Slide 34
Slide 34 text
Variational Formulations
I
The PDE system is very complicated in its coordinate form
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:
@t X (⇠
1
, ⇠
2
, t) + HX (X, g
1
⇠
r⇠ X ) = F(X, ⇢(·, t)),
@t⇢X (⇠
1
, ⇠
2
, t)
1
p
det(g⇠
)
2
X
i=1
@
@⇠i
2
X
j=1
p
det(g⇠
)⇢X (g
1
⇠
)ij@qj HX
!
= 0,
X (⇠
1
, ⇠
2
, 1) = FT (X(⇠
1
, ⇠
2), ⇢(·, 1)), ⇢X (⇠
1
, ⇠
2
, 0) = ⇢0(X(⇠
1
, ⇠
2)).
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Comparing to the PDE formulation, when
F(⇢)
⇢
(x) = F(x, ⇢),
FT (⇢)
⇢
(x) = FT (x, ⇢),
the equivalent variational formulation is easier
to handle
min
⇢,m
Z 1
0
Z
M
⇢(x, t)L
✓
x,
m(x, t)
⇢(x, t)
◆
dMxdt
+
Z 1
0
F(⇢(·, t))dt + FT (⇢(·, 1)),
s.t. @t⇢(x, t) + rM
· m(x, t) = 0, ⇢(·, 0) = ⇢0.
A local coordinate
representation of a
manifold.
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