= Solution
Introduce the velocity variable $v=\dot x$. The <second-order differential equation> becomes the <autonomous planar system>
$$
\boxed{\dot x=v,\qquad \dot v=f(x,v).}
$$
A <critical point of a dynamical system> is a point where both components of the <vector field> vanish: here $v=0$ and $f(x,0)=0$. It is a stationary solution in the <phase plane>. The <Jacobian matrix> at each such point gives the local <linear stability of a planar equilibrium>. For the equations below, a conserved <energy> also determines the nonlinear <phase portrait>; this is important because purely imaginary <eigenvalues> alone do not generally prove that an equilibrium is a <center equilibrium>.
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