First integrals of the axisymmetric shape equation of lipid membranes

doi:10.1088/1674-1056/27/3/038704

Abstract

The shape equation of lipid membranes is a fourth-order partial differential equation. Under the axisymmetric condition, this equation was transformed into a second-order ordinary differential equation (ODE) by Zheng and Liu (Phys. Rev. E 48 2856 (1993)). Here we try to further reduce this second-order ODE to a first-order ODE. First, we invert the usual process of variational calculus, that is, we construct a Lagrangian for which the ODE is the corresponding Euler-Lagrange equation. Then, we seek symmetries of this Lagrangian according to the Noether theorem. Under a certain restriction on Lie groups of the shape equation, we find that the first integral only exists when the shape equation is identical to the Willmore equation, in which case the symmetry leading to the first integral is scale invariance. We also obtain the mechanical interpretation of the first integral by using the membrane stress tensor.

Keywords: lipid membrane shape equation first integral Noether theorem

Received: 30 September 2017
Revised: 07 December 2017
Accepted manuscript online:

PACS:

87.16.D-

(Membranes, bilayers, and vesicles)

Fund:

Project supported by the National Natural Science Foundation of China (Grant No. 11274046) and the National Science Foundation of the United States (Grant No. 1515007).

Corresponding Authors: Zhan-Chun Tu
E-mail: tuzc@bnu.edu.cn

Cite this article:

Yi-Heng Zhang(张一恒), Zachary McDargh, Zhan-Chun Tu(涂展春) First integrals of the axisymmetric shape equation of lipid membranes 2018 Chin. Phys. B 27 038704

[1]	Fernando C M and André N 2014 Ann. Math. 179 683
[2]	Tek S 2007 J. Math. Phys. 48 013505
[3]	Helfrich W 1973 Z. Naturforsch. C 28 693
[4]	Ou-Yang Z C and Helfrich W 1987 Phys. Rev. Lett. 59 2486
[5]	Ou-Yang Z C and Helfrich W 1989 Phys. Rev. A 39 5280
[6]	Konopelchenko B G 1997 Phys. Lett. B 414 58
[7]	Lipowsky R 1991 Nature 349 475
[8]	Landolfi G 2003 J. Phys. A:Math. Theor. 36 11937
[9]	Deuling H J and Helfrich W 1976 Biophysical Journal 16 861
[10]	Seifert U 1997 Adv. Phys. 46 13
[11]	Ou-Yang Z C, Liu J X and Xie Y Z 1999 Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases (Singapore:World Scientific) p. 29
[12]	Capovilla R and Guven J 2002 J. Phys. A:Math. Theor. 35 6233
[13]	Capovilla R, Guven J and Santiago J A 2003 J. Phys. A:Math. Theor. 36 6281
[14]	Capovilla R and Guven J 2004 J. Phys.:Condens. Matter 16 S2187
[15]	Capovilla R, Guven J and Rojas E 2005 J. Phys. A:Math. Theor. 38 8201
[16]	Capovilla R, Guven J and Rojas E 2005 J. Phys. A:Math. Theor. 38 8841
[17]	Tu Z C and Ou-Yang Z C 2004 J. Phys. A:Math. Theor. 37 11407
[18]	Tu Z C 2013 Chin. Phys. B 22 28701
[19]	Zhang Y 2008 Acta Phys. Sin. 57 2649 (in Chinese)
[20]	Liao Y H, Zhu H C, Tang Z M and Huang K M 2015 Chin. Phys. B 24 100204
[21]	Wang J J, Feng J W and Ren C L 2015 Chin. Phys. B 24 088701
[22]	Zheng B, Meng Q T, Robin L B Selingerc, Jonathan V Selingerc and Ye F F 2015 Chin. Phys. B 24 068701
[23]	Chen X J and Liang Q 2017 Chin. Phys. B 26 048701
[24]	Li S L and Zhang S G 2010 Acta Phys. Sin. 59 5202 (in Chinese)
[25]	Zhang P W and Shi A C 2015 Chin. Phys. B 24 128707
[26]	Thomsen G 1923 Abh. Math. Sem. Univ. Hamburg 3 31
[27]	Nitsche J 1993 Q. Appl. Math. 51 363
[28]	Hu J G and Ou-Yang Z C 1993 Phys. Rev. E 47 461
[29]	Zheng W M and Liu J X 1993 Phys. Rev. E 48 2856
[30]	Naito H, Okuda M and Ou-Yang Z C 1995 Phys. Rev. Lett. 74 4345
[31]	Mladenov I M 2002 Eur. Phys. J. B 29 327
[32]	Ou-Yang Z C 1990 Phys. Rev. A 41 4517
[33]	Ou-Yang Z C 1993 Phys. Rev. E 47 747
[34]	Castro-Villarreal P and Guven J 2007 Phys. Rev. E 76 011922
[35]	Zhang S G and Ou-Yang Z C 1996 Phys. Rev. E 53 4206
[36]	Vassilev V M, Djondjorov P A and Mladenov I M 2008 J. Phys. A:Math. Theor. 41 435201
[37]	Zhou X H 2010 Chin. Phys. B 19 58702
[38]	Naito H, Okuda M and Ou-Yang Z C 1993 Phys. Rev. E 48 2304
[39]	Naito H, Okuda M and Ou-Yang Z C 1996 Phys. Rev. E 54 2816
[40]	Ibragimov N H 1969 Theor. Math. Phys. 1 267
[41]	Vassilev V M and Mladenov I M 2004 Proceedings of the Fifth International Conference on Geometry, Integrability and Quantization (Sofia:Softex) p. 246
[42]	Vassilev V M, Djondjorov P A, Atanassov E, Hadzhilazova M T and Mladenov I M 2014 AIP Conference Proceedings 1629 201
[43]	Vassilev V M, Djondjorov P A, Hadzhilazova M T and Mladenov I M 2016 Proceedings of the Seventeenth International Conference on Geometry, Integrability and Quantization (Sofia:Avangard Prima) p. 369
[44]	Müller M M, Deserno M and Guven J 2005 Phys. Rev. E 72 061407

