A fundamental equation of Calculus of Variations which states that if is defined by an Integral of the form

(1) 
where

(2) 
then has a Stationary Value if the EulerLagrange differential equation

(3) 
is satisfied. If time Derivative Notation is replaced instead by space variable notation, the
equation becomes

(4) 
In many physical problems, (the Partial Derivative of with respect to ) turns out to be 0, in
which case a manipulation of the EulerLagrange differential equation reduces to the greatly simplified and
partially integrated form known as the Beltrami Identity,

(5) 
For three independent variables (Arfken 1985, pp. 924944), the equation generalizes to

(6) 
Problems in the Calculus of Variations often can be solved by solution of the appropriate EulerLagrange equation.
To derive the EulerLagrange differential equation, examine
since
. Now, integrate the second term by Parts using
so

(10) 
Combining (7) and (10) then gives

(11) 
But we are varying the path only, not the endpoints, so
and (11) becomes

(12) 
We are finding the Stationary Values such that . These must vanish for any small
change , which gives from (12),

(13) 
This is the EulerLagrange differential equation.
The variation in can also be written in terms of the parameter as
where
and the first, second, etc., variations are
The second variation can be reexpressed using

(21) 
so

(22) 
But

(23) 
Now choose such that

(24) 
and such that

(25) 
so that satisfies

(26) 
It then follows that

(27) 
See also Beltrami Identity, Brachistochrone Problem, Calculus of Variations,
EulerLagrange Derivative
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.
Forsyth, A. R. Calculus of Variations. New York: Dover, pp. 1720 and 29, 1960.
Morse, P. M. and Feshbach, H. ``The Variational Integral and the Euler Equations.'' §3.1 in
Methods of Theoretical Physics, Part I. New York: McGrawHill, pp. 276280, 1953.
© 19969 Eric W. Weisstein
19990525