A fundamental equation of Calculus of Variations which states that if
is defined by an Integral of the form
![\begin{displaymath}
J = \int f(x,y,\dot y)\,dx,
\end{displaymath}](e_2154.gif) |
(1) |
where
![\begin{displaymath}
\dot y \equiv {dy\over dt},
\end{displaymath}](e_2155.gif) |
(2) |
then
has a Stationary Value if the Euler-Lagrange differential equation
![\begin{displaymath}
{\partial f\over \partial y} - {d\over dt}\left({\partial f\over \partial \dot y}\right)= 0
\end{displaymath}](e_2156.gif) |
(3) |
is satisfied. If time Derivative Notation is replaced instead by space variable notation, the
equation becomes
![\begin{displaymath}
{\partial f\over \partial y}-{d\over dx} {\partial f\over \partial y_x}=0.
\end{displaymath}](e_2157.gif) |
(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 and
partially integrated form known as the Beltrami Identity,
![\begin{displaymath}
f-y_x{\partial f\over\partial y_x}=C.
\end{displaymath}](e_2159.gif) |
(5) |
For three independent variables (Arfken 1985, pp. 924-944), the equation generalizes to
![\begin{displaymath}
{\partial f\over \partial u}-{\partial \over \partial x} {\p...
...{\partial \over \partial z}
{\partial f\over \partial u_z}=0.
\end{displaymath}](e_2160.gif) |
(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
![\begin{displaymath}
\int {\partial L\over \partial \dot q} {d(\delta q)\over dt...
...r dt} {\partial L\over \partial \dot q}\,dt}\right)\,\delta q.
\end{displaymath}](e_2167.gif) |
(10) |
Combining (7) and (10) then gives
![\begin{displaymath}
\delta J=\left[{{\partial L\over \partial \dot q} \delta q}\...
...er dt}{\partial L\over \partial \dot q}}\right)\,\delta q\,dt.
\end{displaymath}](e_2168.gif) |
(11) |
But we are varying the path only, not the endpoints, so
and (11) becomes
![\begin{displaymath}
\delta J = \int_{t_1}^{t_2} \left({{\partial L\over \partial...
...er dt}{\partial L\over \partial \dot q}}\right)\,\delta q\,dt.
\end{displaymath}](e_2170.gif) |
(12) |
We are finding the Stationary Values such that
. These must vanish for any small
change
, which gives from (12),
![\begin{displaymath}
{\partial L\over \partial q}-{d\over dt}\left({\partial L\over \partial \dot q}\right)=0.
\end{displaymath}](e_2173.gif) |
(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
![\begin{displaymath}
{d\over dt}(v^2\lambda)=v^2\dot\lambda+2v\dot v\lambda,
\end{displaymath}](e_2187.gif) |
(21) |
so
![\begin{displaymath}
I_2+[v^2\lambda]_2^1=\int_1^2 [v^2(f_{yy}+\dot\lambda)+2v\dot v(f_{y\dot y}+\lambda)+{\dot v}^2f_{\dot y\dot y}]\,dt.
\end{displaymath}](e_2188.gif) |
(22) |
But
![\begin{displaymath}[v^2\lambda]_2^1=0.
\end{displaymath}](e_2189.gif) |
(23) |
Now choose
such that
![\begin{displaymath}
f_{\dot y\dot y}(f_{yy}+\dot\lambda)=(f_{y\dot y}+\lambda)^2
\end{displaymath}](e_2190.gif) |
(24) |
and
such that
![\begin{displaymath}
f_{y\dot y}+\lambda=-{f_{\dot y\dot y}\over z}{dz\over dt}
\end{displaymath}](e_2191.gif) |
(25) |
so that
satisfies
![\begin{displaymath}
f_{\dot y\dot y}\ddot z+\dot f_{\dot y\dot y}\dot z-(f_{yy}-\dot f_{y\dot y})z=0.
\end{displaymath}](e_2192.gif) |
(26) |
It then follows that
![\begin{displaymath}
I_2=\int f_{\dot y\dot y}\left({\dot v+{f_{y\dot y}+\lambda\...
..._{\dot y\dot y}\left({\dot v-{v\over z}{dz\over dt}}\right)^2.
\end{displaymath}](e_2193.gif) |
(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.
© 1996-9 Eric W. Weisstein
1999-05-25