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由牛顿第二定律得到二维各向同性带电谐振子在均匀磁场中运动的运动微分方程, 通过对运动微分方程的直接积分得到系统的两个积分(守恒量). 利用Legendre变换建立守恒量与Lagrange函数间的关系, 从而求得系统的Lagrange函数, 并讨论与守恒量相应的无限小变换的Noether对称性与Lie对称性, 最后求得系统的运动学方程.
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关键词:
- 二维各向同性带电谐振子 /
- 守恒量 /
- Noether对称性 /
- Lie对称性
The kinematic differentiation equations of two-dimensional isotropic harmonic charged oscillator moving in a homogeneous magnetic are obtained by using Newton’s second law. Two integrals (conserved quantities) are obtained by directly integrating the kinematic differentiation equations. The relationship between the Lagrangian and the conserved quantity is established through the Legendre transformation, thereby obtaining a Lagrangian function of the system. The Noether symmetry and Lie symmetry of the infinitesimal transformations corresponding to the conserved quantities are studied. Finally, the kinematical equations of the system are obtained.-
Keywords:
- two-dimensional isotropic harmonic charged oscillator /
- conserved quantities /
- Noether symmetries /
- Lie symmetries
[1] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems(Beijing: Science Press) p103, p303 (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页, 第303页]
[2] Dong W S, Wang B X, Fang J H 2011 Chin. Phys. B 20 010204
[3] Chen R, Xu X J 2012 Chin. Phys. B 21 094510
[4] Fang J H 2010 Chin. Phys. B 19 040301
[5] Wang X X, Han Y L, Zhang M L, Jia L Q 2013 Chin. Phys. B 22 020201
[6] Xie Y L, Jia L Q, Luo S K 2011 Chin. Phys. B 20 010203
[7] Jiang W A, Luo S K 2011 Acta Phys. Sin. 60 060201 (in Chinese) [姜文安, 罗绍凯 2011 物理学报 60 060201]
[8] Han Y L, Sun X T, Zhang Y Y, Jia L Q 2013 Acta Phys. Sin. 62 160201 (in Chinese) [韩月林, 孙现亭, 张耀宇, 贾利群 2013 物理学报 62 160201]
[9] Fang J H, Ding N, Wang P 2007 Chin.Phys. 16 887
[10] Kaushal R S, Gupta S 2001 J. Phys. A: Math. Gen. 34 9879
[11] Kaushal R S, Parashar D, Gupta S 1997 Ann. Phys. 259 233
[12] Lou Z M 2007 Chin. Phys. 16 1182
[13] Lou Z M 2007 Acta Phys. Sin. 56 2475 (in Chinese) [楼智美 2007 物理学报 56 2475]
[14] Haas F, Goedert J 1996 J. Phys. A: Math. Gen. 29 4083
[15] Lou Z M 2005 Acta Phys. Sin. 54 1969 (in Chinese) [楼智美 2005 物理学报 54 1969]
[16] Lou Z M 2005 Acta Phys. Sin. 54 1460 (in Chinese) [楼智美 2005 物理学报 54 1460]
[17] Prelle M J, Singer M F 1983 Trans. Amer. Math. Soc. 279 215
[18] Chandrasekar V K, Senthilvelan M, Lakshmanan M 2006 J. Phys. A: Math.Gen. 39 L69
[19] Lou Z M 2010 Acta Phys. Sin. 59 719 (in Chinese) [楼智美 2010 物理学报 59 719]
[20] Ding G T 2013 Acta Phys. Sin. 62 064502 (in Chinese) [丁光涛 2013 物理学报 62 064502]
[21] Ding G T 2013 Acta Phys. Sin. 62 064501 (in Chinese) [丁光涛 2013 物理学报 62 064501]
[22] López G 1996 Ann. Phys. 251 363
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[1] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems(Beijing: Science Press) p103, p303 (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页, 第303页]
[2] Dong W S, Wang B X, Fang J H 2011 Chin. Phys. B 20 010204
[3] Chen R, Xu X J 2012 Chin. Phys. B 21 094510
[4] Fang J H 2010 Chin. Phys. B 19 040301
[5] Wang X X, Han Y L, Zhang M L, Jia L Q 2013 Chin. Phys. B 22 020201
[6] Xie Y L, Jia L Q, Luo S K 2011 Chin. Phys. B 20 010203
[7] Jiang W A, Luo S K 2011 Acta Phys. Sin. 60 060201 (in Chinese) [姜文安, 罗绍凯 2011 物理学报 60 060201]
[8] Han Y L, Sun X T, Zhang Y Y, Jia L Q 2013 Acta Phys. Sin. 62 160201 (in Chinese) [韩月林, 孙现亭, 张耀宇, 贾利群 2013 物理学报 62 160201]
[9] Fang J H, Ding N, Wang P 2007 Chin.Phys. 16 887
[10] Kaushal R S, Gupta S 2001 J. Phys. A: Math. Gen. 34 9879
[11] Kaushal R S, Parashar D, Gupta S 1997 Ann. Phys. 259 233
[12] Lou Z M 2007 Chin. Phys. 16 1182
[13] Lou Z M 2007 Acta Phys. Sin. 56 2475 (in Chinese) [楼智美 2007 物理学报 56 2475]
[14] Haas F, Goedert J 1996 J. Phys. A: Math. Gen. 29 4083
[15] Lou Z M 2005 Acta Phys. Sin. 54 1969 (in Chinese) [楼智美 2005 物理学报 54 1969]
[16] Lou Z M 2005 Acta Phys. Sin. 54 1460 (in Chinese) [楼智美 2005 物理学报 54 1460]
[17] Prelle M J, Singer M F 1983 Trans. Amer. Math. Soc. 279 215
[18] Chandrasekar V K, Senthilvelan M, Lakshmanan M 2006 J. Phys. A: Math.Gen. 39 L69
[19] Lou Z M 2010 Acta Phys. Sin. 59 719 (in Chinese) [楼智美 2010 物理学报 59 719]
[20] Ding G T 2013 Acta Phys. Sin. 62 064502 (in Chinese) [丁光涛 2013 物理学报 62 064502]
[21] Ding G T 2013 Acta Phys. Sin. 62 064501 (in Chinese) [丁光涛 2013 物理学报 62 064501]
[22] López G 1996 Ann. Phys. 251 363
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