A spherical pendulum is a bob of mass on a light rod of length

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Adiabatic invariants, canonical perturbation theory and integrable systems

Problem

A spherical pendulum is a bob of mass

m

on a light rod of length

\ell_{\rm p}

, free to swing in any direction about a fixed pivot. With

\theta

measured from the downward vertical and

\phi

the angle about it,

H = \frac{p_\theta^2}{2m\ell_{\rm p}^2} + \frac{p_\phi^2}{2m\ell_{\rm p}^2\sin^2\theta} - mg\ell_{\rm p}\cos\theta.

Seen from above, the bob generally traces a rosette that never closes. Someone argues: "A path that never repeats cannot come from an integrable system." Assess the argument. Count the degrees of freedom, find enough constants of the motion, check their Poisson brackets, decide whether the system is integrable, and say what the non-closing rosette does tell you about the motion's two frequencies.

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