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The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations.
The Euler–Lagrange equation serves to extremize action functionals of the form
where a {\displaystyle a} and b {\displaystyle b} are constants and u ′ = d u d x {\displaystyle u'={\frac {du}{dx}}}.
If ∂ L ∂ x = 0 {\displaystyle {\frac {\partial L}{\partial x}}=0} , then the Euler–Lagrange equation reduces to the Beltrami identity,
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