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We study the apsidal precession of a Physical Symmetrical Pendulum (Allais'precession) as a generalization of the precession corresponding to the IdealSpherical Pendulum (Airy's Precession). Based on the Hamilton-Jacobi formalismand using the technics of variation of parameters along with the averagingmethod, we obtain approximate solutions, in terms of which the motion of bothsystems admits a simple geometrical description. The method developed in thispaper is considerably simpler than the standard one in terms of ellipticalfunctions and the numerical agreement with the exact solutions is excellent. Inaddition, the present procedure permits to show clearly the origin of theAiry's and Allais' precession, as well as the effect of the spin of thePhysical Pendulum on the Allais' precession. Further, the method can beextended to the study of the asymmetrical pendulum in which an exact solutionis not possible anymore.

Study of the apsidal precession of the Physical Symmetrical Pendulum