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A concept of fundamental integrals of one-dimensional harmonic oscillator is presented, and other integrals can be constructed by use of fundamental integrals. The above concept and method are extended to multidimensional harmonic oscillators. By directly constructing other integrals from the fundamental integrals, it is proved that there are three independent time-independent integrals for all kinds of two-dimensional harmonic oscillators and there are 2n-1 independent time-independent integrals for n-dimensional harmonic oscillators. The characteristics of the anisotropic two-dimensional harmonic oscillator is discussed when the ratio between two frequencies is rational or irrational number.
[1] Goldstein H, Poole C, Salko J 2005 Classical Mechanics (3rd Ed.) (Beijing: Higher Education Press)
[2] Li D M, Chen C M 2006 Classical Mechanics (Beijing: Higher Education Press) (in Chinese) [李德明, 陈昌民 2006 经典力学 (北京: 高等教育出版社)]
[3] Zeng J Y 2001 Quantum Mechanics (Beijing: Science Press) (in Chinese) [曾谨言 2001 量子力学 (北京: 科学出版社)]
[4] Um C I, Yeon K H, George T F 2002 Phys. Rep. 362 63
[5] Ling R L, Feng J F 2009 Acta Phys. Sin. 58 2164 (in Chinese) [凌瑞良, 冯金福 2009 58 2164]
[6] Hao J H, Huang X Y 1998 Acta Sin. Quant. Opt. 4 49 (in Chinese) [郝剑红, 黄相友 1998 量子光学学报 4 49]
[7] Lou Z M, Mei F X 2012 Acta Phys. Sin. 61 110201 (in Chinese) [楼智美, 梅凤翔 2012 61 110201]
[8] Ding G T 2011 J. Dynam. Control. 9 102 (in Chinese) [丁光涛 2011 动力学与控制学报 9 102]
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[1] Goldstein H, Poole C, Salko J 2005 Classical Mechanics (3rd Ed.) (Beijing: Higher Education Press)
[2] Li D M, Chen C M 2006 Classical Mechanics (Beijing: Higher Education Press) (in Chinese) [李德明, 陈昌民 2006 经典力学 (北京: 高等教育出版社)]
[3] Zeng J Y 2001 Quantum Mechanics (Beijing: Science Press) (in Chinese) [曾谨言 2001 量子力学 (北京: 科学出版社)]
[4] Um C I, Yeon K H, George T F 2002 Phys. Rep. 362 63
[5] Ling R L, Feng J F 2009 Acta Phys. Sin. 58 2164 (in Chinese) [凌瑞良, 冯金福 2009 58 2164]
[6] Hao J H, Huang X Y 1998 Acta Sin. Quant. Opt. 4 49 (in Chinese) [郝剑红, 黄相友 1998 量子光学学报 4 49]
[7] Lou Z M, Mei F X 2012 Acta Phys. Sin. 61 110201 (in Chinese) [楼智美, 梅凤翔 2012 61 110201]
[8] Ding G T 2011 J. Dynam. Control. 9 102 (in Chinese) [丁光涛 2011 动力学与控制学报 9 102]
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