[1]	Curvature-induced lipid segregation Zheng Bin (郑斌), Meng Qing-Tian (孟庆田), Robin L. B. Selinger, Jonathan V. Selinger, Ye Fang-Fu (叶方富). Chin. Phys. B, 2015, 24(6): 068701.
[2]	Challenges in theoretical investigations of configurations of lipid membranes Tu Zhan-Chun (涂展春). Chin. Phys. B, 2013, 22(2): 028701.
[3]	Fractional charges and fractional spins for composite fermions in quantum electrodynamics Wang Yong-Long(王永龙), Lu Wei-Tao(卢伟涛), Jiang Hua(蒋华) Xu Chang-Tan(许长谭), and Pan Hong-Zhe(潘洪哲) . Chin. Phys. B, 2012, 21(7): 070501.
[4]	The approximate conserved quantity of the weakly nonholonomic mechanical-electrical system Liu Xiao-Wei(刘晓巍), Li Yuan-Cheng(李元成), and Xia Li-Li(夏丽莉) . Chin. Phys. B, 2011, 20(7): 070203.
[5]	Traveling wave solutions for two nonlinear evolution equations with nonlinear terms of any order Feng Qing-Hua(冯青华), Meng Fan-Wei(孟凡伟), and Zhang Yao-Ming(张耀明) . Chin. Phys. B, 2011, 20(12): 120202.
[6]	Applications of the first integral method to nonlinear evolution equations Filiz Tacscan and Ahmet Bekir . Chin. Phys. B, 2010, 19(8): 080201.
[7]	Travelling solitary wave solutions for the generalized Burgers--Huxley equation with nonlinear terms of any order Deng Xi-Jun(邓习军), Yan Zi-Zong(燕子宗), and Han Li-Bo(韩立波). Chin. Phys. B, 2009, 18(8): 3169-3173.
[8]	Some new exact solutions to the Burgers--Fisher equation and generalized Burgers--Fisher equation Jiang Lu(姜璐), Guo Yu-Cui(郭玉翠), and Xu Shu-Jiang(徐淑奖). Chin. Phys. B, 2007, 16(9): 2514-2522.
[9]	Discrete variational principle and first integrals for Lagrange--Maxwell mechanico-electrical systems Fu Jing-Li(傅景礼), Dai Gui-Dong(戴桂冬), Salvador Jimènez(萨尔瓦多·希梅尼斯), and Tang Yi-Fa(唐贻发). Chin. Phys. B, 2007, 16(3): 570-577.
[10]	The discrete variational principle and the first integrals of Birkhoff systems Zhang Hong-Bin(张宏彬), Chen Li-Qun(陈立群), Gu Shu-Long(顾书龙), and Liu Chuan-Zhang(柳传长). Chin. Phys. B, 2007, 16(3): 582-587.
[11]	A Birkhoff-Noether method of solving differential equations Shang Mei(尚玫), Guo Yong-Xin(郭永新), and Mei Feng-Xiang (梅凤翔). Chin. Phys. B, 2007, 16(2): 292-295.
[12]	First integrals and stability of second-order differential equations Xu Xue-Jun (许学军), Mei Feng-Xiang (梅凤翔). Chin. Phys. B, 2006, 15(6): 1134-1136.
[13]	Direct method of finding first integral of two-dimensional autonomous systems in polar coordinates Lou Zhi-Mei (楼智美), Wang Wen-Long (汪文珑). Chin. Phys. B, 2006, 15(5): 895-898.
[14]	The discrete variational principle in Hamiltonian formalism and first integrals Zhang Hong-Bin (张宏彬), Chen Li-Qun (陈立群), Liu Rong-Wan (刘荣万). Chin. Phys. B, 2005, 14(6): 1063-1068.
[15]	Discrete variational principle and the first integrals of the conservative holonomic systems in event space Zhang Hong-Bin (张宏彬), Chen Li-Qun (陈礼群), Liu Rong-Wan (刘荣万). Chin. Phys. B, 2005, 14(5): 888-892.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

First integrals of the axisymmetric shape equation of lipid membranes

Cite this article:

Online attention