The rotor and the pendulum

Holm, D. D. and J. E. Marsden

In Symplectic Geometry and Mathematical Physics, (P. Donato and C. Duval and J. Elhadad and G. M. Tuynman, eds.), (1991), Birkhäuser, Boston, 189-203

Abstract:

We show that Euler's equations for a free rigid body, and for a rigid body with a controlled feedback torque each reduce to the classical simple pendulum equation under an explicit cylindrical coordinate change of variables. These examples illustrate several ideas in Hamiltonian mechanics: Lie-Poisson reduction, cotangent bundle reduction, singular Lie-Poisson maps, deformations of Lie algebras, brackets on $\mathbb {R}$³ , 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.