= Normal modes of a spring-mounted pendulum cart
{title2=$mML^2\omega^4-\big[kML^2+(m+M)MgL\big]\omega^2+kMgL=0$}
A cart of <mass> $m$ moves horizontally on a <spring> of stiffness $k$, carrying a <simple pendulum> of bob <mass> $M$ and length $L$. Let $x$ be cart displacement and $\phi$ the small angle from downward vertical. The quadratic <Lagrangian> is
$$
\mathcal L=\frac12(m+M)\dot x^2+ML\dot x\dot\phi+\frac12ML^2\dot\phi^2-\frac12kx^2-\frac12MgL\phi^2.
$$
The kinetic cross term comes from the bob's horizontal <velocity> $\dot x+L\dot\phi$ to first order. The <generalized eigenvalue problem> has mass matrix $\begin{pmatrix}m+M&ML\\ML&ML^2\end{pmatrix}$ and restoring matrix $\operatorname{diag}(k,MgL)$. Their characteristic <determinant> gives the displayed polynomial in $\omega^2$. For $m=M=L=g=1,k=2$, the squared <angular frequencies> are $2\mp\sqrt2$, with <mode shapes> $(x,\phi)=(1,\pm\sqrt2)$. The bob and cart move in phase in the lower <normal mode> and in opposite phases in the higher <normal mode>.
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