Generalized Lagrangian of the parametric Foucault pendulum with dissipative forces

Abstract

The Foucault Pendulum (FP) is a classical mechanical system, particularly interesting because of its sensibility to the rotational motion of Earth, for which has been installed in many scientific laboratories and museums. However, its motion faces two difficulties. The first is connected to its decreasing energy, forcing the maintenance staff to restart the movement every day, and the second, that the rotation speed of the plane of oscillation does not match the value predicted by theory. To understand the observed motion, the analytical description of the spherical pendulum, free of any dissipative process, where the rotation of the Earth along the vertical, depending on the latitude of the observation point is superposed, is utilized in the literature. To truly solve this situation, we propose a Lagrangian that is simultaneously sensitive to the properties of the spherical pendulum and to the rotation of the coordinate system. However, the resulting equations can be solved only under certain limiting conditions, connected to the conservation of certain expressions for energy and for the projection of angular momentum along the vertical. A term corresponding to the parametric energy feed, consisting of a change in the length of the wire, in phase with the movement of the pendulum (periodical) was introduced. The new Lagrangian leads to a system of differential equations that can not be solved by analytical methods, and thus, the Runge–Kutta method is employed. To characterize the accuracy of our method, we applied it first to the differential equations that can be solved analytically, in order to see the numerical difference. After demonstrating that such differences are negligible, we focused on determining how accurately are satisfied the assumptions leading to the analytically solvable problem, i.e., the constancy of certain expressions for the energy and for the projection of the angular momentum along the vertical, which are shown to be smaller than 35 and 10 parts in a million. Finally, we solve the problem that includes both dissipative forces and parametric energizing. We have installed an actual FP in the Centro Educativo y Cultural Manuel Gomez Morín del Estado de Querétaro in Central Mexico, whose motion is made up of a complex cycle, consisting of 57 and 25 oscillations, the last ones under the parametric energy feed, to experimentally support our model.