CHAPTER 3
Solving for the Scalar
Magnetic Potential
3.1 THE "VARIATIONAL" FORMULATION
We now treat the problem we arrived at for what it is, an
equation,
to be
studied and solved as such: Given a bounded domain D, a number I
(the mmf), and a function t-t (the permeability), subject to the conditions
0 < l.t 0 < ~ < bt 1 of Eq. (18) in Chapter 2,
(1)
find ~
~
(I)I=
{(4) E
(I}" (4)=
0 on sh0, q)= I
on
Shl }
such that
fD ~ grad q0. grad q0' = 0 V q0' e ~0.
S b
FIGURE 3.1. The situation, reduced to its meaningful geometrical elements.
All potentials q0 and test functions q0' belong to the encompassing
linear space • of piecewise smooth functions on D (cf. 2.4.2), and the
geometrical elements of this formulation, surface S = S h u S b partition
h h h . . h
S = S o u S 1 of the magnetic wall S (Fig. 3.1), are all that we abstract
from the concrete situation we had at the beginning of Chapter 2. We
note that the magnetic energy (or rather, coenergy, cf. Remark 2.6) of h =
grad q0, that is,
61
62 CHAPTER 3 Solving for the Scalar Magnetic Potential
1 fD~
Igrad
I 2,
F( 0) =
is finite for all elements of cI). The function F, the type of which is
FIELD ~ REAL,
and more precisely, (I) --~ IR, is called the
(co)energy
functional.
Remark 3.1. The use of the quaint term "functional" (due to Hadamard),
not as an adjective here but as a somewhat redundant synonym for
"function", serves as a reminder that the argument of F is not a simple
real- or vector-valued variable, but a point in a space of infinite dimension,
representative of a field. This is part of the "functional" point of view
advocated here: One
may
treat complex objects like fields as mere "points"
in a properly defined functional space. 0
Function F is quadratic with respect to q0, so this is an analogue, in
infinite dimension, of what is called a
quadratic form
in linear algebra.
Quadratic forms have associated polar forms. Here, by analogy, we
define the
polar form
of F as Y-(q0, ~) = ~D ~t grad q). grad ~¢, a bilinear
function of two arguments, that reduces to F, up to a factor 2, when both
arguments take the same value.
The left-hand side of (1) is thus y- (q0, q0'). This cannot be devoid of
significance, and will show us the way: In spite of the dimension being
infinite, let us try to apply to the problem at hand the body of knowledge
about quadratic forms. There is in particular the following trick, in which
only the linearity properties are used, not the particular way F was
defined: For any real 5~,
(2) 0 < F(q0 + k~) : F(q0) + 5~ y- (q0, ~)
+ ~2
F(~) V ~ ~ ~.
One may derive from this, for instance, the Cauchy-Schwarz inequality,
by noticing that the discriminant of this binomial function of ~ must be
nonpositive, and hence
y- (q0, ~) < 2 [F(q))] 1/2 [F(I]/)] 1/2,
with equality only if ~1/= aq0 + b, with a and b real, a > 0. Here we
shall use (2) for a slightly different purpose:
Proposition 3.1.
Problem
(1)
is equivalent to
(3)
Find (p
~ (D I
such that
F(q))< F(~) V ~ ~ cI) ~,
the
coenergy minimization
problem.
Proof.
Look again at Fig. 2.8, and at Fig. 3.2 below. If q0 solves (3), then
F(q0) < F(tp + ;~q0') for all q0' in cI) °, hence 5~ 7 (q0, q0') + 5~2 F(q0') > 0 for all
3.1 THE "VARIATIONAL" FORMULATION 63
e IR, which implies (the discriminant, again 1) that y-(% q0') = 0 for all
q0' in ¢0, which is (1). Conversely, if q0 solves (1), and ~ belongs to ¢~,
then (cf. (2)) F(~)= F(q0) + Y-(q0, ~- q0) + F(~- q0)= F(q0) + F(~- q0) _> F(q0),
since t~-q0 is in ¢0 and F(~- q0) > 0. ~)
This confirms our intuitive expectation that the physical potential
should be the one, among all eligible potentials, that minimizes the
coenergy. Problem (3) is called the variational form of the problem. In the
tradition of mathematical physics, a problem has been cast in variational
form when it has been reduced to the minimization of some function
subject to some definite conditions, called "constraints". The constraint,
here, is that q0 must belong to the affine subspace ¢i (an affine constraint,
therefore). Such problems in the past were the concern of the calculus of
variations, which explains the terminology. Nowadays, Problem (1) is
often described as being "in variational form", but this is an abuse of
language, for such a weak formulation does not necessarily correspond
to a minimization problem: In harmonic-regime high-frequency problems,
for instance, a complex-valued functional is stationarized, not minimized.
