User:Bob K31416/BI

The Beltrami identity is a mathematical relation used in the calculus of variations to find a function $u(x)$ that minimizes a functional, and is
 * $$L-u'\frac{\partial L}{\partial u'}=C \, ,$$

where
 * $C$ is a constant,

It is derived from the Euler-Lagrange equation for the case of $u&prime; = du/dx,$ not appearing explicitly in $L = L[ u(x), u&prime;(x) ]$. The Euler-Lagrange equation applies to functionals of the form
 * $$I[u]=\int_a^b L[x,u(x),u'(x)] \, dx \, ,$$

where $x$ are constants and $L$. For the case of $a, b$, the Euler-Lagrange equation reduces to the Beltrami identity,
 * $$L-u'\frac{\partial L}{\partial u'}=C \, ,$$

where $u&prime;(x) = du / dx$ is a constant.

The Beltrami identity is a simplified and less general version of the Euler-Lagrange equation in the calculus of variations. The Euler-Lagrange equation applies to functionals of the form
 * $$I[u]=\int_a^b L[x,u(x),u'(x)] \, dx \, ,$$

where $∂L / ∂x = 0$ are constants and $C$. For the case of $a, b$, the Euler-Lagrange equation reduces to the Beltrami identity,
 * $$L-u'\frac{\partial L}{\partial u'}=C \, ,$$

where $u&prime;(x) = du / dx$ is a constant.