% miko:type: atomic
% miko:title: Hamiltonian Mechanics
% miko:summary: Dynamics on phase space using coordinates and conjugate momenta.
% miko:tags: mechanics
% miko:difficulty: undergrad
% miko:related: lagrangian-mechanics, euler-lagrange-equation, poisson-bracket
% miko:keywords: Hamiltonian, phase space, Legendre transform

\documentclass{article}
\usepackage{amsmath}
\title{Hamiltonian Mechanics}
\begin{document}
\maketitle

\section{Legendre Transform}

Define conjugate momenta

$$
p_i = \frac{\partial L}{\partial \dot q_i}.
$$

The Hamiltonian is

$$
H(q,p,t)=\sum_i p_i\dot q_i - L(q,\dot q,t).
$$

\section{Hamilton's Equations}

\begin{equation}
\dot q_i = \frac{\partial H}{\partial p_i},
\qquad
\dot p_i = -\frac{\partial H}{\partial q_i}.
\end{equation}

This form makes conservation laws and phase-space geometry visible.

\end{document}
