angular velocity of Earth onto the normal direction to Earth, which implies that the trace of the plane of oscillation will undergo parallel transport. After 24 hours, the difference between initial and final orientations of the trace in the Earth frame is , which corresponds to the value given by the Gauss–Bonnet theorem. is also called the holonomy or geometric phase of the pendulum. When analyzing earthbound motions, the Earth frame is not an inertial frame, but rotates about the local vertical at an effective rate of radians per day. A simple method employing parallel transport within cones tangent to the Earth's surface can be used to describe the rotation angle of the swing plane of Foucault's pendulum. From the perspective of an Earth-bound coordinate system (the measuring circle and spectator are Earth-bounded, also if terrain reaction to Coriolis force is not perceived by spectator when he moves), using a rectangular coordinate system with its -axis pointing east and its -axis pointing north, the precession of the pendulum is due to the Coriolis force (other fictitious forces as gravity and centrifugal force have not direct precession component, Euler's force is low because Earth's rotation speed is nearly constant). Consider a planar pendulum with constant natural frequency in the small angle approximation. There are two forces acting on the pendulum bob: the restoring force provided by gravity and the wire, and the Coriolis force (the centrifugal force, opposed to the gravitational restoring force, can be neglected). The Coriolis force at latitude is horizontal in the small angle approximation and is given by where is the rotational frequency of Earth, is the component of the Coriolis force in the -direction and is the component of the Coriolis force in the -direction. The restoring force, in the small-angle approximation and neglecting centrifugal force, is given by Using Newton's laws of motion this leads to the system of equations Switching to complex coordinates , the equations read To first order in this equation has the solution If time is measured in days, then and the pendulum rotates by an angle of during one day. Related physical systems Many physical systems precess in a similar manner to a Foucault pendulum. As early as 1836, the Scottish mathematician Edward Sang contrived and explained the precession of a spinning top. In 1851, Charles Wheatstone described an apparatus that consists of a vibrating spring that is mounted on