% miko:type: atomic
% miko:title: Euler-Lagrange Equation
% miko:summary: The differential equation obtained by making the action stationary.
% miko:tags: mechanics
% miko:difficulty: undergrad
% miko:related: lagrangian-mechanics, hamiltonian-mechanics, noether-theorem
% miko:keywords: variational principle, action, generalized coordinates

\documentclass{article}
\usepackage{amsmath}
\title{Euler-Lagrange Equation}
\begin{document}
\maketitle

\section{Statement}

For an action functional

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

stationarity under variations with fixed endpoints gives

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

\section{Meaning}

The equation is Newton's law written in coordinates adapted to constraints.
It is the local form of the principle of stationary action.

\end{document}
