% miko:type: atomic
% miko:title: Small Oscillations
% miko:summary: Linearized motion near stable equilibrium and the normal-mode structure.
% miko:tags: mechanics
% miko:difficulty: undergrad
% miko:related: lagrangian-mechanics, euler-lagrange-equation, quantum-harmonic-oscillator
% miko:keywords: normal modes, equilibrium, harmonic approximation

\documentclass{article}
\usepackage{amsmath}
\title{Small Oscillations}
\begin{document}
\maketitle

\section{Expansion Near Equilibrium}

Near a stable equilibrium $q_0$, expand the potential:

$$
V(q)\simeq V(q_0)+\frac{1}{2}k(q-q_0)^2.
$$

\section{Linear Equation}

The motion becomes

$$
m\ddot x+kx=0,
$$

with angular frequency

$$
\omega=\sqrt{\frac{k}{m}}.
$$

\section{Normal Modes}

For many coupled coordinates, diagonalizing the quadratic form reveals
independent normal modes.

\end{document}
