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Multi-Function Variation
Suppose that we wish to maximize or minimize the functional
 |
(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,
 |
(B.62) |
for
, which determine the
functions
. If
does not
explicitly depend on the function
then the
th Euler-Lagrange
equation simplifies to
 |
(B.63) |
Likewise, if
does not explicitly depend on
then all
Euler-Lagrange equations simplify to
 |
(B.64) |
for
.
Richard Fitzpatrick
2016-01-25