Many Faces of Elastica

“This book provides a treatment of a beautiful area of mathematics and its applications which ties together aspects of classical differential geometry of surfaces and the calculus of variations. … The reader will find in this book a useful introduction to some of the relevant underlying mathematics; there is a nice introduction to the differential geometry of curves and surfaces and certain aspects of the calculus of variations.” (John MuCuan, Mathematical Reviews, April, 2018)

• Preface.- List of Symbols.- Geometry and Variational Calculus.- Plane Curves.- Space Curves.- Surfaces.- Variational Calculus.- Planar Curves Whose 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.- Sturmian Spirals.- Generalized Sturm Spirals.- Serret Curves.- Cassinian Ovals.- Biological Membranes.- Subject Matter and Biological Membranes.- Types of Membranes.- Functions of Biomembranes.- Chemical Composition and Physical Properties of Biomembranes.- Membrane Models and Methods for the Study of Biomembranes.- Model Membrane Structures.- Surface Tension and Equilibrium.- Mechanical Equilibrium.- Tensions and Geometry.- Delaunay Surfaces.- Mylar Balloon and Elastic Curves.- Whewell Parameterization.- Elastic Sturmian Spirals.- Alternative Parametrization of Elastic Spirals.- Zero Tension.- Geometry of the Rotating Liquid Drop.- Introduction.- Geometry and Surface Invariants.- Parametrization via Legendre's Integrals.- Parameterization via Weierstrass's Functions.- Geodesics on the Drop.- Questions for Future Work.- Equations of Equilibrium States of Membranes.- Canham Model.- Key Assumptions in the Model.- Helfrich and Deuling Model.- Ou-Yang and Helfrich Model.- Symmetries of the Shape Equation.- Exact Solutions and Applications.- Unduloids and Nerve Fibers.- Mathematical Model of the Cole Experiment.- Fusion of Membranes.- Cylindrical Membranes.- Beyond Delaunay Surfaces.- Epilogue.- A Elliptical Integrals and Functions.- A.1 Jacobian Elliptic Functions.- A.2 Weierstrassian Elliptic Functions.- References.- Index.