  Soft elastic capsules which are driven through a viscous fluid undergo shape
deformation coupled to their motion. We introduce an iterative solution scheme
which couples hydrodynamic boundary integral methods and elastic shape
equations to find the stationary axisymmetric shape and the velocity of an
elastic capsule moving in a viscous fluid at low Reynolds numbers. We use this
approach to systematically study dynamical shape transitions of capsules with
Hookean stretching and bending energies and spherical rest shape sedimenting
under the influence of gravity or centrifugal forces. We find three types of
possible axisymmetric stationary shapes for sedimenting capsules with fixed
volume: a pseudospherical state, a pear-shaped state, and buckled shapes.
Capsule shapes are controlled by two dimensionless parameters, the
F\"oppl-von-K\'arm\'an number characterizing the elastic properties and a Bond
number characterizing the driving force. For increasing gravitational force the
spherical shape transforms into a pear shape. For very large bending rigidity
(very small F\"oppl-von-K\'arm\'an number) this transition is discontinuous
with shape hysteresis. The corresponding transition line terminates, however,
in a critical point, such that the discontinuous transition is not present at
typical F\"oppl-von-K\'arm\'an numbers of synthetic capsules. In an additional
bifurcation, buckled shapes occur upon increasing the gravitational force. This
type of instability should be observable for generic synthetic capsules. All
shape bifurcations can be resolved in the force-velocity relation of
sedimenting capsules, where up to three capsule shapes with different
velocities can occur for the same driving force. All three types of possible
axisymmetric stationary shapes are stable with respect to rotation during
sedimentation.
