释义 |
Euler-Lagrange Differential EquationA 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 Euler-Lagrange differential equation
 | (3) |
is satisfied. If time Derivative Notation is replaced instead by space variable notation, theequation 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 Euler-Lagrange differential equation reduces to the greatly simplified andpartially integrated form known as the Beltrami Identity,
 | (5) |
For three independent variables (Arfken 1985, pp. 924-944), the equation generalizes to
 | (6) |
Problems in the Calculus of Variations often can be solved by solution of the appropriate Euler-Lagrange equation.
To derive the Euler-Lagrange 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 smallchange , which gives from (12),
 | (13) |
This is the Euler-Lagrange 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 re-expressed 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,Euler-Lagrange 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. 17-20 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: McGraw-Hill, pp. 276-280, 1953. |