Abstract
In this paper, the Lie symmetries and conserved quantities of the relative motion systems on time scales are proposed and studied. The Lagrange equations with delta derivatives on time scales are derived. By defining the infinitesimal transformations generators and using the invariance of differential equations under infinitesimal transformations, the determining equations of the Lie symmetries on time scales are established. Then, the structural equation and the form of conserved quantities of the Lie symmetries are given. Lie symmetries of the relative motion systems in discrete and continuous systems were discussed, respectively. Finally, an example is given to illustrate the applications of the conclusion.
Similar content being viewed by others
References
B Aulbach and S Hilger Qualitative Theory of Differential Equations, Szeged (1988)
F M Atici, D C Biles and A Lebedinsky Math. Comput. Model. 43 718 (2006)
Z Bartosiewicz IFAC Proceedings Volumes, 38 435 (2005)
A Peterson and C Allan Dynamic Equations on Time Scales, (Boston: Birkhäuser) (2001)
C D Ahlbrandt and C Morian J. Comput. Appl. Math. 141 35 (2002)
M Bohner Dyn. Syst. Appl. 13 339 (2004)
A C F Rui and D F M Torres Math. Control T&Finance. (2008). https://doi.org/10.1007/978-3-540-69532-5_9
Z Bartosiewicz, N Martins and D F M Torres Eur. J. Control17 9 (2011)
Z Bartosiewicz and D F M Torres J. Math. Anal. Appl. 342 1220 (2008)
N Martins and D F M Torres Appl. Math. Lett. 23 1432 (2010)
D F M Torres Maths36 33–38 (2003)
P P Cai, J L Fu and Y X Guo Sci. China-Phys. Mech. Astron. 5 1017 (2013)
C J Song and Y Zhang J. Math. Phys. 56 10 (2015)
X H Zhai and Y Zhang Commun. Nonlinear Sci. Numer Simul.52 32 (2017)
A I Lur’e Analytical Mechanics (Moscow:GIFML) (1961) (in Russian)
L Q Chen Mech. Mater. Struct. 14 63 (1992)
R W Liu Coll. Phys. 12 3 (1993)
R W Liu, J L Fu and F X Mei J. Beijing Inst. Technol. (Eng. Ed.) 7 221 (1998)
F X Mei and H B Wu Acta Phys. Sin. 58 5919 (2009)
X X Wang, X T Sun and M L Zhang Acta Phys. Sin. 61 64501 (2012)
X W Zhang Acta Phys. Sin. 55 2610 (2006)
M Lutzky J. Phys. A Math. Gen. 12 973 (1979)
R W Liu and J L Fu Appl. Math. Mech. 20 635 (1999)
J L Fu, B Y Chen and H Fu Sci. China Phys. Mech. Astron.54 288–295 (2011)
D Levi and P Winternitz Phys. Lett. A152 335 (1991)
R P Agarwal, M Bohner and A Peterson J. Comput. Appl. Math. 141 1 (2002)
P P Cai, J L Fu, S Duan and F Y Hong China-Phys. (2012)
P P Cai and J L Fu Rep. Math. Phys. 79 279 (2017)
Acknowledgements
This work has been supported by the National Natural Science Foundation of China (Grant No. 11472247).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gong, SN., Gao, HF. & Fu, JL. Lie symmetries of the relative motion systems on time scales. Indian J Phys 94, 371–377 (2020). https://doi.org/10.1007/s12648-019-01486-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12648-019-01486-8