Skip to main content\(\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\bivec}[1]{\overset{\curvearrowleft}{#1}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 1.1 Hamiltonian Noether’s Theorem
202605020001 |
Hamiltonian mechanics,
Noether's theorem
Subsection 1.1.1 Conserved in time
Claim 1.1.1.
A quantity \(Q\) is said to be conserved in time if
\begin{equation*}
\{Q,H\}=0
\end{equation*}
Two Interpretations:
-
\(Q\) is a constant along the flow of \(H\text{:}\)
\begin{equation*}
\frac{dQ}{dt}=0
\end{equation*}
-
\(H\) is
invariant along the
flow of
\(Q\) (up to a minus sign). In other words,
\(Q\) generates a symmetry of
\(H\text{.}\)
Take-Home Message.
\begin{equation*}
\frac{dQ}{dt}=0
\Leftrightarrow
Q \text{ a of } H
\end{equation*}
