Beltrami identity

The Beltrami identity, named after Eugenio Beltrami, is a simplified and less general version of the Euler–Lagrange equation in the calculus of variations.

The Euler–Lagrange equation serves to extremize action functionals of the form

where $a, b$ are constants and $u'(x) = du / dx$ .

For the special case of $\partial L / \partial x = 0$ , the Euler–Lagrange equation reduces to the Beltrami identity,

$L-u'{\frac {\partial L}{\partial u'}}=C\,,$

where $C$ is a constant.

The following derivation of the Beltrami identity starts with the Euler–Lagrange equation,

Multiplying both sides by $u'$ ,

According to the chain rule,

where $u'' = du'/ dx = d 2 u / dx 2$ .

Rearranging this yields

Thus, substituting this expression for $u' \partial L /\partial u$ into the second equation of this derivation,

By the product rule, the last term is re-expressed as

and rearranging,

For the case of $\partial L / \partial x = 0$ , this reduces to

so that taking the antiderivative results in the Beltrami identity,