Classical models of affinely-rigid bodies with “thickness” in degenerate dimension. (English) Zbl 1171.70003

Kovalchuk, Vasyl; Rożko, Ewa Eliza

J. Geom. Symmetry Phys. 14, 51-65 (2009).

Summary: The paper is devoted to such situations when the material space of 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, \(m<n\). 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, we obtain some special solutions, namely the stationary ellipses which are analogous to ellipsoidal figures of equilibrium well-known in astro- and geophysics, e.g. in the theory of the Earth’s shape.

MSC:

70B10

Kinematics of a rigid body

Keywords:

one-dimensional oscillations; stationary ellipses