## Journal of Geometry and Symmetry in Physics

- J. Geom. Symmetry Phys.
- Volume 17 (2010), 45-86.

### Euler's Elastica and Beyond

#### Abstract

In 1691, James (Jacob) Bernoulli proposed a problem called *elastica problem*: What shape of elastica, an ideal thin elastic rod on a plane, is allowed? Daniel Bernoulli discovered its energy functional, *Euler-Bernoulli energy function*, and the minimal principle of the elastica. Using it, Euler essentially solved the problem in 1744 by developing the variational method, elliptic integral theory and so on. This article starts with a review of its mathematical meaning and historical background. After that we present one of its extensions, statistical mechanics of elastica as a model of the DNA and the large polymers. We will call it a quantized elastica, and show that it is connected with the modified Korteweg-de Vries hierarchy, loop space, submanifold Dirac operators, moduli spaces of the real hyperelliptic curves and so on. By reviewing the other extensions of the elastica problem, we will see that elastica is in the center of mathematics even now.

#### Article information

**Source**

J. Geom. Symmetry Phys., Volume 17 (2010), 45-86.

**Dates**

First available in Project Euclid: 24 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.jgsp/1495591319

**Digital Object Identifier**

doi:10.7546/jgsp-17-2010-45-86

**Mathematical Reviews number (MathSciNet)**

MR2642065

**Zentralblatt MATH identifier**

1209.37086

#### Citation

Matsutani, Shigeki. Euler's Elastica and Beyond. J. Geom. Symmetry Phys. 17 (2010), 45--86. doi:10.7546/jgsp-17-2010-45-86. https://projecteuclid.org/euclid.jgsp/1495591319