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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

x ¨-λzz ˙=0,z ¨+λzx ˙=0,λ>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.

MSC:
53A04Curves in Euclidean space
74B20Nonlinear elasticity