= Two-mass three-spring normal modes
{title2=$\omega_\pm^2=k(m_1+m_2\pm\sqrt{m_1^2-m_1m_2+m_2^2})/(m_1m_2)$}
Two positive masses linked in series between fixed walls by three identical <springs> of <spring constant> $k$ have <equations of motion> $m_1\ddot x_1=-2kx_1+kx_2$ and $m_2\ddot x_2=kx_1-2kx_2$. A <normal mode> $x_j=A_j\cos\omega t$ exists when $(2k-m_1\omega^2)(2k-m_2\omega^2)-k^2=0$, giving the displayed <angular frequencies>. Its amplitude ratio is $A_2/A_1=(2k-m_1\omega^2)/k$: positive for the lower <angular frequency>, negative for the higher one. Equal masses give amplitude ratios $+1,-1$ and squared <angular frequencies> $k/m,3k/m$.
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