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Section 1.1 Hamiltnian Noether’s Theroem
Note ID: 202605020001 |
Tags: <Hamiltonian mechanics>,
<Noether's theroem>
Definition 1.1.1. Conserved in time.
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).
i.e,
\(Q\) generates
\(H\text{.}\)
\begin{equation*}
\frac{dQ}{dt}=0 \, \Leftrightarrow \, Q \, \text{generates a symmetry of} \, H
\end{equation*}
