Conformal Invariance and Conserved Quantities for Lagrange Equation of Thin Elastic Rod

• Authors: Peng Wang , Hui-Rong Feng , Zhi-Mei Lou
• Languages of publication: EN

Abstracts

EN

Basing on the analytical mechanics methods, the Lagrangian equations of thin elastic rod is constructed. The definition of conformal invariance for the Lagrange mechanics of elastic rod is given. The criterions that conformal invariance of elastic rod is the Lie symmetry are obtained based on the Lie point transformation group. The structure equation and conserved quantity deduced from conformal invariance of elastic rod are constructed. Take twist rod as an example to illustrate the application of the results got in this paper.

Keywords

EN

conformal invariance   Lie symmetry   conserved quantities   thin elastic rod   Lagrange equation  

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2017
• Volume: 131
• Issue: 2
• Pages: 283-287

Dates

published

2017-02

received

2016-11-01

References

• [1] F. Mei, Symmetries and Conserved Quantities of Constrained Mechanical Systems, Beijing Institute of Technology Press, Beijing 2004 (in Chinese)
• [2] R. de Ritis, G. Marmo, G. Platania, C. Rubano, P. Scudellaro, C. Stornaiolo, Phys. Rev. D 42, 1091 (1990), doi: 10.1103/PhysRevD.42.1091
• [3] V. Dorodnitsyn, Appl. Num. Math. 39, 307 (2001), doi: 10.1016/S0168-9274(00)00041-6
• [4] D. Levi, P. Winternitz, J. Phys. A Math. Gen. 39, 1 (2006), doi: 10.1088/0305-4470/39/2/R01
• [5] P. Wang, Nonlin. Dyn. 68, 53 (2012), doi: 10.1007/s11071-011-0203-3
• [6] P. Wang, H.J. Zhu, Acta Phys. Pol. A 119, 298 (2011), doi: 10.12693/APhysPolA.119.298
• [7] Z.X. Long, Y. Zhang, Acta Mech. 225, 77 (2013), doi: 10.1007/s00707-013-0956-5
• [8] M. Lutzky, Int. J. Non-Lin. Mech. 34, 387 (1999), doi: 10.1016/S0020-7462(98)00024-9
• [9] F.X. Mei, Acta Mech. 141, 135 (2000), doi: 10.1007/BF01268673
• [10] F.X. Mei, Chin. Phys. 10, 177 (2001), doi: 10.1088/1009-1963/10/3/301
• [11] J.H. Fang, Chin. Phys. B 19, 040301 (2010), doi: 10.1088/1674-1056/19/4/040301
• [12] L.Q. Jia, X.X. Wang, M.L. Zhang, Y.L. Han, Nonlin. Dyn. 69, 1807 (2012), doi: 10.1007/s11071-012-0387-1
• [13] S.K. Luo, Z.J. Li, W. Peng, L. Li, Acta Mech. 224, 71 (2013), doi: 10.1007/s00707-012-0733-x
• [14] S.K. Luo, Y.L. Xu, Int. J. Theor. Phys. 54, 572 (2015), doi: 10.1007/s10773-014-2249-8
• [15] S.K. Luo, Z.J. Li, L. Li, Acta Mech. 223, 2621 (2012), doi: 10.1007/s00707-012-0729-6
• [16] F.X. Mei, H.B. Wu, Y.F. Zhang, Int. J. Dyn. Control. 2, 285 (2014), doi: 10.1007/s40435-013-0043-8
• [17] Z.M. Lou, Acta Phys. Sin. 62, 220201 (2013) (in Chinese), doi: 10.7498/aps.62.220201
• [18] C.J. Song, Y. Zhang, Int. J. Theor. Phys. 54, 2481 (2015), doi: 10.1007/s10773-014-2475-0
