  Two questions related to elastic motions are raised and addressed. First: in
which theoretical framework can the equations of motion be written for an
elastic half-space put into uniform rotation? It is seen that nonlinear finite
elasticity provides such a framework for incompressible solids. Second: how can
finite amplitude exact solutions be generated? It is seen that for some finite
amplitude transverse waves in rotating incompressible elastic solids with
general shear response, the solutions are obtained by reduction of the
equations of motion to a system of ordinary differential equations equivalent
to the system governing the central motion problem of classical mechanics. In
the special case of circularly-polarized harmonic progressive waves, the
dispersion equation is solved in closed form for a variety of shear responses,
including nonlinear models for rubberlike and soft biological tissues. A
fruitful analogy with the motion of a nonlinear string is pointed out.
