  We investigate the problem of determining the shape of a rotating celestial
object - e.g., a comet or an asteroid - under its own gravitational field. More
specifically, we consider an object symmetric with respect to one axis - such
as a dumbbell - that rotates around a second axis perpendicular to the symmetry
axis. We assume that the object can be modeled as an incompressible fluid of
constant mass density, which is regarded as a first approximation of an
aggregate of particles.
  In the literature, the gravitational field of a body is often described as a
multipolar expansion involving spherical coordinates (Kaula, 1966). In this
work we describe the shape in terms of cylindrical coordinates, which are most
naturally adapted to the symmetry of the body, and we express the gravitational
potential generated by the rotating body as a simple formula in terms of
elliptic integrals. An equilibrium shape occurs when the gravitational
potential energy and the rotational kinetic energy at the surface of the body
balance each other out. Such an equilibrium shape can be derived as a solution
of an optimization problem, which can be found via the variational method. We
give an example where we apply this method to a two-parameter family of
dumbbell shapes, and find approximate numerical solutions to the corresponding
optimization problem.