• [19] A.S. Galiullin, G.G. Gafarov, R.P. Malaishka, A.M. Khwan, Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems, UFN, Moscow 1997
• [20] F.X. Mei, J.F. Xie, T.Q. Gang, Acta Mech. Sin. 24, 583 (2008), doi: 10.1007/s10409-008-0176-8
• [21] C. Liu, F.X. Mei, Y.X. Guo, Acta Phys. Sin. 57, 6704 (2008) (in Chinese), doi: 10.7498/aps.63.140201
• [22] X.W. Chen, Y.M. Li, Y.H. Zhao, Chin. Phys. B 18, 3139 (2009), doi: 10.1088/1674-1056/18/8/007
• [23] Y.P. Luo, J.L. Fu, Chin. Phys. B 19, 090303 (2010), doi: 10.1088/1674-1056/19/9/090303
• [24] J.L. Cai, Int. J. Theor. Phys. 49, 201 (2010), doi: 10.1007/s10773-009-0193-9
• [25] J.L. Cai, Nonlin. Dyn. 69, 487 (2012), doi: 10.1007/s11071-011-0279-9
• [26] Y.Y. Zhang, F. Zhang, Y.L. Han, L.Q. Jia, Nonlin. Dyn. 77, 521 (2014), doi: 10.1007/s11071-014-1314-4
• [27] E.E. Zajac, Trans. ASME J. Appl. Mech. 29, 136 (1962), doi: 10.1115/1.3636445
• [28] G.H.M. Van der Heijden, Proc. R. Soc. Lond. A 457, 695 (2001), doi: 10.1098/rspa.2000.0688
• [29] C.J. Benham, Proc. Natl. Acad. Sci. USA 74, 2937 (1977), doi: 10.1073/pnas.74.6.2397
• [30] Y.Z. Liu, Nonlinear Mechanics of Thin Elastic Rod-Theoretical Basis of Mechanical Model of DNA, Tsinghua Press and Springer, Beijing 2006 (in Chinese)
• [31] Y.M. Shi, J.E. Hearst, J. Chem. Phys. 101, 5186 (1994), doi: 10.1063/1.468506
• [32] R.S. Manning, K.A. Rogers, J.H. Maddocks, Proc. R. Soc. Lond. A 454, 3047 (1998), doi: 10.1098/rspa.1998.0291
• [33] A. Goriely, M. Tabor, Phys. Rev. Lett. 77, 3537 (1996), doi: 10.1103/PhysRevLett.77.3537
• [34] I. Tobias, D. Swigon, B.D. Coleman, Phys. Rev. E 61, 747 (2000), doi: 10.1103/PhysRevE.61.747
• [35] Y. Xue, Y.Z. Liu, Acta Phys. Sin. 5, 6737 (2009) (in Chinese), doi: 10.3321/j.issn:1000-3290.2009.10.012
• [36] Y.Z. Liu, Y. Xue, Appl. Math. Mech. Engl. Ed. 32, 603 (2011), doi: 10.1007/s10483-011-1442-8
• [37] B.D. Coleman, E.H. Dill, D. Swigon, Archiv. Ration. Mechan. Anal. 129, 147 (1995), doi: 10.1007/BF00379919
• [38] J.H. Maddocks, D.J. Dichmann, J. Elastic. 34, 83 (1994), doi: 10.1007/BF00042427
• [39] J.L. Fu, W.J. Zhao, Y.Q. Weng, Chin. Phys. 17, 2361 (2008), doi: 10.1088/1674-1056/17/7/007
• [40] P. Jung, S. Leyendecker, J. Linn, M. Ortiz, Int. J. Num. Meth. Eng. 85, 31 (2011), doi: 10.1002/nme.2950
• [41] Y. Xue, P. Wang, Acta Phys. Sin. 60, 114501 (2011) (in Chinese), doi: 10.7498/aps.60.114501
• [42] P. Wang, Y. Xue, Y.L. Liu, Chin. Phys. B 21, 070203 (2012), doi: 10.1088/1674-1056/21/7/070203
• [43] P. Wang, Y. Xue, Y.L. Liu, Chin. Phys. B 22, 104503 (2013), doi: 10.1088/1674-1056/22/10/104503
• [44] P. Wang, Y. Xue, Nonlin. Dyn. 83, 1815 (2016), doi: 10.1007/s11071-015-2448-8
• [45] Y. Xue, Y.Z. Liu, L.Q. Chen, Chin. J. Theor. Appl. Mech. 37, 485 (2005) (in Chinese), doi: 10.3321/j.issn:0459-1879.2005.04.014

Identifiers

DOI

10.12693/APhysPolA.131.283