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By introducing fundamental integrals of one-dimensional linear damped oscillators the other first integrals can be constructed, including time-irrelevant integrals. The above method is extended to multidimensional systems, in order to construct different integrals of two-dimensional and n-dimensional linear damped oscillators. It is proved that there are three independent time-irrelevant integrals for all kinds of two-dimensional linear damped oscillators, and 2n-1 independent time-irrelevant integrals for n-dimensional linear damped oscillators. Using the transformation of variables the first integrals of linear damped oscillator transform into ones of harmonic oscillator.
[1] Goldstein H, Poole C, Salko J 2005 Classical Mechanics (3rd Ed.) (Beijing: Higher Education Press)
[2] Li D M, Cheng 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] Lou Z M, Mei F X 2012 Acta Phys. Sin. 61 110201 (in Chinese) [楼智美, 梅凤翔 2012 物理学报 61 110201]
[4] 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)]
[5] Dekker H 1981 Phys. Rep. 80 1
[6] Um C I, Yean K H, George T F 2002 Phys. Rep. 362 63
[7] Pen H W 1980 Acta Phys. Sin. 29 1084 (in Chinese) [彭桓武 1980 物理学报 29 1084]
[8] Zhu R Z 1981 Acta Phys. Sin. 30 1410 (in Chinese) [朱如曾 1981 物理学报 30 1410]
[9] Ling R L, Feng J F 1998 Acta Phys. Sin. 47 1952 (in Chinese) [凌瑞良, 冯金福 1998 物理学报 47 1952]
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[1] Goldstein H, Poole C, Salko J 2005 Classical Mechanics (3rd Ed.) (Beijing: Higher Education Press)
[2] Li D M, Cheng 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] Lou Z M, Mei F X 2012 Acta Phys. Sin. 61 110201 (in Chinese) [楼智美, 梅凤翔 2012 物理学报 61 110201]
[4] 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)]
[5] Dekker H 1981 Phys. Rep. 80 1
[6] Um C I, Yean K H, George T F 2002 Phys. Rep. 362 63
[7] Pen H W 1980 Acta Phys. Sin. 29 1084 (in Chinese) [彭桓武 1980 物理学报 29 1084]
[8] Zhu R Z 1981 Acta Phys. Sin. 30 1410 (in Chinese) [朱如曾 1981 物理学报 30 1410]
[9] Ling R L, Feng J F 1998 Acta Phys. Sin. 47 1952 (in Chinese) [凌瑞良, 冯金福 1998 物理学报 47 1952]
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