A simple pendulum of length m and bob mass kg swings with small amplitude rad (so )

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

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

Problem

A simple pendulum of length ℓ=0.80\ell=0.80 m and bob mass m=0.20m=0.20 kg swings with small amplitude θ0=0.10\theta_0=0.10 rad (so sin⁡θ≈θ\sin\theta\approx\theta). For small oscillations its Hamiltonian is H=pθ22mℓ2+12mgℓθ2H=\dfrac{p_\theta^2}{2m\ell^2}+\dfrac12mg\ell\theta^2, with g=9.8 m s−2g=9.8\,\mathrm{m\,s^{-2}}. ((a)) Write the Hamilton-Jacobi equation for the characteristic function W(θ)W(\theta). ((b)) This has the same form as the oscillator with effective mass M=mℓ2M=m\ell^2 and effective stiffness κ=mgℓ\kappa=mg\ell. Using ω0=κ/M\omega_0=\sqrt{\kappa/M} and E=12κθ02E=\tfrac12\kappa\theta_0^2, compute the action variable I=E/ω0I=E/\omega_0. ((c)) Give the frequency ν=∂H/∂I\nu=\partial H/\partial I.

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