= Spring-supported pendulum normal modes
{title2=$\omega_\pm^2=\kappa/2+q\pm\sqrt{\kappa^2/4+q^2}$}
A pendulum bob and a horizontally moving support of equal <mass> $m$, with support spring <spring constant> $k$, have the quadratic <potential energy> $kx_1^2/2+mg(x_2-x_1)^2/(2l)$. With $\kappa=k/m$, $q=g/l$, the <normal mode> <eigenvectors> can be normalized to $(1,r_\pm)$, where $r_\pm=(\kappa/2\mp\sqrt{\kappa^2/4+q^2})/q$. The lower mode is in phase and the upper mode in opposite phase. The positive <eigenvalues> of the symmetric stiffness-per-mass <matrix> give the displayed squared <angular frequencies>.
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