A block of mass kg on a frictionless horizontal spring of stiffness oscillates 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 block of mass m=0.50m = 0.50 kg on a frictionless horizontal spring of stiffness κ=200 N m−1\kappa = 200\,\mathrm{N\,m^{-1}} oscillates with amplitude A=0.12A = 0.12 m. ((a)) Compute the total energy EE. ((b)) The loop traced in the (q,p)(q,p) plane is an ellipse of semi-axis AA along qq and 2mE\sqrt{2mE} along pp. Use its area to compute the action variable I=12π∮p dqI = \dfrac{1}{2\pi}\oint p\,\mathrm{d}q. ((c)) Hence give the frequency ν=∂H/∂I\nu = \partial H/\partial I, and check it against ω=κ/m\omega = \sqrt{\kappa/m}.

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