
Symmetry in Nonlinear Mathematical Physics  2009
Vasyl Kovalchuk (Institute of Fundamental Technological Research, Warsaw, Poland)
Stationary ellipses as special solutions for flat affinelyrigid bodies with 'thickness' in degenerate dimension
Abstract:
We mainly concentrate on the case of flat affinelyrigid bodies with 'thickness' in degenerate dimension, when the additional constraints are implied on the motion of the affinelyrigid body, i.e., that the 'thickness' performs onedimensional oscillations orthogonal to the twodimensional central plane of the flat body. This orthogonality is known in the theory of plates and shells as the KirchhoffLove 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.

