22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 15: Variational Techniques
1. Variational Procedure
2. Variational formulation of MHD
3. Energy Principle
4. Extended Energy Principle
Variational Procedure
1. Alternate representation of a differential equation
2. Consider classic eigenvalue problem
d ⎛ dy ⎞
+ ( λ − g) y = 0
F
dx ⎜⎝ dx ⎟⎠
y ( 0 ) = y (1) = 0
F = F (x) , g = g (x)
3. Methods of solution
a. inspection if F, g simple
b. computer
c. power series (not useful usually)
d. fourier series (not useful usually)
e. variational procedure (allows guess and a solution)
λ is more accurate than guess for y(x)
4. Variational principle: multiply by
⎡
d ⎛ dy ⎞
∫ ⎢⎣y dx ⎜⎝ F dx ⎟⎠ + ( λ − g) y
⎡
y dx ⎢ →
0
⎣
∫
1
∫ ∫
1
0
⎤
⎥ below
⎦
2⎤
⎥ dx = 0
⎦
1
2
+ ⎡⎢ −Fy' + λy2 − gy2 ⎤⎥ dx + Fy y' = 0
0
⎣
⎦
∫
⎛
'2
⎜ Fy
∫
λ= ⎝
+ gy2 ⎞⎟ dx
⎠
∫ y dx
2
(1)
5. Why is this variational? Substitute all allowable y(x) into (1). When resulting
λ exhibits an extremum (maximum, minimum, saddle pt) then λ and y are
actual eigenvalue and eigenfunction.
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 13
Page 1 of 7
6. Proof: assume y0 = ( x ) is substituted into (1) yielding λ0 , Modify y a little bit
by a small but arbitrary perturbation.
y ( x ) = y0 ( x ) + δy ( x )
δy ( 0) = δy (1) = 0
This produces a change in λ = λ0 + δλ given by
δλ =
∫
⎡
'2
2⎤
⎢F ( y0 + δy ) + g ( y0 + δy ) ⎥ dx
⎣
⎦
−
2
( y0 + δy ) dx
∫
⎛
'2
0
∫ ⎝⎜ Fy
+ gy20 ⎞⎟ dx
⎠
∫ y dx
2
0
7. For small δy
⎛
'2
0
∫ ⎜ Fy
δλ = ⎝
∫
+ gy20 ⎞⎟ dx
⎠
−
2
y0dx
2
2 ⎛⎜ Fy'0 + gy20 ⎞⎟ dx
⎝
⎠
−
∫
∫
y20dx
⎛
'2
0
∫ ⎜⎝ Fy
∫
+ gy20 ⎞⎟ dx
⎠
+2
2
y0dx
∫ (Fy δy + gy δy ) dx
∫ y dx
'
0
'
0
2
0
∫ y δydx
∫ y dx
0
2
0
λ0
integrate by parts
=
∫
2 ⎡Fy'0 δy' + gy0 δy − λ0 y0 δy ⎤ dx
⎣
⎦
∫ y dx
2
0
( )
∫
'
⎡
⎤
−2 dx δy ⎢ Fy'0 + ( λ0 − g) y0 ⎥
⎣
⎦
∂λ =
2
y0dx
∫
8. At an extremum δλ = 0 for arbitrary δy , implying
(Fy ) + ( λ
'
0
'
0
− g) y0 = 0
This is equivalent to original problem
9. One can multiply original equation by h(x) g(x) when h is arbitrary resulting
expression
λ=
∫
⎧⎪
⎡⎛
'2
'
⎨hFy + ⎢⎜ hg − Fh
⎢⎣⎝
⎩⎪
( )
1
∫ hy dx
2
⎞ ⎤ 2 ⎫⎪
⎟ ⎥ y ⎬ dx
⎠ ⎥⎦ ⎭⎪
2
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 13
Page 2 of 7
is correct mathematical expression but is not a variational principle unless
h=1. Varying y gives
( )
'⎤
⎡
Fh' ⎥
d ⎛
dy ⎞ ⎢
+ λh − hg +
y=0
hF
dx ⎜⎝
dx ⎟⎠ ⎢
2 ⎥
⎢
⎥
⎣
⎦
( )
'⎤
⎡
Fh' ⎥
d ⎛ dy ⎞
h' dy ⎢
or
+F
+ λ −g+
y=0
F
dx ⎝⎜ dx ⎠⎟
h dx ⎢
2h ⎥
⎢
⎥
⎣
⎦
10. Since δλ = 0 when y coincides with a true eigenfunction, this implies that an
estimate for λ using a guess (trial function) for g is more accurate than the
trial function itself.
11. Proof: Write y = y0 + δy where y0 is true eigenfunction and δy is the error in
the guess, assumed of order ∈ . Substitute into (1) yields
λ=
=
N0 + N1 + N2
N + N1 + N2
= 0
D0 + D1 + D2
D0
⎡
⎤
D
D
D2
⎢1 − 1 − 2 + 12 + ...⎥
D0 D0 D0
⎢⎣
⎥⎦
N0
N0 ⎤ 1 ⎡
N0
ND
D2 ⎤
1 ⎡
+
+ N0 12 ⎥ + …
⎢N2 − 1 1 − D2
⎢N1 − D1
⎥+
D0
D0 D0 ⎣
D0 ⎦ D0 ⎢⎣
D0
D0 ⎦⎥
0
= λ0
( δλ = 0 )
1
⎡N2 − D2 λ0 ⎦⎤
D0 ⎣
∫ dx ⎢⎣F ( δy ) + (g − λ
+
∫ dx y
⎡
= λ0
+
'
2
0
) ( δy )2 ⎤⎥
⎦
2
0
( )
= λ0 + 0 ∈2
( )
Thus, error is λ is 0 ∈2
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
while error in y is 0 (∈)
Lecture 13
Page 3 of 7
Generalized Boundary Conditions
1. Problem of practical importance
(Fy ) − gy = 0
'
'
y (0) = 1
y' (1) = Ay (1)
2. Simple Variational Principle >
⎛
'2
∫ ⎜ Fy
λ= ⎝
new B.C.
