Total gravitational energy of a slightly ellipsoidal trilayer planet

We use a perturbational technique to compute the total gravitational energy of a slightly ellipsoidal trilayer planet as a function of the two sets of Euler angles. A second order computation is required since the torque is proportional to the product of the ellipticities of the inner core and the mantle. Although we focus on ellipsoidal perturbations, the intermediate analytical expressions are valid for arbitrary small deformations of the spherical configuration. The primary application of the expression for the total gravitational energy is in the Lagrangian formulation of dynamics. As a by-product, we determine the gravitationally stable equilibrium orientation of the rigid inner core. Due to symmetry, the six coaxial configurations are equilibrium. We show how to identify the stable configuration and to prove its uniqueness.