A one-degree-of-freedom system has Hamiltonian

PHYSICS · Multi-part · 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 one-degree-of-freedom system has Hamiltonian H(q,p;λ(t))H(q,p;\lambda(t)), with λ\lambda a slowly varying parameter, and action I(E,λ)=12π∮p dqI(E,\lambda) = \frac{1}{2\pi}\oint p\,\mathrm{d}q evaluated at fixed λ\lambda. ((a)) Starting from dE/dt=(∂H/∂λ)λ˙\mathrm{d}E/\mathrm{d}t = (\partial H/\partial\lambda)\dot\lambda and the fact that the period of the motion is 2π ∂I/∂E2\pi\,\partial I/\partial E, show that the cycle average ⟨dI/dt⟩\langle\mathrm{d}I/\mathrm{d}t\rangle vanishes to first order in ∣λ˙/λ∣/ν|\dot\lambda/\lambda|/\nu. ((b)) A simple pendulum of length ℓ0=0.60 m\ell_0 = 0.60\,\mathrm{m} swinging with small angular amplitude θ0=12∘\theta_0 = 12^\circ has its length increased steadily to ℓ1=1.20 m\ell_1 = 1.20\,\mathrm{m} over 40 s40\,\mathrm{s}, with g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}. Use the result of (a) to find the new angular amplitude θ1\theta_1. ((c)) Evaluate the slowness ratio ∣ℓ˙p/ℓp∣/ω|\dot\ell_{\rm p}/\ell_{\rm p}|/\omega at the start and at the end of the lengthening, to justify the use of the adiabatic result.

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