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{\nopagenumbers
~ \vskip 2 truecm
\centerline{\grand COUNTING SYMMETRY BREAKING SOLUTIONS}
\bigskip
\centerline{\grand TO SYMMETRIC VARIATIONAL PROBLEMS}
\vskip 3 truecm
\centerline{Giuseppe Gaeta}
\bigskip
\centerline{\it Department of Mathematical Sciences}
\centerline{\it Loughborough University of Technology}
\centerline{\it Loughborough LE11 3TU (England)}
\centerline{\tt G.Gaeta@lut.ac.uk}
\footnote{}{{\tt To appear in {\it Int. J. Theor. Phys.} January 1996}}
\vskip 2 truecm
{\bf Summary}
{By combining the Michel's geometric theory of symmetry breaking and classical
result from variational analysis, we obtain a lower bound on critical
points with
given symmetry $H \sse G$ of a potential symmetric under $G$. The result is
obtained
by applying Ljusternik-Schnirelman category in the group orbit space, and can be
extendeded along the same lines to more general situations.}
\vfill\eject}
\pageno=1
{\bf 1. Introduction}
It is well known that Michel's theorem [1] (see also [2,3] for more detailed
discussion, and [4-9] for applications in Theoretical Physics) permits to obtain
generic minimal symmetry breaking solutions to symmetric variational
problems on the
basis of a study of the geometry of group action, and in particular of the {\it
isotropy stratification} [1-3] of the orbit space.
Here, by ``minimal symmetry breaking'' we mean a breaking of the symmetry
from $G$ to
$G_0$ with $G_0$ a maximal isotropy subgroup of $G$; in the Michel theory
language
this would be more precisely expressed by saying that $G_0$ corresponds to the
symmetry type of a maximally singular stratum. By ``generic'' solution, we
mean a
solution which depends only on the symmetry properties of the problem, but
not on the
particular form of the potential.
Thus, Michel theory provides {\it model-independent} solutions to variational
problems\nota{$^1$}{Actually, Michel theory is able to deal with more general
classes of problems with symmetry; however, here we are mainly interested in the
variational case.}. The best known application of Michel theorem, and the
one which
motivated Michel's work, is maybe the one -- by Michel himself and Radicati
-- to the
spontaneous symmetry breaking in the SU(3) theory of elementary particles
[4-7], giving
raise to the octet\nota{$^2$}{See also the contemporaneous results of
Cabibbo and
Maiani [8]}.
The purpose of the present note is to remark that, by the same approach, one can
also count the model-independent number of (non necessarily minimal)
symmetry breaking
solutions. For a variational problem, critical points come in group orbits,
so that we
will actually count critical orbits with a given symmetry type, as
explained below.
We will state our result in the case of a finite dimensional smooth
manifold $M$ on
which is defined the smooth action of a compact Lie group $G$ (one says
then that $M$
is a smooth $G$-manifold). However, our results would extend to a much more
general
situation (e.g. Banach manifolds), provided some technical conditions are met; this
extension would essentially parallel the extension of Michel theorem to the
``Symmetric Criticality Principle'' of Palais [10,11]\nota{$^3$}{It is
interesting to
notice that Cabibbo and Maiani, in the work mentioned above [8], had used
an argument
essentially amounting to the symmetric criticality principle, although they
did not go
to the generality of Michel theorem. The strange story of the argument will be
discussed elsewhere.}, and will not be discussed here.
It is remarkable that our result can be stated, in the mathematical
language, in terms
of the Ljusternik-Schnirelman (LS) category (see e.g. [12]), which -- to
conform to
physicists' language -- we will call {\it LS index}. Actually, progresses
in this
direction were obtained in the mathematical literature, but these focus on
considering
equivariant LS index in the manifold $M$ rather than -- as we do here --
ordinary LS
index in the orbit space $\Om = M/G$, see e.g. [13-16].
In the following, we will first briefly recall Michel's construction of isotropy
stratification, and the key lemma used by Michel in order to prove his
theorem [1]. Our
present result will then follow from Michel's discussion, essentially by
just applying
it (and the LS category) to the problem at hand here.
It is worth pointing out that the Michel construction is discussed in more
detail in
[2,3,9,17-19], to which we refer for further precisions. This same -- or some
strongly related -- construction, and to some extent the same lemma, was
then used to
obtain further results in the field of nonlinear dynamics -- such as the
equivariant
branching lemma and the reduction lemma -- some of which became fundamental
tools in
understanding bifurcation theory and symmetry breaking (see e.g.
[17,20-26]). One
should also mention the deep results of Field and Richardson [27-30], who
extended
Michel's theory and clarified the role of the Weyl group in this context.
