= Beltrami identity for a second-derivative functional
{c}
{title2=$H=y''F_{y''}+y'(F_{y'}-(F_{y''})')-F,\qquad H'=-F_x$}
For a smooth <functional> $\int F(x,y,y',y'')\,dx$, let $P=F_{y'}$ and $Q=F_{y''}$. Along an <extremal>, the <Euler-Lagrange equation for a second-derivative functional> is $F_y-P'+Q''=0$. Differentiating $H=y''Q+y'(P-Q')-F$ by the <chain rule> gives
$$
H'=y'(P'-Q''-F_y)-F_x=-F_x.
$$
Consequently $H$ is constant whenever $F$ has no explicit dependence on $x$. This extends the <Beltrami identity> to a second-derivative variational problem. If $F$ is independent of $y''$, the expression reduces to $y'F_{y'}-F$.
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