∫ y dx
− gy2 ⎞⎟ dx − f yy'
x =1
⎠
∫
(2)
y2 dx
3. Let y = y0 + δy, λ = λ0 + δλ
( ) + ( λ − g) y ⎤⎥⎦ dx + F ( y δy − y δy )
∫ y dx
⎡
2 δy ⎢ Fy'0
⎣
δλ = −
∫
'
0
'
0
0
'
1
2
0
4. δλ vanishes if
a. y0 satisfies original ODE
b. y0 and δy satisfY y'0 (1) = Ay0 (1) , δy'0 (1) = Aδy (1)
5. Using trial functions which satisfy g' (1) = Ag (1) is not unexpected but can be
cumbersome in practice.
6. Alternate, more practical, more elegant varational principle. Replace y' (1) in
(2) with Ay (1) . Then
⎛
'2
1
⎜ Fy
∫
λ= ⎝
+ gy2 ⎞⎟ dx + Afy2
1
⎠
∫y
2
dx
7. Let y = y0 + δy, and λ = λ0 + δλ
( ) + (λ
⎡
−2 δy ⎢ fy'0
⎣
δλ =
∫
'
0
(
⎤
− g) y0 ⎥ dx + 2Fδy y'0 − Ay0
⎦
∫y
2
0
)
dx
8. δλ vanishes if
a. original ODE satisfied
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 13
Page 4 of 7
b. y'0 = Ay0
If we choose trial functions which allow y (1) to float freely, then variational
principle forces Ay (1) = y' (1) . This is a natural boundary condition.
Practical Applications
1. “Exact” solution and eigenvalue. Let
y=
∑
complete set of orthornormal functions
an Yn ( x )
2. Then
λ=
∑a a W
∑a
n m
2
n
nm
,
Wnm =
∫ dx ⎡⎣Fy y
' '
n m
+ gynym ⎤ dx
⎦
3. Minimize with respect to the an. This is equivalent to finding the eigenvalues
⎛ N ⎞
of W . Simple numerical procedure. In the limit N → ∞ ⎜
⎟ , we obtain exact
⎜
⎟
⎝ 1 ⎠
eigenvalue and eigenfunction. Since Yn is a complete set.
∑
4. Send “estimate” of eigenvalue. Gives a trial function
y = y ( x, c1 , c2 , c3 )
⎡e.g. y = x (1 − x ) ⎡1 + c x + c x2 + c x3 ⎤ ⎤
1
2
3
⎣
⎦⎦
⎣
(
)
2
or y = x 1 − x2 e−c1x ⎡1 + c2 xc3 ⎤
⎣
⎦
Evaluate λ = λ ( c1 , c2 , c3 )
Find c1 , c2 , c3 to extremize λ
∂λ
=0
∂c1
∂λ
=0
∂c2
∂λ
=0
∂c3
Substitute c1 , c2 , c3 back into λ to get good estimate.
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 13
Page 5 of 7
Application of the Variational Principle to MHD
1. Generalize previous analysis to include 3-D and sectors
2. Conceptually the same
( ) ∫ ξ* ⋅ dr
−ω2ρξ = F ξ >
δW
K
ω2 =
K=
2
1
ρ ξ dr
2
∫
δW = −
1 *
ξ ⋅ F ξ dr
2
()
∫
ω2 → ω2 + δω2
3. Proof: Let ξ → ξ + δξ ,
2
2
ω + δω =
(
)
( ) ( ) (
)
K ( ξ , ξ ) + K ( δξ , ξ ) + K ( ξ , δξ ) + K ( δξ , δξ )
*
*
*
δW ξ , ξ + δW δξ* , ξ + δW ξ , δξ + δW δξ , δξ
*
*
*
*
4. Assume δξ , δω2 are small
2
δω =
(
) (
K (ξ , ξ)
*
*
δW ξ , δ ξ + δW δξ , ξ
*
) − δW ( ξ , ξ) ⎡⎢⎣K ( δξ , ξ) + K ( ξ , δ ξ )⎤⎥⎦
K (ξ , ξ)
K (ξ , ξ)
*
*
*
*
*
ω2
=
(
*
)
(
*
) (
K (ξ , ξ)
)
*
(
δW δξ , ξ − ω2K δξ , ξ + δW ξ , δξ − ω2K ξ* , δξ
)
*
(
)
*
(
*
Use self adjoint property: K ξ , ∂ξ = K δξ, ξ
(
*
)
(
)
*
δW ξ , δξ = δW δξ, ξ
)
Then
δω2 =
∫ dr δξ
*
( )
*
*
⋅ ⎡F ( ξ ) + ω2ρξ ⎤ + δξ ⎡⎢F ξ + ω2ρξ ⎤⎥ K
⎣
⎦
⎣
⎦
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 13
Page 6 of 7
If δξ is arbitrary and δω2 = 0 (extremum) then
()
ω2ρξ = −F ξ
Simple Interpretation
H2 = −ω2K + δW = 0
conservation of energy
potential energy
kinetic energy
22.615, MHD Theory of Fusion Systems
Prof. Freidberg
Lecture 13
Page 7 of 7