An identity in calculus of variations discovered in 1868 by Beltrami. The Euler-Lagrange differential equation is
 |
(1)
|
Now, examine the derivative of 
with respect to 
 |
(2)
|
Solving for the 
term gives
 |
(3)
|
Now, multiplying (1) by 
gives
 |
(4)
|
Substituting (3) into (4) then gives
 |
(5)
|
 |
(6)
|
This form is especially useful if 
, since in that case
 |
(7)
|
which immediately gives
 |
(8)
|
where 
is a constant of integration (Weinstock 1974, pp. 24-25; Arfken 1985, pp. 928-929;
Fox 1988, pp. 8-9).
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.
