The motion of a rigid body in an inviscid incompressible fluid of inhomogeneous density is considered. The size of the body is taken small with respect to the length scale of the density variations; its shape is otherwise arbitrary. The force and the torque acting on the body in an arbitrary motion are derived from Hamilton's principle of least action, thus offering a variational derivation of Kirchhoff's equations, supplemented by the terms due to the density gradient. The force and the torque due to a density gradient are proportional to the gradient and quadratic in the velocity and the angular velocity. The same coefficients are shown to govern both the inertial behaviour of the body, i.e. the response to accelerations, and the effects of density gradients. The free motion of spheres and two-dimensional circular cylinders is shown to obey a condition akin to the Fermat principle in optics.