Let $(M,\omega)$ be a symplectic manifold, let $F,H\in C^\infty(M)$, and let $X_H\in\mathfrak X(M)$ be the Hamiltonian vector field determined by
paragraph
admin
\begin{align*}
\iota_{X_H}\omega=dH.
\end{align*}
latex_env
admin
Let $\varphi:D\to M$ be the maximal flow of $X_H$, where $D\subset \mathbb R\times M$ is the open flow domain and $\varphi_t(p):=\varphi(t,p)$. Use the Poisson bracket convention
paragraph
admin
\begin{align*}
\{F,H\}=dF(X_H).
\end{align*}
latex_env
admin
Then $F$ is conserved along the Hamiltonian flow of $H$, meaning that for every $p\in M$ the map $t\mapsto F(\varphi_t(p))$ is constant on every [connected component](/page/Connected%20Component) of $\{t\in\mathbb R:(t,p)\in D\}$, if and only if