Contents. Introduction List of notation

Transcription

1 Contents Introduction List of notation i vii 1 Geometry and variational calculus Planar curves Polar coordinate system Curvature Frenet-Serret equations Space curves Unit tangent vector Tangent line and normal plane Curvature Principal normal vector Principal and oscular plane Binormal and moving trihedron Torsion Frenet-Serret equations of space curves Main theorem in the local theory of curves Surfaces Surfaces and parametric curves First fundamental form Normal vector to the surface A second fundamental form Integrabilty conditions Variational calculus Euler-Lagrange equation First integrals of the Euler-Lagrange equation Euler-Lagrange equation for two independent variables Euler-Lagrange equation for Lagrangians containing second order derivatives 48

2 2 Planar curves which curvature depends only on the distance from a fixed point The moving frame associated with a plane curve Integration Bernoulli s lemniscates Relationship between the lemniscate and the elastica Spirals Sturm spirals Generalized Sturm spirals The case when σ> The case when 0 <σ < The sub-case when 0 <σ <1 and c = Serret curves Generalized Serret curves Cassinian ovals Alternative parameterizations 66 3 Biological membranes Subject matter and biological membranes Types of membranes Functions of biomembranes Chemical composition and physical properties of biomembranes Molecular structure and physicochemical properties of membrane lipids Membrane proteins and glycoproteins Membrane models and methods for the study of biomembranes Modern theories Model membrane structures Lipid associates Model artificial membranes 85 4 Surface tension and balance Mechanical equilibrium Laplace Young equation Axially symmetric membranes Stresses 92

3 4.1.4 The case w = Forms and the corresponding surfaces Delaunay surfaces Nodoids and unduloids Intrinsic equation of the profile curves of Delaunay surfaces Some useful formulas Delaunay construction Nodary Undulary Polyester balloon and elastic curves Bending energy Original formulation and treatment of the problem about elastic curves Parametric representation of curvature of Elastica Intrinsic equation of the Elastica Form of a hanging chain One-dimensional membranes Euler s elasticas Whewell parameterization Introduction Equilibrium equations Elastics with tension Geometry of the rotating liquid drop Surface invariants Parameterization by Legendrian integrals Parameterization by Weierstrassian functions Intrinsic equation of the profile curves Geodesic curves Non-resolved issues Equations of equilibrium states of membranes Canham model Key assumptions in the model Helfrich and Deuling model Model of Ou-Yang and Helfrich Basic formulas and definitions 148

4 5.3.2 Equation of the form Symmetries of the form equation Cartesian coordinates Group-invariant solutions Conformal coordinates Lie equations Determining system of equations Exact solutions and applications Unduloids and nerve fibers Introduction Model Parameterization Parameters of the nerve fibers Sensitivity of the equilibrium forms on the parameters Mathematical model of the Cole experiment Cole model Yoneda method Nodoids and the compression of the spherical eggs Fusion of membranes Stalk model Mathematics of the stalk model Geometric and energetic aspects Cylindrical membranes Translational-invariant solutions Analytical solutions Closure conditions Self-intersections Hele-Shaw cells Beyond Delaunay s surfaces Parametric equations 192

5 Epilogue 197 Appendix A 199 A.1 Elliptic integrals and functions 199 A.2 Jacobian elliptic functions 200 A.3 Weierstrassian elliptic functions 204 Bibliography 207