One pendulum to run them all

Abstract

The analytical solution of the three--dimensional linear pendulum in a rotating frame of reference is obtained, including Coriolis and centrifugal accelerations, and expressed in terms of initial conditions. This result offers the possibility of treating Foucault and Bravais pendula as trajectories of the very same system of equations, each of them with particular initial conditions. We compare with the common two--dimensional approximations in textbooks. A previously unnoticed pattern in the three--dimensional Foucault pendulum attractor is presented.