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

Variational formulation of the earth's elastic-gravitational deformations under low regularity conditions