Classical Models of Affinely-Rigid Bodies with “Thickness” in Degenerate Dimension

Abstract

The special interest is devoted to such situations when the mate- rial space of our object with affine degrees of freedom has generally lower dimension than the one of the physical space. In other words when we have $m$-dimensional affinely-rigid body moving in the $n$-dimensional physical space, $m \lt n$. We mainly concentrate on the physical situation $m = 2, n = 3$ when “thickness” of flat bodies performs one-dimensional oscillations orthogonal to the two-dimensional central plane of the body. For the isotropic case in two “flat” dimensions 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, are obtained.