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UNIVERSITY OF BUCHAREST
FACULTY OF PHYSICS Guest
2024-11-22 1:33 |
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Conference: Bucharest University Faculty of Physics 2005 Meeting
Section: Theoretical Physics and Applied Mathematics Seminar
Title:
On the classical motion of the spherical pendulum
Authors:
Tudor A. Marian
Affiliation:
Deptartament of Physics,University of Bucharest
E-mail
t.marian@b.astral.ro
Keywords:
spherical pendullum, lagrange equation, conservation laws, turning points, latitudine oscillation
Abstract:
We consider the motion of a spherical mathematical pendulum in the framework of Lagrangeian Mechanics.
The polar axis is taken in the downward vertical direction.
It is well known that conservation of the energy and that of the vertical component of the angular momentum combine to lead to a cubic equation for both latitudes of the turning points.
We find their exact expressions by applying the Cardano formulae.
The period of the motion between the upper and lower turning-point parallels is expressed in terms of a Gauss hypergeometric function whose variable is determined by the roots of the above-mentioned cubic polynomial.
After a careful discussion of the obtained formulae,
we test them by writing a first-order perturbation correction to a horizontal motion of the pendulum. In fact, we evaluate only the correction to the uniform circular motion of a conical pendulum consisting of the small-amplitude harmonic oscillations of its latitude:
their period is then compared to that derived to the same order from the exact expression of the period of the latitude oscillations.
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