For the sake of definiteness, I'll refer to (1) as "the weak form" and to (3)
as "the variational form".
(I~} $
jJ
z~
j~J~0
/
~J
FIGURE
3.2. Geometry of the variational method (~ = q0 + q0').
Conversely, however, variational problems with affine constraints
have as a rule a weak form, which can be derived by consideration of the
directional derivative 2 of F at point % By definition, the latter is the
linear map
(4)
t~ ~ lim ~ -~0 [F(qo + Kt~)- F(qo)] / ;~.
If q0 yields the minimum, the directional derivative of F should vanish
1Alternatively, first divide by ;~, then let ~ go to 0.
2Known as the G~teaux derivative.
64 CHAPTER 3 Solving for the Scalar Magnetic Potential
at q0, for all directions that satisfy the constraint. The condition obtained
that way is called the Euler equation of the variational problem.
Here, (4) is the map tg --~ y-(q0, ~g), after (2). Therefore, Problem (1),
which expresses the cancellation of this derivative in all directions parallel
to q~0, is the Euler equation of the coenergy minimization problem (3).
Exercise 3.1. Find variational forms for Problems (2.34) and (2.36).
In the space q~* of the last chapter (Exer. 2.9), which is visualized as
ordinary space in Fig. 3.2, we may define a norm, IIq011, = (2F(q))) 1/2 =
[~D ~t I grad
q)[2]1/2,
hence a notion of distance" The distance in energy of
two potentials is d,(q0, tg) = IIq0 - ~gll
- [fD ~
I grad(q0- ~g)l
211/2.
The
variational problem can then be described as the search for this potential
in q~i that is closest to the origin, in energy: in other words, the projection
of the origin on q~i.
Moreover, this norm stems from a scalar product, which is here, by
definition, (q0, ~g),
= fD ~
grad q0. grad ~g (= y(q0, ~g), the polar form), with
the norm IIq011, defined as [(q0,
q))~t] 1/2. The
weak form also then takes on
a geometrical interpretation: It says that vector q0 is orthogonal to q~0,
which amounts to saying (Fig. 3.2) that point q0 is the orthogonal projection
of the origin on q~i. The relation we have found while proving Prop. 3.1,
(5) F(/[/) -- F(q)) q- F(l[/- q)) V 1[/E (I)I,
if q0 is the solution, then appears as nothing but the Pythagoras theorem,
in a functional space of infinite dimension.
Exercise 3.2. Why the reference to q~*, and not to q~ ?
This is our first encounter with a functional space: an affine space, the
elements of which can usually be interpreted as functions or vector fields,
equipped with a notion of distance. When, as here, this distance comes
from a scalar product on the associated vector space, we have a pre-
Hilbertian space. (Why "pre" will soon be explained.) The existence of
this metric structure (scalar product, distance) then allows one to speak
with validity of the "closeness" of two fields, of their orthogonality, of
converging sequences, of the continuity of various mappings, and so
forth. For instance (and just for familiarization, for this is a trivial result),
if we call q0(I) the solution of (1) or (3), considered as a function of the
mmf I, we have
Proposition 3.2. The mapping I --~ q0(I) is continuous in the energy metric.
Proof. By (1), q0(I)= I q0(1), hence IIq0(I)ll, = I II IIq0(1)ll, that is, IIq~(I)ll, <
C I II for all I, where C is a constant, thus satisfying the criterion for
continuity of linear operators. O
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