The Foucault pendulum viewed as a spherical pendulum placed in a rotating reference system

Abstract

For a pendulum of a given length, installed at any fixed latitude, the motion depends on three variables: the total mechanical energy, connected with the maximum angular distance of the bob from the vertical; the projection of the angular momentum along the vertical, connected with the shape of its orbit along the surface of a sphere, which may be described as a veering ellipse recalling the orbit of Mercury; and the angular velocity, Ω, of the rotating reference system given by that of the Earth. When the value of one of these variables is negligible, the ensuing motion has already been calculated in analytical grounds: a plummet, a Foucault pendulum, or a spherical pendulum including as a special case the so-called plane pendulum. This paper offers a Lagrangian approach in the case when Ω is the sidereal angular velocity of the Earth. To our knowledge, these are the first analytical solutions for any boundary conditions. When the boundary conditions are such that the bob passes through the vertical (i.e., the direction defined by the plummet), the behavior of the pendulum is as expected: the veering velocity of the plane equals the angular velocity of the Earth times the sine of the latitude. This state is reached when the projection of its angular momentum along the vertical vanishes. However, if that projection takes any other value, the motion is produced along a veering ellipse and the time rate of advance of the major axis is the sum of the previous result plus the spherical pendulum contribution. Our solutions include, as special cases, those of the spherical pendulum, or the ideal Foucault pendulum with finite amplitude.