhoffian system were given; then two formulations and solutions of the inverse problems of Lie symmetries for the system were presented. The results show that the same Birkhoffian(Birkhoff's functions) and
first integral can correspond to different Birkhoff's functions(Birkhoffian) and different Lie symmetries, and can also correspond to the same Lie symmetry and different Birkhoff's functions(Birkhoffian).

Key words: Birkhoffian system, Lie symmetry, inverse problem

CLC Number:

O316

LONG Zi-xuan, ZHANG Yi. Two formulations and solutions of the inverse problems for Lie symmetries in dynamics of a Birkhoffian system[J]. Journal of East China Normal University(Natural Sc, 2012, 2012(3): 49-55.

References

{1} LUTZKY M. Dynamical symmetries and conserved quantities[J]. J Phys A: Math Gen, 1979, 12(7): 973-981.
{2} 赵跃宇. 非保守力学系统的Lie对称性和守恒量[J]. 力学学报, 1994, 26(3): 380-384.
{3} 梅凤翔. Birkhoff系统的Lie对称性和守恒律[J]. 科学通报, 1998, 43(18)：1937-1939.
{4} FANG J H. Lie symmetries and conserved quantities of second-order nonholonomic mechanical system[J]. Applied Mathematics and Mechanics, 2002, 23(9): 1105-1110.
{5} 傅景礼, 王新民. 相对论性Birkhoff系统的Lie对称性和守恒量[J]. 物理学报, 2000, 49(6)：1023-1027.
{6} 张毅. Birkhoff系统的一类Lie对称性守恒量[J]. 物理学报, 2002, 51(3)：461-464.
{7} 张宏彬, 陈立群, 顾书龙. Birkhoff 系统的一般Lie对称性和非Noether守恒量[J]. 力学学报, 2004, 36(2)：254-256.
{8} LUO S K. New types of the Lie symmetries and conserved quantities for a relativistic Hamilton system[J]. Chinese Physics Letters, 2003, 20(5): 597-599.
{9} ZHANG Y. Symmetries and conserved quantities of generalized Birkhoffian systems[J]. Journal of Southeast University(English Edition), 2010, 26(1): 146-150.
{10} 梅凤翔. 李群和李代数对约束力学系统的应用[M]. 北京：科学出版社, 1999.
{11} 梅凤翔．非完整动力学逆问题的基本解法[J]．力学学报, 1991, 23(2)：252-256.
{12} LIU F L, MEI F X. Formulation and solution for inverse problem of nonholonomic dynamics[J]. Applied Mathematics and Mechanics, 1993, 14(4)：327-332.
{13} 张永发, 梅凤翔．Birkhoff系统动力学逆问题的两种提法和解法[J]. 北京理工大学学报, 1996, 16(4)：352-356.
{14} LI G C, MEI F X. An inverse problem in analytical dynamics[J]. Chinese Physics, 2006, 15(8)：1669-1671.
{15} 丁光涛．Noether-Birkhoff动力学逆问题[J]．中国科学：物理学力学天文学, 2010, 40(2): 1514-1520.
{16} 梅凤翔. 动力学逆问题[M]. 北京：国防工业出版社, 2009.
{17} 梅凤翔, 史荣昌, 张永发, 等. Birkhoff系统动力学[M]. 北京：北京理工大学出版社, 1996.
{18} 梅凤翔. 约束力学系统的对称性与守恒量[M]. 北京：北京理工大学出版社, 2004.

[1]	DAI Hui-ju, LI Lian-zhong, WANG Qi, SHA An. Lie symmetry analysis, Bäcklund transformation and exact solutions for a class of fourth-order partial differential equations [J]. Journal of East China Normal University(Natural Sc, 2019, 2019(1): 24-31.
[2]	ZHANG Li-xiang, LIU Han-ze, XIN Xiang-peng. Symmetry reductions, exact solutions and conservation laws of a class of forth-order partial differential equations [J]. Journal of East China Normal University(Natural Sc, 2017, 2017(6): 50-62.
[3]	CUI Jin-chao, LIAO Cui-cui, MEI Feng-xiang. Four kinds of gradient representations of autonomous Birkhoffian systems [J]. Journal of East China Normal University(Natural Sc, 2017, (3): 94-98.

Two formulations and solutions of the inverse problems for Lie symmetries in dynamics of a Birkhoffian system

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 3

Recommended Articles

Metrics

Comments