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Multi-Function Variation
Suppose that we wish to maximize or minimize the functional
![$\displaystyle I = \int_a^b F(y_1,y_2,\cdots,y_{\cal F},y_1',y_2',\cdots,y_{\cal F}',x) dx.$](img3025.png) |
(B.61) |
Here, the integrand
is now a functional of the
independent
functions
, for
. A fairly straightforward extension of the
analysis in Section B.5 yields
separate Euler-Lagrange equations,
![$\displaystyle \frac{d}{dx}\!\left(\frac{\partial F}{\partial y_i'}\right)-\frac{\partial F}{\partial y_i} = 0,$](img3027.png) |
(B.62) |
for
, which determine the
functions
. If
does not
explicitly depend on the function
then the
th Euler-Lagrange
equation simplifies to
![$\displaystyle \frac{\partial F}{\partial y_k'} = {\rm constant}.$](img3028.png) |
(B.63) |
Likewise, if
does not explicitly depend on
then all
Euler-Lagrange equations simplify to
![$\displaystyle y_i' \frac{\partial F}{\partial y_i'} - F = {\rm constant},$](img3029.png) |
(B.64) |
for
.
Richard Fitzpatrick
2016-01-25