Explicit parameterization of Euler’s elastica. (English) Zbl 1196.53004

Mladenov, Ivaïlo M. (ed.), Proceedings of the 9th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 8–13, 2007. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-42-4/pbk). 175-186 (2008).
The authors deal with the dynamic system modeling of a novel explicit parametrization of Euler’s elastica. It has the form of a system of nonlinear coupled ordinary differential equations
\[ \ddot x-\lambda z\dot z=0,\quad \ddot z+\lambda z\dot x=0,\quad \lambda>0. \]
The solution \((x,z)\) is expressed using the complete elliptic integral of the first kind. A geometrical and mechanical analysis of the solution is performed.
For the entire collection see [Zbl 1154.17001].


53A04 Curves in Euclidean and related spaces
74B20 Nonlinear elasticity