  We present a construction of the action, in the framework of the calculus of
variations and Sobolev spaces, describing deformations and the oscillations of
a uniformly rotating, elastic and self-gravitating earth. We establish the
Fr\'{e}chet differentiability of the action under minimal regularity
assumptions, which constrain the possible composition of an earth model. Thus
we obtain well-defined Euler-Lagrange equations, weakly and strongly, that is,
the system of elastic-gravitational equations.
