Beltrami identity

When a variational integrand

f (z, z^{'})

has no explicit dependence on the independent variable, every extremal satisfies

f - z^{'} f_{z^{'}} = constant .

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- Beltrami identity for several dependent variables Beltrami identity

Beltrami identity for several dependent variables ( $L - \sum_{j} q_{j}^{'} \partial_{q_{j}^{'}} L = constant$ )

If the integrand

L (q_{1}, \dots, q_{m}, q_{1}^{'}, \dots, 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_{j} q_{j}^{'} (\partial_{q_{j}} L - d (\partial_{q_{j}^{'}} L) / d z) = 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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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.

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