Closed Elastic Curves and Rods I: Euler’s Elastica

Acknowledgements Joel Langer, Case Western Reserve Univ. Manuel Barros, Univ. of Granada, Spain Oscar Garay, Univ. del Pais Vasco, Spain Rongpei Huang, East China Normal Univ. Thomas Ivey, College of Charleston, S.C. Vel Jurdjevic, University of Toronto, Canada Anders Linner, Northern Illinois Univ. Daniel Steinberg, Dim Sum Thinking

Classical Bernoulli-Euler Elastica

Bending Energy Functionals

Deriving the Euler Equations

The Euler-Lagrange Equations

Use the Frenet Equations

Use Rotational Symmetry

Killing Fields

Conserved Quantities

Planar Elastic Curves

Cylindrical Coordinates

Non-planar elastic curves

A little bit of elliptic integrals

The Figure-Eight elastica

Finding Closed Curves

Closed Elastic Curves and Rods II: Elastica in a Space Form

Riemannian Geometry

Curves in M

Some Useful Formulas

Proof of (3)

Euler-Lagrange Equations

Solutions

Killing Fields

Elastica on the 2-sphere

Wavelike elastic curves on the sphere

Classifying wavelike elastic curves on the sphere