= Solution
The <functional derivative> of the action is
$$
\frac{\delta A}{\delta\mathbf x}
=\frac{\partial\mathcal L}{\partial\mathbf x}
-\frac d{dt}\frac{\partial\mathcal L}{\partial\dot{\mathbf x}}.
$$
For the <Hamiltonian function>
$$
H=\dot{\mathbf x}\mathbin\cdot
\frac{\partial\mathcal L}{\partial\dot{\mathbf x}}-\mathcal L,
$$
and a time-independent <Lagrangian>,
$$
\frac{dH}{dt}
=\dot{\mathbf x}\mathbin\cdot
\left(\frac d{dt}\frac{\partial\mathcal L}{\partial\dot{\mathbf x}}
-\frac{\partial\mathcal L}{\partial\mathbf x}\right).
$$
Therefore
$$
\boxed{\dot{\mathbf x}\mathbin\cdot
\frac{\delta A}{\delta\mathbf x}=-\frac{dH}{dt}.}
$$
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