  Rocking rigid bodies appear in several shapes in everyday life: As furniture
like rocking chairs and rocking cradles or as toys like rocking horses or
tilting dolls. The familiar rocking motion of these objects, a non-linear
combination of a rigid rotation and a translation of the center of mass, gives
rise to a number of interesting dynamical properties. However, their study has
received little attention in the literature.
  This work presents a comprehensive introduction to the dynamics of rocking
rigid bodies, including a concise derivation of the equations of motion as well
as a general inversion procedure to construct rocking rigid body shapes with
specified dynamical properties. Moreover, two novel rigid body shapes are
derived - the tautochrone shape and the brachistochrone shape - which represent
an intriguing generalization of the well-know tautochrone and brachistochrone
curves. In particular, tautochrone shapes offer an alternative construction of
a tautochrone pendulum, in addition to Huygens' cycloid pendulum solution.
