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Section 1.3 Hamiltonian Noether’s Theorem
202605020001 |
Hamiltonian mechanics Noether's theorem
Subsection 1.3.1 Conserved in time
Claim 1.3.1.
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*}