It should be mentioned that we make no use of Morse theory [16,31,32], so
that we
cannot reproduce results that intrinsically need it (but one could apply
Morse theory
to the present construction, i.e. in $\Om$). Also, we do not discuss here
periodic
solutions, which have been recently studied by paralleling Michel's
approach [33,34]
(see also [35]), or can be studied by $S^1$-equivariant index [13-16].
\bigskip \bigskip
{\bf 2. Isotropy stratification and orbit space}
\bigskip
Let $M$ be a finite dimensional smooth manifold, and $G$ a compact Lie
group; let a
smooth action (not necessarily linear) of $G$ on $M$ be defined, i.e. let
$M$ be a
$G$-manifold. To any point $x \in M$ is then associated a subgroup $G_x
\sse G$, its
{\it isotropy subgroup}:
$$ G_x \: = \{ g \in G \: gx = x \} . \eqno(1) $$
In many cases -- and anyway in the present context -- subgroups conjugate in $G$
should be seen as physically equivalent\nota{$^4$}{E.g., for $G=SO(3)$, the
subgroups $H_\ell = SO(2)$ of rotation around the axis $\ell$.}, so that we will
consider equivalence classes of subgroups under conjugation in $G$, called {\it
symmetry types}:
$$ [H] \: = \{ K \sse G \: K = g H g^{-1} ~,~ g \in G \} . \eqno(2) $$
If there exist $G_1 \in [H]$, $G_2 \in [K]$ such that $G_1 \ss G_2$, we say
that the
symmetry type $[K]$ is higher than the symmetry type $[H]$, or $[H] < [K]$.
We can then consider an equivalence relation in $M$ as follows: the
equivalence class
of $x$, called the {\it stratum}\nota{$^5$}{This is the latin word for layer
(plural: {\it strata}).} of $x$, denoted by $\s [x]$, is defined as
$$ \s [x] \: = \{ y \in M \: G_y \in [ G_x ] \} ~\equiv ~ \{ y \in M \: G_y
= g G_x
g^{-1} ~,~ g \in G \} . \eqno(3) $$
The set $\s [x]$ is a smooth submanifold of $M$ [2,3,36,37].
Let us also consider another equivalence relation on $M$ given by
$G$-action: the
equivalence classes are the $G$-orbits, denoted by $\om [x]$,
$$ \om [x] \: = \{ Gx \} ~\equiv ~ \{ y \in M \: y = gx ~,~ g \in G \} .
\eqno(4) $$
It is quite obvious, but important, that
$$ \om [x] \sse \s [x] \eqno(5) $$
(this simply follows from the fact that $y=gx$ implies $G_y = g G_x g^{-1}$).
One can also consider the {\it orbit space}
$$ \Om \: = M / G \eqno(6) $$
(which with the present assumptions on $M$ and $G$ is a semialgebraic
manifold [3,9,36-38]), and define a
stratification on this: that this is possible, is indeed granted by (5). We will
denote isotropy strata in
$\Om$ by
$\S$, and have
$$ \S [\om ] \: = \{ \nu \in \Om \: \exists x \in \om ,~ \exists y \in \nu
\: y \in \s
[x] \} . \eqno(7) $$
Again, $\S [\om]$ is a smooth submanifold of $\Om$.
An important point is that the group-subgroup relation is reflected into a
bordering
relation for strata. This means that if $G_x , G_y$ are isotropy subgroups
of $G$ and
$\om , \nu \in \Om$ with $x \in \om$, $y \in \nu$, then $\S [\om ] \sse
\partial (
\S [ \nu ] )$. Conversely, $\partial ( \S [ \nu ] )$ belongs to the union
of strata
$\S [\om_i ]$ with symmetry types $[K] \ge [G_y]$.
\bigskip\bigskip
{\bf 3. Equivariant flows and invariance of strata}
\bigskip
The relevance of the above construction and definitions, in this context,
is due to
the following result. Consider a vector field $f : M \to \T M$ on $M$, or
equivalently
the dynamical system on $M$ defined by $f$,
$$ {\dot x} = f(x) , \eqno(8) $$
and suppose that $f$ is $G$-equivariant, i.e.
$$ f (gx) = (\dd g \cdot f ) (x) \eqno(9) $$
where $\dd g$ denotes the action of $g$ on the tangent space (recall the
action of $G$
is not necessarily linear). Then [1] necessarily
$$ f : x \to \T_x \s [x] . \eqno(10) $$
In the case of a variational $G$-invariant problem, defined by a smooth
potential $F :
M \to R$, i.e. a potential which satisfies
$$ F( gx) = F(x) ~~~ \forall g \in G ,~ \forall x \in M , \eqno(11) $$
the gradient $f := \grad F$ of $F$ satisfies (9), and therefore (10). This
means
that, given a stratum $\s \sse M$, we can consider the restriction $F_\s$
of $F$ to
$\s$: critical points of $F_\s$ are then granted to be critical
points\nota{$^6$}{It should be stressed that this does not extend to stability
matters: e.g. a minimum of $F_\s$ will not necessarily be a minimum of $F$:
it could
also be a saddle point.} of $F$ as well [1,10,11].
