The Many Faces of Elastica

1. Front Matter

   Pages i-xv

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2. Geometry and Variational Calculus

   • Ivaïlo M. Mladenov, Mariana Hadzhilazova

   Pages 1-46

3. Planar Curves Whose Curvature Depends Only on the Distance From a Fixed Point

   • Ivaïlo M. Mladenov, Mariana Hadzhilazova

   Pages 47-68

4. Biological Membranes

   • Ivaïlo M. Mladenov, Mariana Hadzhilazova

   Pages 69-82

5. Surface Tension and Equilibrium

   • Ivaïlo M. Mladenov, Mariana Hadzhilazova

   Pages 83-149

6. Equations of Equilibrium States of Membranes

   • Ivaïlo M. Mladenov, Mariana Hadzhilazova

   Pages 151-166

7. Exact Solutions and Applications

   • Ivaïlo M. Mladenov, Mariana Hadzhilazova

   Pages 167-199

8. Back Matter

   Pages 201-212

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This book provides an introduction to the mathematical aspects of Euler's elastic theory and its application. The approach is rigorous, as well as visually depicted, and can be easily digested. The first few chapters introduce the needed mathematical concepts from geometry and variational calculus. The formal definitions and proofs are always illustrated through complete derivations and concrete examples. In this way, the reader becomes acquainted with Cassinian ovals, Sturmian spirals, co-Lemniscates, the nodary and the undulary, Delaunay surfaces, and their generalizations. The remaining chapters discuss the modeling of membranes, mylar balloons, rotating liquid drops, Hele-Shaw cells, nerve fibers, Cole's experiments, and membrane fusion. The book is geared towards applied mathematicians, physicists and engineers interested in Elastica Theory and its applications.

• Bernoulli’s lemniscates
• Biological membranes
• Canham model
• Cassinian ovals
• Cole model
• Delaunay surfaces
• Deuling model
• Frenet-Serret equations
• Hele-Shaw cells
• Helfrich model
• Laplace-Young equation
• Whewell parameterization
• Yoneda method
• rotating liquid drop
• local theory of curves
• Sturmian spirals
• nodoid
• unduloid
• mylar balloon
• Stalk model

“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)

• Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia, Bulgaria

  Ivaïlo M. Mladenov, Mariana Hadzhilazova

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