% miko:type: atomic
% miko:title: Central Force Motion
% miko:summary: Motion under a force directed along the line from a fixed center, with angular momentum conservation.
% miko:tags: mechanics
% miko:difficulty: undergrad
% miko:related: lagrangian-mechanics, noether-theorem, hamiltonian-mechanics
% miko:keywords: central force, angular momentum, effective potential

\documentclass{article}
\usepackage{amsmath}
\title{Central Force Motion}
\begin{document}
\maketitle

\section{Setup}

For a central potential $V(r)$, the Lagrangian in polar coordinates is

$$
L=\frac{1}{2}m(\dot r^2+r^2\dot\theta^2)-V(r).
$$

\section{Angular Momentum}

The coordinate $\theta$ is cyclic, so

$$
\ell = \frac{\partial L}{\partial \dot\theta}=mr^2\dot\theta
$$

is conserved.

\section{Effective Potential}

The radial motion can be written as one-dimensional motion in

$$
V_{\rm eff}(r)=V(r)+\frac{\ell^2}{2mr^2}.
$$

The second term is the centrifugal barrier.

\end{document}
