= Time derivative of a conserved quantity for an autonomous Hamiltonian
{title2=$D_tf=0,\quad H_t=0\ \Longrightarrow\ D_t(f_t)=0$}
For a time-independent <Hamiltonian> $H$, the explicit time <partial derivative> of a smooth time-dependent <conserved quantity> is again a <conserved quantity>. Here conservation means the identity $D_tf=f_t+\{f,H\}=0$ on <phase space>, with the standard <Poisson bracket>. Differentiating this identity at fixed canonical coordinates gives
$$
f_{tt}+\{f_t,H\}+\{f,H_t\}=0.
$$
Because $H_t=0$, this is exactly $D_t(f_t)=0$, proving the assertion. Iteration gives the same result for all higher explicit time <partial derivatives> for which the differentiations are defined. By <Hamiltonian energy balance>, requiring $H$ itself to be a <conserved quantity> on all trajectories is equivalent to $H_t=0$.
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