Beltrami Identity
المؤلف:
Arfken, G
المصدر:
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.
الجزء والصفحة:
...
12-10-2018
2214
Beltrami Identity
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.
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.
Fox, C. An Introduction to the Calculus of Variations. New York: Dover, 1988.
Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, 1974.
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