First integrals of the axisymmetric shape equation of lipid membranes

• 来源期刊: 中国物理B（英文版）
• 关键词: lipid membrane shape equation first integral Noether theorem
• 年，卷（期）: 2018,（3）
• 页码范围: 551-559
• 页数: 9页
• 语种: 英文
• DOI: 10.1088/1674-1056/27/3/038704

摘要

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.