  We study the dynamics of an elastic body whose shape and position evolve due
to the gravitational forces exerted by a pointlike planet. We work in the
quadrupole approximation. We consider the solution in which the center of mass
of the body moves on a circular orbit, and the body rotates in a synchronous
way about its axis, so that it always shows the same face to the planet as the
Moon does with the Earth. We prove that if any internal deformation of the body
dissipates some energy, then such an orbit is locally asymptotically stable.
The proof is based on the construction of a suitable system of coordinates and
on the use of LaSalle's principle. A large part of the paper is devoted to the
analysis of the kinematics of an elastic body interacting with a gravitational
field. We think this could have some interest in itself.
