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 space|