IN NONLINEAR FUNCTIONAL ANALYSIS
TOMOAKI CHIBA AND HIROYUKI NAGAHAMA
1 INTRODUCTION: THE CURIE PRINCIPLE The Curie symmetry principle (Curie, 1894) is the causality relation between the symmetry of the cause and that of the effect. The principle is composed of three parts:
2 NONLINEAR FUNCTIONAL ANALYSIS Sattinger (1979) described the method
to analyze functional in association with group theory. Nonlinear functional
equations are generally represented in the form
(1) F(m
, w) = 0 where F: R´B®B
and w = w(m)
(mÎR,wÎ
B). It denotes that w is defined on a Banach space B
and depends on a real parameter (bifurcation parameter) and that F
maps w to new w
with the change of m. Sattinger investigated
the functional equations which mapping F
is covariant under group G,
i.e. (2) F(m
, gw) = gF(m , w) where G
and g are set and its element,
respectively. In this paper, we consider for unitary group that preserves
the inner product invariant. This assumption doesn’t loose generality in
considering most of real phenomena. Since unitary group is isomorphic with
general linear group that is a set of linear mapping, we can take advantage
of knowledge and techniques in linear algebra. With application of group
representation theory, we can define "standard decomposition" (Sattinger,
1979). We decompose Hilbert space H
here since we have to analyze functional equations on H
to deal with unitary group. Then H
is decomposed into some H^{a}
which corresponds to each irreducible element g_{a}ÎG
and H^{a}
intersects orthogonally one another. Therefore, we can now treat variables
by its symmetrical compositions, e.g. w
=
(w_{1f 1}, w_{2f
2}, ¼ , w_{qf
q}) where (f_{1},
f_{2, ¼
,} f_{q})
is a symmetrical basis. Each projection P^{a}
onto H^{a}
can also be defined (Sattinger, 1979; Fujii and Yamaguti, 1980). Especially,
a=
1 is assigned to isotoropy group and T_{g}P^{1
}=
P^{1}
for the transformations
T_{g}
(gaÎG).
Furthermore, we define Fréchet derivative F´=
¶
F/¶ w to investigate local mapping
of functional in the neighborhood of a point
(m_{0,
w0)}
at which F´ is defined.
If there exists reducible F´(m_{0},
w_{0}),
implicit function theorem assures that eq.(1) has a unique solution in
the neighborhood of (m_{0,
w0)}.
If F is irreducible at
a certain point, this point is singular and several solutions exist in
its neighborhood. For simplicity, we consider about only simple, i.e. not
multiple singular points in this paper.
3 SYMMETRY IN FUNCTIONAL SYSTEMS First we consider the case thatF
is reducible. Since T_{g}P^{1
}=
P^{1}, T_{g}w
=
w for w = w_{1}ÎH^{1}.
With this Ginvariant
property, symmetry preserving of Gcovariant
mapping in the subspace is proved. This is to say that for all w_{1}ÎH^{1},^{F(}m,
w_{1})
= P^{a}F(m,
w_{1})
is derived from eq.(2). Therefore, P^{a}F(m,
w_{1})
= 0 (a =
2, 3, ¼
, q). Since P^{1}
reflects isotoropy group, both relations indicate that the solution path
of eq.(1) is enclosed in the subspace H^{1}
unless F becomes irreducible.
In other words, the symmetry of the system does not change on each solution
path. Next we consider the case that F
is irreducible. From eq.(2), Fréchet derivative F´
satisfies the following equation F´(m,
T_{g}w)×T_{g
}=
T_{g}F´(m,
w)
("g
ÎG,
"w
ÎH).
Multiplying by the group characters and summing for "g
ÎG,
we get F´(m,
w_{1})P^{a
}=
P^{a}F´(m,
w_{1})
(a =
1, ¼
, q; "w_{1
}_{Î}H^{1}).
According
to linear algebra, it denotes that F´
can be represented as a diagonal matrix form, i.e. F´(m
,w) = (F_{1}´,
F_{2}´, ¼,
F_{q}´). Since the basis represents the symmetry,
each eigenvalue Fa´
corresponds to a certain symmetrical element of the system. According to
Sattinger (1979), the symmetry corresponding Fa´
= 0 dominates the new solution path, e.g. the path bifurcating from the
singular point. Because the new solution path is more stable (Fujii and
Yamaguti, 1980), the system follows the new path. Therefore, if eigenfunction
of Fa´
= 0 doesn’t correspond the symmetry dominated the system until reaching
the singular point, the symmetry of the system has not been held yet. Since
the old path belongs to the space corresponding the isotoropy group, the
new path belongs in most case to the space corresponding the other subgroup
with the lower symmetry than that of decomposed original space. The system
is "dissymetrical" at the singular point in this sense.
4 RECONSIDERATION OF CURIE PRINCIPLE Finally, we consider the Curie principle
in functional analysis. We can derive another Fréchet derivative =
?F/?m
in the space with symmetrical basis by the same procedures described above.
Because eq.(1) denotes the relation w
=
w(m),
we can refer m and w
as the cause and the effect, respectively. Then, we can refer
and F´ as representing
the symmetries of the cause and the effect, respectively. Once F_{a}´
not corresponding to the existing component of
becomes to 0, the symmetry of
the cause is no longer preserved in the symmetry of the effect. This case
is just the "breaking of the Curie symmetry principle". In contrast, even
if F_{a}´
corresponding the existing component of becomes
to 0, the symmetry of the cause
is held in the effect. The Curie symmetry principle is preserved in this
case. Of course, the principle is preserved if F´
is reducible. In this sense, both preserving and breaking of the Curie
symmetry principle occur in a same nonlinear system.
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