= Energy balance for an autonomous Lagrangian
{title2=$\dot H=-\dot x\,\delta A/\delta x$}
For an autonomous <Lagrangian> $L(x,\dot x)$, define its <action> $A=\int L\,dt$ and <Hamiltonian> $H=\dot xL_{\dot x}-L$. The <chain rule> gives
$$
\dot H=\dot x\left(\frac d{dt}L_{\dot x}-L_x\right)
=-\dot x\frac{\delta A}{\delta x}.
$$
Thus the identity holds along forced trajectories as well as solutions of the homogeneous <Euler-Lagrange equation>. Explicit time dependence adds $-\partial_tL$; stochastic work requires a consistent regularization or the <Stratonovich chain rule>.
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