= Solution
A <central force> confines nonradial motion to a <plane>. In <plane polar coordinates>, the <Lagrangian> for $V(r)=\beta r^k$ is
$$
\mathcal L=\frac12m(\dot r^2+r^2\dot\phi^2)-\beta r^k.
$$
The <azimuth> $\phi$ is a <cyclic coordinate>, so $L=mr^2\dot\phi$ is conserved <angular momentum>. The Lagrangian has no explicit <time> dependence, giving conserved <energy>
$$
\boxed{E=\frac12m\dot r^2+\frac{L^2}{2mr^2}+\beta r^k=\frac12m\dot r^2+V_{\mathrm{eff}}(r),\qquad
V_{\mathrm{eff}}(r)=\frac{L^2}{2mr^2}+\beta r^k.}
$$
At fixed $L$, the radial <equation of motion> is $m\ddot r=-V_{\mathrm{eff}}'(r)$. The <effective potential> packages the transverse <kinetic energy> as a centrifugal term alongside the actual <potential energy>. It gives a one-dimensional radial problem without introducing an extra physical interaction.
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