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The damped brachistochrone problem and that with non-zero initial velocity are studied. Based on the discussion of these problems, one may take theorems of motion as constraints for some systems, and whether the constraints are holonomic or nonholonomic is related to the fact that the differential theorems of motion are integrable or non-integrable.
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Keywords:
- variational calculus /
- brachistochrone /
- theorem of motion /
- constraint
[1] Courant R, Hilbert D (translated by Qian M, Guo D R) 1958 Methods of Mathematical Physics Vol. I (Beijing: Science Press) pp129-135 (in Chinese) [柯朗R, 希伯尔特D 著(钱敏郭敦仁译)1958 数学物理方法卷I (北京: 科学出版社) 第129–135 页]
[2] Goldstein H, Poole C, Safko J 2005 Classical Mechanics (3rd Ed.) (Beijing: Higher Education Press)
[3] Smirnov V E (translated by Chen C Z) 1958 Textbook of Higher Mathematics Vol. IV-I (Beijing: People Education Press) pp232-234 (in Chinese) [斯米尔诺夫V E 著 (陈传璋译)1958 高等数学教程第四卷第一分册(北京: 人民教育出版社) 第232–234 页]
[4] Zhang H B, Chen L Q, Liu R W 2005 Chin. Phys. B 14 1063
[5] Zhang H B, Chen L Q, Gu S L, Liu C Z 2007 Chin. Phys. 16 582
[6] Musielak Z E 2008 J. Phys. A: Math. Their. 41 055205
[7] Cieslinski J L, Nikiciuk T 2010 J. Phys. A: Math. Their. 43 175205
[8] Ding G T 2011 Acta Phys. Sin. 60 044503 (in Chinese)[丁光涛2011 物理学报60 044503]
[9] Ding G T 2012 Acta Phys. Sin. 61 020204 (in Chinese)[丁光涛2012 物理学报61 020204]
[10] Ding G T 2013 Acta Phys. Sin. 62 064501 (in Chinese)[丁光涛2013 物理学报62 064501]
[11] Luo S K, Zhang Y F 2008 Advances in the Study of Dynamics of Constrained Systems (Beijing: Science Press) (in Chinese) [罗绍凯, 张永发2008 约束系统动力学研究进展(北京: 科学出版社)]
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[1] Courant R, Hilbert D (translated by Qian M, Guo D R) 1958 Methods of Mathematical Physics Vol. I (Beijing: Science Press) pp129-135 (in Chinese) [柯朗R, 希伯尔特D 著(钱敏郭敦仁译)1958 数学物理方法卷I (北京: 科学出版社) 第129–135 页]
[2] Goldstein H, Poole C, Safko J 2005 Classical Mechanics (3rd Ed.) (Beijing: Higher Education Press)
[3] Smirnov V E (translated by Chen C Z) 1958 Textbook of Higher Mathematics Vol. IV-I (Beijing: People Education Press) pp232-234 (in Chinese) [斯米尔诺夫V E 著 (陈传璋译)1958 高等数学教程第四卷第一分册(北京: 人民教育出版社) 第232–234 页]
[4] Zhang H B, Chen L Q, Liu R W 2005 Chin. Phys. B 14 1063
[5] Zhang H B, Chen L Q, Gu S L, Liu C Z 2007 Chin. Phys. 16 582
[6] Musielak Z E 2008 J. Phys. A: Math. Their. 41 055205
[7] Cieslinski J L, Nikiciuk T 2010 J. Phys. A: Math. Their. 43 175205
[8] Ding G T 2011 Acta Phys. Sin. 60 044503 (in Chinese)[丁光涛2011 物理学报60 044503]
[9] Ding G T 2012 Acta Phys. Sin. 61 020204 (in Chinese)[丁光涛2012 物理学报61 020204]
[10] Ding G T 2013 Acta Phys. Sin. 62 064501 (in Chinese)[丁光涛2013 物理学报62 064501]
[11] Luo S K, Zhang Y F 2008 Advances in the Study of Dynamics of Constrained Systems (Beijing: Science Press) (in Chinese) [罗绍凯, 张永发2008 约束系统动力学研究进展(北京: 科学出版社)]
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