Beltrami identity

The Beltrami identity is an identity in the calculus of variations. It says that a function "u" which is an extremal of the integral

:$I(u)=int\_a^b\; f(x,u,u\text{'})\; ,\; dx$

satisfies the differential equation

:$frac\{d\}\{dx\}left(f-u\text{'}frac\{partial\; f\}\{partial\; u\text{'}\}\; ight)-frac\{partial\; f\}\{partial\; x\}=0.$

Proof

The Euler-Lagrange equation tells that

:$frac\{partial\; f\}\{partial\; u\}-frac\{d\}\{dx\}frac\{partial\; f\}\{partial\; u\text{'}\}=0.$

Now consider the total differential of functional $f(x,u,u\text{'})$. Substituting the Euler-Lagrange equation into it, we have

:

Therefore,

:$frac\{d\}\{dx\}left(f-u\text{'}frac\{partial\; f\}\{partial\; u\text{'}\}\; ight)-frac\{partial\; f\}\{partial\; x\}=0.$

Application

In case the functional "f" is independent of "x", then the Beltrami identity can be simplified into

:

Using the above form is an easier approach to solve for the optimal function "u" than directly applying the Euler-Lagrange equation.