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Beltrami Identity

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

{\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 $x$
{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
{\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 $y_x$ gives
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
{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)

-{\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 $f_x = 0$, since in that case
{d\over dx}\left({f-y_x {\partial f\over \partial y_x}}\right)=0,
\end{displaymath} (7)

which immediately gives
f-y_x {\partial f\over \partial y_x} = C,
\end{displaymath} (8)

where $C$ 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.

See also Brachistochrone Problem, Calculus of Variations, Euler-Lagrange Differential Equation, Surface of Revolution

© 1996-9 Eric W. Weisstein