Closed Elastic Curves and Closed Elastic Curves and RodsRods

I: Euler’s ElasticaI: Euler’s Elastica

AcknowledgementsAcknowledgements

• Joel Langer, Case Western Reserve Univ.Joel Langer, Case Western Reserve Univ.

• Manuel Barros, Univ. of Granada, SpainManuel Barros, Univ. of Granada, Spain

• Oscar Garay, Univ. del Pais Vasco, SpainOscar Garay, Univ. del Pais Vasco, Spain

• Rongpei Huang, East China Normal Univ.Rongpei Huang, East China Normal Univ.

• Thomas Ivey, College of Charleston, S.C.Thomas Ivey, College of Charleston, S.C.

• Vel Jurdjevic, University of Toronto, CanadaVel Jurdjevic, University of Toronto, Canada

• Anders Linner, Northern Illinois Univ.Anders Linner, Northern Illinois Univ.

• Daniel Steinberg, Dim Sum ThinkingDaniel 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 Closed Elastic Curves and RodsRods

II: Elastica in a Space FormII: 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