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Research Papers

# Study of the Apsidal Precession of the Physical Symmetrical Pendulum

[+] Author and Article Information
Héctor R. Maya

Departamento de Física,
Montería, Colombia;
Departamento de Física,
Bogotá, Colombia
e-mail: hrmaya@hotmail.com

Rodolfo A. Diaz

Departamento de Física,
Bogotá, Colombia

William J. Herrera

Departamento de Física,
Bogotá, Colombia
e-mail: jherreraw@unal.edu.co

From the initial conditions that we shall use, we have $θ¯$ ≲ 10−1 and for the dimensionless momenta Pa ≲ 10–3. Hence, it is logical to consider H1 as a perturbation with respect to H0.

This is a reasonable ansatz since those variables are constant when we use the nonperturbed Hamiltonian H0. The time evolution of these variables arises from the introduction of the (much smaller) Hamiltonian H1.

In addition, it can be seen from Eqs. (A2a) and (A2b) that only the functions $cosϕ$ and $sinϕ$ are important but not $ϕ$ by itself, and such functions are smooth.

We are using the convention of positive direction of the Z-axis in the direction of the gravitational field (i.e., downward).

1In the most usual scenario of the symmetrical top, the spin is the dominant part of its motion.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 15, 2014; final manuscript received December 20, 2014; accepted manuscript posted December 31, 2014; published online January 13, 2015. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 82(2), 021008 (Feb 01, 2015) (12 pages) Paper No: JAM-14-1313; doi: 10.1115/1.4029470 History: Received July 15, 2014; Revised December 20, 2014; Accepted December 31, 2014; Online January 13, 2015

## Abstract

We study the apsidal precession of a physical symmetrical pendulum (PSP) (Allais’ precession) as a generalization of the precession corresponding to the ideal spherical pendulum (ISP) (Airy’s precession). Based on the Hamilton–Jacobi formalism and using the techniques of variation of parameters along with the averaging method, we obtain approximate analytical 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 could be extended to the study of the asymmetrical pendulum in which an exact analytical solution is not possible anymore.

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## Figures

Fig. 1

Inertial system XYZ and the spherical pendulum. The angular coordinates θ and ϕ are shown, as well as the weight and tension.

Fig. 2

(a) Graphical form of the approximate and exact solutions (dashed and solid lines, respectively) of the ISP for θ with 300 ≤ τ ≤ 310. These graphics cannot be distinguished from each other. (b) Graphical form of the approximate values of the angle ϕ coming from Eq. (40) and the exact values (dashed and solid lines, respectively), for 500 ≤ τ ≤ 510. The approximate and exact solutions are superposed. In addition, the straight line corresponds to the linear approximation given by Eq. (43).

Fig. 3

Projection of the trajectory of the ISP on the XY plane for 30nT ≤ t ≤ (30n + 1)T with n = 0, 1, and 2, using the approximate solution (47) and the exact one (dashed and solid lines, respectively). The approximate and exact solutions cannot be distinguished.

Fig. 4

PSP hung on an edge. θ, Φ, and ψ are the Euler angles used in the description of the motion. The spherical coordinate ϕ is shown as well as its relation with the Euler angle Φ.

Fig. 5

Plot of the approximate solution (88) and the exact one (dashed and solid lines, respectively) for the coordinate of nutation θ¯(sinθ) of the PSP as a function of time, for 500 ≤ t ≤ 504 s. The exact solution is superposed to the approximate one.

Fig. 6

(a) Approximate and exact solutions (dashed and solid lines, respectively) for the azimuthal angle Φ of the PSP, for 500 ≤ t ≤ 504 s, (b) approximate and exact solutions (dashed and solid lines, respectively) for the spin angle ψ of the PSP, for 500 ≤ t ≤ 504 s. The straight lines correspond to the linear approximations for each one of these angles. The exact solutions cannot be distinguished from the approximate ones.

Fig. 7

Projections of the trajectory of the CM of the PSP on the XY plane, obtained with the approximate and exact solutions (dashed and solid lines, respectively), for time intervals of a period T, (a) for 0 ≤ t ≤ T: (T). (b) For 29 T ≤ t ≤ 30 T: (30 T). (c) for 59 T ≤ t ≤ 60 T: (60 T). In all cases, the exact solutions are superposed to the approximate ones.

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