% miko:type: atomic
% miko:title: Poisson Bracket
% miko:summary: The algebraic operation controlling time evolution in Hamiltonian mechanics.
% miko:tags: mechanics
% miko:difficulty: undergrad
% miko:related: hamiltonian-mechanics, canonical-momentum, commutator
% miko:keywords: symplectic structure, phase space, observable

\documentclass{article}
\usepackage{amsmath}
\title{Poisson Bracket}
\begin{document}
\maketitle

\section{Definition}

For phase-space functions $f(q,p)$ and $g(q,p)$,

$$
\{f,g\} = \sum_i\left(
\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}
- \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}
\right).
$$

\section{Time Evolution}

If $H$ is the Hamiltonian,

$$
\frac{df}{dt} = \{f,H\} + \frac{\partial f}{\partial t}.
$$

\end{document}
