= Solution
For an autonomous <Lagrangian>, the <Euler-Lagrange equation> uses
$$
\frac{\delta A}{\delta x}=L_x-\frac{d}{dt}L_{\dot x},
\qquad H=\dot xL_{\dot x}-L.
$$
Apply the <chain rule> to the <Hamiltonian> along an arbitrary smooth path, without assuming the unforced <Euler-Lagrange equation>:
$$
\frac{dH}{dt}
=\ddot xL_{\dot x}+\dot x\frac{d}{dt}L_{\dot x}
-L_x\dot x-L_{\dot x}\ddot x
=-\dot x\frac{\delta A}{\delta x}.
$$
Consequently the <energy balance for an autonomous Lagrangian> is
$$
\boxed{-\int_{t_1}^{t_2}\dot x\frac{\delta A}{\delta x(t)}\,dt=H_2-H_1}.
$$
With explicit time dependence, an additional $-\partial_tL$ enters $dH/dt$. For ideal <Gaussian white noise>, the identity is understood through smooth-noise regularization or the <Stratonovich chain rule>. The original PDF correctly differentiates $L$ with respect to $\dot x$ in $H$; the supplied TeX's derivative with respect to $x$, its endpoint $t^2$, and its ordinary derivative of $A$ are transcription errors.
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