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The energy and the two partial energies of two-dimensional anisotropic harmonic oscillator are conserved quantities, but only two of them are independent. The system possesses the third independent conserved quantity when the 1/2 is a rational number. The extended Prelle-Singer method is used to find the third independent conserved quantity for the five typical two-dimensional anisotropic harmonic oscillators. The Noether symmetry and the Lie symmetry of the third independent conserved quantities are also discussed.
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Keywords:
- extended P-S method /
- two-dimensional anisotropic harmonic oscillator /
- conserved quantity /
- symmetry
[1] Zeng J Y 2001 Quantum Mechanics (Beijing: Science Press) p447 (in Chinese) [曾谨言 2001 量子力学 (北京: 科学出版社) 第447页]
[2] Lou Z M 2002 College Phys. 21 18 (in Chinese) [楼智美 2002 大学物理 21 18]
[3] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) p103, 303 (in Chinese) [梅凤翔 1999 李群和李代数对约束力学系统的应用(北京: 科学出版社) 第103, 303页]
[4] Shang M, Chen X W 2006 Chin. Phys. 15 2788
[5] Fang J H, Liu Y K, Zhang X N 2008 Chin. Phys. B 17 1962
[6] Fang J H 2010 Chin. Phys. B 19 040301
[7] Lou Z M 2006 Chin. Phys. 15 891
[8] Xie Y L, Jia L Q, Luo S K 2011 Chin. Phys. B 20 010203
[9] Ge W K 2008 Acta Phys. Sin. 57 6714 (in Chinese) [葛伟宽 2008 57 6714]
[10] Lou Z M 2005 Acta Phys. Sin. 54 1015 (in Chinese) [楼智美 2005 54 1015]
[11] Haas F, Goedert J 1996 J. Phys. A: Math. Gen. 29 4083
[12] Lou Z M 2005 Acta Phys. Sin. 54 1969 (in Chinese) [楼智美 2005 54 1969]
[13] Kaushal R S, Gupta S 2001 J. Phys. A: Math. Gen. 34 9879
[14] Kaushal R S, Parashar D, Gupta S 1997 Ann. Phys. 259 233
[15] Lou Z M 2007 Chin. Phys. 16 1182
[16] Annamalai A, Tamizhmani K M 1994 Nonlinear Math. Phys. 1 309
[17] Ge W K, Mei F X 2001 Acta Armamentarii 22 241 (in Chinese) [葛伟宽, 梅凤翔 2001 兵工学报 22 241]
[18] Lou Z M, Wang W L 2006 Chin. Phys. 15 895
[19] Prelle M J, Singer M F 1983 Trans. Amer. Math. Soc. 279 215
[20] Guha P, Choudhury A G, Khanra B 2009 J. Phys. A: Math. Theor. 42 115206
[21] Duarte L G S, Duarte S E S, da Mota L A C P, Skea J E F 2001 J. Phys. A: Math. Gen. 34 3015
[22] Chandrasekar V K, Senthilvelan M, Lakshmanan M 2006 J. Phys. A: Math. Gen. 39 L69
[23] Chandrasekar V K, Senthilvelan M, Lakshmanan M 2005 J. Nonlinear Math. Phys. 12 184
[24] Chandrasekar V K, Senthilvelan M, Lakshmanan M 2006 J. Math. Phys. 47 023508
[25] Lou Z M 2010 Acta Phys. Sin. 59 719 (in Chinese) [楼智美 2010 59 719]
[26] Lou Z M 2010 Acta Phys. Sin. 59 3633 (in Chinese) [楼智美 2010 59 3633]
-
[1] Zeng J Y 2001 Quantum Mechanics (Beijing: Science Press) p447 (in Chinese) [曾谨言 2001 量子力学 (北京: 科学出版社) 第447页]
[2] Lou Z M 2002 College Phys. 21 18 (in Chinese) [楼智美 2002 大学物理 21 18]
[3] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) p103, 303 (in Chinese) [梅凤翔 1999 李群和李代数对约束力学系统的应用(北京: 科学出版社) 第103, 303页]
[4] Shang M, Chen X W 2006 Chin. Phys. 15 2788
[5] Fang J H, Liu Y K, Zhang X N 2008 Chin. Phys. B 17 1962
[6] Fang J H 2010 Chin. Phys. B 19 040301
[7] Lou Z M 2006 Chin. Phys. 15 891
[8] Xie Y L, Jia L Q, Luo S K 2011 Chin. Phys. B 20 010203
[9] Ge W K 2008 Acta Phys. Sin. 57 6714 (in Chinese) [葛伟宽 2008 57 6714]
[10] Lou Z M 2005 Acta Phys. Sin. 54 1015 (in Chinese) [楼智美 2005 54 1015]
[11] Haas F, Goedert J 1996 J. Phys. A: Math. Gen. 29 4083
[12] Lou Z M 2005 Acta Phys. Sin. 54 1969 (in Chinese) [楼智美 2005 54 1969]
[13] Kaushal R S, Gupta S 2001 J. Phys. A: Math. Gen. 34 9879
[14] Kaushal R S, Parashar D, Gupta S 1997 Ann. Phys. 259 233
[15] Lou Z M 2007 Chin. Phys. 16 1182
[16] Annamalai A, Tamizhmani K M 1994 Nonlinear Math. Phys. 1 309
[17] Ge W K, Mei F X 2001 Acta Armamentarii 22 241 (in Chinese) [葛伟宽, 梅凤翔 2001 兵工学报 22 241]
[18] Lou Z M, Wang W L 2006 Chin. Phys. 15 895
[19] Prelle M J, Singer M F 1983 Trans. Amer. Math. Soc. 279 215
[20] Guha P, Choudhury A G, Khanra B 2009 J. Phys. A: Math. Theor. 42 115206
[21] Duarte L G S, Duarte S E S, da Mota L A C P, Skea J E F 2001 J. Phys. A: Math. Gen. 34 3015
[22] Chandrasekar V K, Senthilvelan M, Lakshmanan M 2006 J. Phys. A: Math. Gen. 39 L69
[23] Chandrasekar V K, Senthilvelan M, Lakshmanan M 2005 J. Nonlinear Math. Phys. 12 184
[24] Chandrasekar V K, Senthilvelan M, Lakshmanan M 2006 J. Math. Phys. 47 023508
[25] Lou Z M 2010 Acta Phys. Sin. 59 719 (in Chinese) [楼智美 2010 59 719]
[26] Lou Z M 2010 Acta Phys. Sin. 59 3633 (in Chinese) [楼智美 2010 59 3633]
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