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Chin. Phys. B, 2018, Vol. 27(2): 020502    DOI: 10.1088/1674-1056/27/2/020502
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Generalized Chaplygin equations for nonholonomic systems on time scales

Shi-Xin Jin(金世欣)1, Yi Zhang(张毅)2
1. School of Science, Nanjing University of Science and Technology, Nanjing 210094, China;
2. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China
Abstract  The generalized Chaplygin equations for nonholonomic systems on time scales are proposed and studied. The Hamilton principle for nonholonomic systems on time scales is established, and the corresponding generalized Chaplygin equations are deduced. The reduced Chaplygin equations are also presented. Two special cases of the generalized Chaplygin equations on time scales, where the time scales are equal to the set of real numbers and the integer set, are discussed. Finally, several examples are given to illustrate the application of the results.
Keywords:  nonholonomic system      generalized Chaplygin equations      time scales     
Received:  15 July 2017      Published:  05 February 2018
PACS:  05.45.-a (Nonlinear dynamics and chaos)  
  11.10.Ef (Lagrangian and Hamiltonian approach)  
  02.30.Hq (Ordinary differential equations)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11572212 and 11272227) and the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province, China (Grant No. KYLX16-0414).
Corresponding Authors:  Yi Zhang     E-mail:  zhy@mail.usts.edu.cn
About author:  05.45.-a; 11.10.Ef; 02.30.Hq

Cite this article: 

Shi-Xin Jin(金世欣), Yi Zhang(张毅) Generalized Chaplygin equations for nonholonomic systems on time scales 2018 Chin. Phys. B 27 020502

[1] Hilger S 1988 "Ein Mαβkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten", Ph. D. thesis (Universit Würzburg)
[2] Ahlbrand D, Bohner M and Ridenhour J 2000 J. Math. Anal. Appl. 250 561
[3] Bohner M and Peterson A 2001 Dynamic Equations on Time Scale:An Introduction with Applications (Boston:Birkhäuser)
[4] Bohner M and Peterson A 2003 Advances in Dynamic Equations on Time Scales (Boston:Birkhäuser)
[5] Agarwal R, Martin B, et al. 2004 J. Comput. Appl. Math. 141 1
[6] Bohner M and Guseinov G S H 2004 Dyn. Sys. Appl. 13 351
[7] MAtici F, Biles D C and Lebedinsky A 2006 Math. Comput. Definition 43 718
[8] Zhang H T and Li Y K 2009 Commun. Nonlinear Sci. Numer. Simul. 14 19
[9] Abdeljawad T, Jarad F and Baleanu D 2009 Adv. Differ. Eqs. 30 840386
[10] Hilscher R Š and Zeidan V 2012 Nonlinear Anal. 75 932
[11] Ferreira R A C and Torres D F M 2007 Math. Control Theory Fin. 136 145
[12] Bohner M 2004 Dyn. Sys. Appl. 13 339
[13] Martins N and Torres D F M 2009 Nonlinear Anal. 71 e763
[14] Torres D F M 2010 Int. J. Sim. Mult. Des. Optim. 4 11
[15] Bartosiewicz Z, Martins N and Torres D F M 2010 Eur. J. Control 17 9
[16] Malinowska A B and Ammi M R S 2014 Int. J. Differ. Eqs. 9 87
[17] Guo Z H and Gao P Y 1990 Acta Mech. Sin. 22 185(in Chinese)
[18] Chen B 1991 Acta Mech. Sin. 23 379(in Chinese)
[19] Guo Z H and Gao P Y 1992 Acta Mech. Sin. 24 253(in Chinese)
[20] Cai P P, Fu J L and Guo Y X 2013 Sci. China G-Phys. Mech. Astron. 56 1017
[21] Zu Q H and Zhu J Q 2016 J. Math. Phys. 57 082701
[22] Song C J and Zhang Y 2015 J. Math. Phys. 56 102701
[23] Jin S X and Zhang Y 2017 Chin. Phys. B 26 014501
[24] Martins N and Torres D F M 2010 Appl. Math. Lett. 23 1432
[25] Malinowska A B and Martins N 2013 Abstr. Appl. Anal. 2013 1728
[26] Carlson D A 2015 J. Optim. Theory Appl. 166 351
[27] Zhang Y 2016 Chin. Quar. Mech. 37 214(in Chinese)
[28] Jin S X and Zhang Y 2017 Chin. Quar. Mech. 38 447(in Chinese)
[29] Mei F X 2013 Analytical Mechanics (Beijing:Beijing Institute of Technology Press) (in Chinese)
[30] Chen X W and Mei F X 2016 Mech. Res. Commun. 76 91
[31] Guo H Y, Li Y Q, et al. 2002 Commun. Theor. Phys. 37 1
[32] Mei F X, Liu D and Luo Y 1991 Advanced Analytical Mechanics (Beijing:Beijing Institute of Technology Press) (in Chinese)
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