Nonholonomic System - Examples - The Foucault Pendulum

The Foucault Pendulum

The classic example of a nonholonomic system is the Foucault pendulum. In the local coordinate frame the pendulum is swinging in a vertical plane with a particular orientation with respect to geographic north at the outset of the path. The implicit trajectory of the system is the line of latitude on the earth where the pendulum is located. Even though the pendulum is stationary in the earth frame, it is moving in a frame referred to the sun and rotating in synchrony with the Earth's rate of revolution, so that the only apparent motion of the pendulum is that caused by the rotation of the earth. This latter frame is considered to be an inertial reference frame, although it too is non-inertial in more subtle ways. The earth frame is well known to be non-inertial, a fact made perceivable by the apparent presence of centrifugal and Coriolis forces.

Motion along the line of latitude is parameterized by the passage of time, and the Foucault pendulum's plane of oscillation appears to rotate about the local vertical axis as time passes. The angle of rotation of this plane at a time t with respect to the initial orientation is the anholonomy of the system. The anholonomy induced by a complete circuit of latitude is proportional to the solid angle subtended by that circle of latitude. The path need not be constrained to latitude circles. For example, the pendulum might be mounted in an airplane. The anholonomy is still proportional to the solid angle subtended by the path, which may now be quite irregular. The Foucault pendulum is a physical example of parallel transport.