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Euler’s elastica and beyond. (English) Zbl 1209.37086

In 1691, James (Jacob) Bernoulli proposed the elastica problem, which was essentially solved by Euler in 1744: What shape can be obtained when an elastica, an ideal infinitesimally thin elastic rod, is bent without stretching on a plane?

The author reviews the studies and classification of Bernoulli and Euler from an historic point of view and indicates connections with elliptic curves, lemniscate, nonlinear integrable differential equations, etc.

The second part of the paper summarizes the author’s work over two decades on the quantized elastica problem, i.e., statistical mechanics of elastica in a heat bath, which serves as a model of the DNA. In this context the modified Korteweg–de Vries hierarchy, loop space, and the submanifold Dirac operator appear.

MSC:
37K20Relations of infinite-dimensional systems with algebraic geometry, etc.
82D60Polymers (statistical mechanics)
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q53KdV-like (Korteweg-de Vries) equations
74G65Energy minimization (equilibrium problems in solid mechanics)
58E50Applications of variational methods in infinite-dimensional spaces
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics