A circle construction based on elastostatics and hydrodynamics (Q2368304)

In a note on hydrodynamics \textit{C. Darwin} [Proc. Camb. Phil. Soc. 49, 342-354 (1953)] pointed out that ``if a solid cylinder is moving through an infinite, incompressible fluid then the particles of the fluid move along `elastica' curves'' (i.e. along curves satisfying Euler's differential equation for the static equilibrium of slender, elastic beams). In this note we would like to show that the converse is also true: If the homoclinic solution of the ``elastica'' is considered as a quasi-static travelling wave, then the particles of the beam move along semi-circles. Moreover, the homoclinic elastica is the only planar curve to produce this behaviour when considered as a travelling wave.