% miko:type: complete
% miko:title: Complete Classical Mechanics
% miko:title_ja: 古典力学 Complete Note
% miko:summary: A compact test complete note generated directly from a LaTeX source file.
% miko:summary_ja: LaTeX ソースから直接生成される Complete note のテスト。
% miko:tags: mechanics
% miko:difficulty: undergrad
% miko:related: lagrangian-mechanics, lorentz-transformation
% miko:keywords: action, Lagrangian, Hamiltonian

\documentclass{article}
\usepackage{amsmath}
\title{Complete Classical Mechanics}

\begin{document}
\maketitle

\section{Purpose}

This is a test note written as a normal \LaTeX{} file inside the project.
It should appear in the Complete notes section without manually moving it into
the Astro content folder.

\section{Action Principle}

The starting point is the action

$$
S[q] = \int_{t_1}^{t_2} L(q,\dot q,t)\,dt.
$$

The physical path is selected by the stationary condition

$$
\delta S = 0.
$$

\section{Euler--Lagrange Equation}

For generalized coordinates $q_i$, this gives

\begin{equation}
\frac{d}{dt}\frac{\partial L}{\partial \dot q_i}
- \frac{\partial L}{\partial q_i} = 0.
\end{equation}

\section{Map Connections}

This complete note should connect to the atomic notes on Lagrangian mechanics
and relativity through its metadata.

\end{document}
