The Rotor and the Pendulum

We show that Euler's equations for a free rigid body, and for a rigid body with a controlled feedback torque each red lice to the classical simple pendulum equation under an explicit cylindrical coordinate change of variables. These examples illustrate several ideas in Hamiltonian mechanics: LiePoisson reduction, cotangent bundle reduction, singular Lie-Poisson maps, deformations of Lie algebras, brackets R3, simplifications obtained by utilizing the representation-dependence of Lie-Poisson reduction, and controlling instability by inducing global bifurcations among a set of equilibria using a control parameter.