  We study the apsidal precession of a Physical Symmetrical Pendulum (Allais'
precession) as a generalization of the precession corresponding to the Ideal
Spherical Pendulum (Airy's Precession). Based on the Hamilton-Jacobi formalism
and using the technics of variation of parameters along with the averaging
method, we obtain approximate solutions, in terms of which the motion of both
systems admits a simple geometrical description. The method developed in this
paper is considerably simpler than the standard one in terms of elliptical
functions and the numerical agreement with the exact solutions is excellent. In
addition, the present procedure permits to show clearly the origin of the
Airy's and Allais' precession, as well as the effect of the spin of the
Physical Pendulum on the Allais' precession. Further, the method can be
extended to the study of the asymmetrical pendulum in which an exact solution
is not possible anymore.
