A heavy symmetric top separates in Euler angles with

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Hamilton-Jacobi theory and action-angle variables

Problem

A heavy symmetric top separates in Euler angles with W=pϕϕ+pψψ+Wθ(θ)W = p_\phi\phi + p_\psi\psi + W_\theta(\theta), and the θ\theta equation is Chapter 7's energy equation in the effective potential Veff(θ)V_{\rm eff}(\theta). For a particular top undergoing steady precession with small nutation about it, Chapter 7's formulas give a mean precession rate νφ=2.0 rad s−1\nu_\varphi = 2.0\,\mathrm{rad\,s^{-1}} and, from ∂H/∂Iθ\partial H/\partial I_\theta, a nutation frequency νθ=5.0 rad s−1\nu_\theta = 5.0\,\mathrm{rad\,s^{-1}}. ((a)) State, in one sentence, which of Chapter 8's ideas (developed there for the orbit of a central-force particle) applies here to the pair (θ,φ)(\theta,\varphi) of the top, and why the separated Hamilton-Jacobi equation makes that idea available for the top as well as for the orbit. ((b)) Express νφ/νθ\nu_\varphi/\nu_\theta as a ratio of the smallest whole numbers. Predict whether the top's (θ,φ mod 2π)(\theta,\varphi \bmod 2\pi) configuration ever exactly repeats and, if so, after how many nutation periods, and by how many complete turns φ\varphi has then advanced. ((c)) Check the prediction of part (b) directly: compute the total angle φ\varphi advances during that many nutation periods and confirm it is a whole number of full turns.

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