= Beltrami identity for several dependent variables
{c}
{title2=$L-\sum_jq_j'\partial_{q_j'}L=\mathrm{constant}$}
If the <integrand> $L(q_1,\ldots,q_m,q_1',\ldots,q_m')$ has no explicit dependence on its independent variable $z$, its <Euler-Lagrange equations> imply the displayed first integral. Indeed, differentiate the left side and cancel the second-derivative terms to obtain $\sum_jq_j'(\partial_{q_j}L-d(\partial_{q_j'}L)/dz)=0$. This is the several-variable form of the <Beltrami identity>. Applied to <Fermat principle>, it expresses conservation of the component of the <optical momentum of a ray> along a translation-invariant direction.
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