Beltrami Identity

An identity in Calculus of Variations discovered in 1868 by Beltrami. The Euler-Lagrange Differential Equation is

$\begin{displaymath} {\partial f \over \partial y} - {d\over dx} \left({\partial f \over \partial y_x}\right) = 0. \end{displaymath}$

(1)

Now, examine the Derivative of

$\begin{displaymath} {df \over dx} = {\partial f \over \partial y}y_x+{\partial f\over\partial y_x}y_{xx} + {\partial f\over \partial x}. \end{displaymath}$

(2)

Solving for the $\partial f/\partial y$ term gives

$\begin{displaymath} {\partial f \over \partial y}y_x={df \over dx}-{\partial f\over\partial y_x}y_{xx} - {\partial f\over \partial x}. \end{displaymath}$

(3)

Now, multiplying (1) by

gives

$\begin{displaymath} y_x{\partial f \over \partial y} - y_x {d\over dx}\left({\partial f \over \partial y_x}\right)= 0. \end{displaymath}$

(4)

Substituting (3) into (4) then gives

$\begin{displaymath} {df\over dx}-{\partial f\over \partial y_x}y_{xx}-{\partial ... ...y_x {d\over dx}\left({\partial f\over \partial y_x}\right) = 0 \end{displaymath}$

(5)

$\begin{displaymath} -{\partial f\over \partial x} + {d\over dx}\left({f-y_x {\partial f\over \partial y_x}}\right)=0. \end{displaymath}$

(6)

This form is especially useful if

, since in that case

$\begin{displaymath} {d\over dx}\left({f-y_x {\partial f\over \partial y_x}}\right)=0, \end{displaymath}$

(7)

which immediately gives

$\begin{displaymath} f-y_x {\partial f\over \partial y_x} = C, \end{displaymath}$

(8)

where

is a constant of integration.

The Beltrami identity greatly simplifies the solution for the minimal Area Surface of Revolution about a given axis between two specified points. It also allows straightforward solution of the Brachistochrone Problem.