Stationary ellipses as special solutions for flat affinely-rigid bodies with 'thickness' in degenerate dimension

Abstract:
We mainly concentrate on the case of flat affinely-rigid bodies with 'thickness' in degenerate dimension, when the additional constraints are implied on the motion of the affinely-rigid body, i.e., that the 'thickness' performs one-dimensional oscillations orthogonal to the two-dimensional central plane of the flat body. This orthogonality is known in the theory of plates and shells as the Kirchhoff-Love condition. Obtained equations of motion are strongly nonlinear and in a general case there is hardly a hope to solve them analytically. Nevertheless, there exists a way to imaging some features of the phase portrait of such a dynamical system, i.e., we have obtained some special solutions, namely, the stationary ellipses, which are analogous to the ellipsoidal figures of equilibrium well known in astro- and geophysics, e.g., in the theory of the Earth's shape.