Classical Mechanics · Constraints and Lagrangian mechanics
Rayleigh dissipation supplies the correct damping sign
Problem
A generalized coordinate has $L=m\dot q^2/2-kq^2/2$ and Rayleigh function $\mathcal R=c\dot q^2/2$. Use the dissipative Euler-Lagrange equation to derive the motion and the mechanical-energy rate. For $m=2.00\,\mathrm{kg}$, $k=18.0\,\mathrm{N/m}$, $c=4.00\,\mathrm{kg/s}$, $q=0.300\,\mathrm m$, and $\dot q=-0.500\,\mathrm{m/s}$, compute $\ddot q$ and $\dot E$. Check the $c\to0$ limit.
Hint
Differentiate the Rayleigh function with respect to generalized velocity.
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