Euler-Lagrange Differential Equation
المؤلف:
Arfken, G
المصدر:
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.
الجزء والصفحة:
...
13-7-2018
4029
Euler-Lagrange Differential Equation
The Euler-Lagrange differential equation is the fundamental equation of calculus of variations. It 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-derivative notation 
, the equation becomes
 |
(4)
|
The Euler-Lagrange differential equation is implemented as EulerEquations[f, u[x], x] in the Wolfram Languagepackage VariationalMethods` .
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 and partially 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
![int(partialL)/(partialq^.)(d(deltaq))/(dt)dt=int(partialL)/(partialq^.)d(deltaq)=[(partialL)/(partialq^.)deltaq]_(t_1)^(t_2)-int_(t_1)^(t_2)(d/(dt)(partialL)/(partialq^.)dt)deltaq.](http://mathworld.wolfram.com/images/equations/Euler-LagrangeDifferentialEquation/NumberedEquation7.gif) |
(13)
|
Combining (◇) and (◇) then gives
![deltaJ=[(partialL)/(partialq^.)deltaq]_(t_1)^(t_2)+int_(t_1)^(t_2)((partialL)/(partialq)-d/(dt)(partialL)/(partialq^.))deltaqdt.](http://mathworld.wolfram.com/images/equations/Euler-LagrangeDifferentialEquation/NumberedEquation8.gif) |
(14)
|
But we are varying the path only, not the endpoints, so 
and (14) becomes
 |
(15)
|
We are finding the stationary values such that 
. These must vanish for any small change 
, which gives from (15),
 |
(16)
|
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
 |
(25)
|
so
![I_2+[v^2lambda]_2^1=int_1^2[v^2(f_(yy)+lambda^.)+2vv^.(f_(yy^.)+lambda)+v^.^2f_(y^.y^.)]dt.](http://mathworld.wolfram.com/images/equations/Euler-LagrangeDifferentialEquation/NumberedEquation12.gif) |
(26)
|
But
![[v^2lambda]_2^1=0.](http://mathworld.wolfram.com/images/equations/Euler-LagrangeDifferentialEquation/NumberedEquation13.gif) |
(27)
|
Now choose 
such that
 |
(28)
|
and 
such that
 |
(29)
|
so that 
satisfies
 |
(30)
|
It then follows that
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.
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 44, 1980.
Lanczos, C. The Variational Principles of Mechanics, 4th ed. New York: Dover, pp. 53 and 61, 1986.
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.
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