Consider now
S=∫X∫YL(x,y,f(x),f(y),∇f(x),∇f(y)) dydx
where
X=Y
. Using the same prescription to vary
f
as above,
df=vdϵ
For extremal
S
, we obtain
dϵdS=∫X∫Yv(x)+v(y)[∂f(x)∂L−i∑∂xi∂(∂fi′(x)∂L)][∂f(y)∂L−i∑∂yi∂(∂fi′(y)∂L)]dydx=0
Since
v
is arbitrary up to boundary conditions, we may treat the values of
v
at
x
and
y
as independent when
x=y
, but we may not do so
when
x=y
. When the event
x=y
has measure 0, we arrive at the
independent equations:
∫Y[∂f(x)∂L−i∑∂xi∂(∂fi′(x)∂L)]dy=0∫X[∂f(y)∂L−i∑∂yi∂(∂fi′(y)∂L)]dx=0