 Beltrami identity

When a variational integrand $f(z,z^{\prime })$ has no explicit dependence on the independent variable, every extremal satisfies

The Beltrami identity is a mathematical result related to the calculus of variations, particularly in the context of classical mechanics and fluid dynamics. It is named after the Italian mathematician Ernesto Beltrami. In the calculus of variations, the Beltrami identity provides a necessary condition for a functional to be extremized.