The Euler elastica

The famous Euler elastica [1 - 5] was considered (1997-2005) and it was shown that apart from the known static equilibrium configurations there exist also dynamic equilibrium configurations. In the latter case, the form of elastic curve remains the same, and the bent rod rotates about the vertical axe. The energy of deformation does not change in this motion. Note that we do not speak about the rigid motion of a rod, since the clamped end of the rod remains fixed. This means that the curvilinear equilibrium configuration in the Euler elastica is unstable, contrary to the common point of view. On the other hand, this conclusion is not confirmed by experiments. Thus there appears a paradox requiring its explanation.
  1. Zhilin P.A., Sergeyev A.D., Tovstik T.P. Nonlinear theory of rods and its application // Proc. of XXIV Summer School - Seminar “Nonlinear Oscillations in Mechanical Systems”. St. Petersburg. 1997. P. 313-337. (In Russian.)
  2. Zhilin P.A. Dynamic Forms of Equilibrium Bar Compressed by a Dead Force // Proc. of 1997 1st Int. Conf. Control of Oscillations and Chaos. Vol. 3. P. 399-402.
  3. Zhilin P.A. Advanced problems in mechanics. Selection of articles. V. 1. St. Petersburg. Edition of IPME RAS. 2006. 306 p. (In Russian.)
  4. Zhilin P.A. Advanced problems in mechanics. Selection of articles. V. 2. St. Petersburg. Edition of IPME RAS. 2006. 271 p.
  5. Zhilin P.A.Applied mechanics. Theory of thin elastic rods. Tutorial book. St. Petersburg State Polytechnical University. 2007. 101 p. (In Russian.)