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}](b_390.gif) |
(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}](b_391.gif) |
(2) |
Solving for the
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}](b_393.gif) |
(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}](b_395.gif) |
(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}](b_396.gif) |
(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}](b_397.gif) |
(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}](b_399.gif) |
(7) |
which immediately gives
![\begin{displaymath}
f-y_x {\partial f\over \partial y_x} = C,
\end{displaymath}](b_400.gif) |
(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.
See also Brachistochrone Problem, Calculus of Variations, Euler-Lagrange Differential Equation,
Surface of Revolution
© 1996-9 Eric W. Weisstein
1999-05-26