Classical Mechanics · Small oscillations and normal modes
A coordinate-dependent mass freezes at linear order
Problem
Consider $L=\tfrac12m(1+\alpha q^2)\dot q^2-\tfrac12kq^2$, where $m,k>0$ and $\alpha$ has units $q^{-2}$. Derive the exact Euler-Lagrange equation and linearize about $q=0$. Explain why $\alpha$ does not alter the small-oscillation frequency. Evaluate the frequency for $m=2.00\,\mathrm{kg}$ and $k=18.0\,\mathrm{N/m}$.
Hint
$d[ m(1+\alpha q^2)\dot q]/dt$ contains two terms proportional to $\alpha$.
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