Notice that (11) also grants that critical points of $F$ come in
$G$-orbits, and that
we can think $F$ as being defined on $\Om$ rather than on $M$; we use then
the notation
$\Phi ( \om )$, with $\Phi : \Om \to R$ and
$$ \Phi ( \om ) = F(x) ~~~~ x \in \om . \eqno(12) $$
Consider now the vector field $\phi$ induced by $f$ on $\Om$; if $f = \grad
F$, then
$\phi = \grad_\om \Phi$, where $\grad_\om$ is the gradient with respect to $\om
$ \nota{$^7$}{This could be precisely defined, e.g. by using a Hilbert minimal
integrity basis [2,3,9,36-38].}. As critical points of $F$ come in
$G$-orbits, we can
equivalently study critical points of $\Phi$.
Applying (10) to $f = \grad F$, we have that
$$ ( \grad_\om \Phi ) (\om ) \in \T_\om \S [\om ] . \eqno(13) $$
Therefore, we can consider the restriction $\Phi_\S$ of $\Phi$ to any
stratum $\S \sse
\Om$; critical points of $\Phi_\S$ will also be critical points of $\Phi$,
corresponding to critical points of $F$ with symmetry type
$$ [H] = [G_x ] ~~,~ x \in \om ~,~ \om \in \S . \eqno(14) $$
It should be stressed that, in general, the strata $\s$ or $\S$ are {\it
not} closed
manifolds; thus, although (10) ensures that the solution $x(t)$ of (8) with
initial
datum $x(0) = x_0$ lies in $\s_0 = \s [x_0 ]$, even if $x(t)$ admits a
limit $x_*$
for $t \to \infty$, this could very well not lie in $\s_0$, but only in
$\bar{\s_0}$,
i.e. it could be $x_* \in \partial \s_0$, $x_* \not\in \s_0$.
Similar considerations also hold for the flow $\phi$, and in particular the
gradient
flow $\phi = \grad_\om \Phi$, in $\Om$ considered in (13) and in the following.
Finally, it should be mentioned that (10) could be made more precise
[22-24]: indeed,
if $M_H$ is the subset of $M$ left fixed by $H \sse G$, i.e.
$$ M_H \: = \{ x \in M \: H \sse G_x \} , \eqno(15) $$
and writing $M_x$ for $M_{G_x}$, we have that if $f$ satisfies (9)
then\nota{$^8$}{This fact is related to the slice theorem, see e.g. [3,36,37]).}
$$ f : x \in \T_x M_x \sse \T_x \s [x] . \eqno(16) $$
Notice however that in this way we loose track of the physically desirable
identification of conjugate subgroups.
\bigskip \bigskip
{\bf 4. Counting critical orbits, and symmetry breaking solutions}
\bigskip
Let us now introduce the following notation: we denote by $\S^H$ the
stratum in $\Om$
whose orbits have symmetry type $[H]$,
$$ \S^H \: = \S [\om ] ~~~,~ \om \: \forall x \in \om ,~ H \in [ G_x ] ;
\eqno(17) $$
and we denote $\Phi_{\S^H}$ simply by $\Phi_H$ for ease of notation.
The above discussion, which followed Michel's construction [1,2,3,9,17-19],
shows in
particular that:
{\bf Lemma 1.} The number $n[H]$ of critical $G$-orbits of $F$ in $M$ with
symmetry
type $[H]$, is equal to the number of critical points of $\Phi_H$.
Indeed, we have seen that critical $G$-orbits of $F$ are critical points of
$\Phi$; also, by definition critical orbits of $F$ with symmetry type $[H]$ are
critical points of $\Phi$ which lie in $\S^H$. But, as we have seen before,
these
are also critical points of $\Phi_H$: hence the lemma is just a
reformulation of
the previous discussion.
Due to the considerations appearing after equation (14), we are also
interested in
considering the set $\La^H$ corresponding to the union of the $\S^K$ with
symmetry
type $[H]$ or higher,
$$ \La^H := \bigcup_{[H] \le [K]} \S^K . \eqno(18) $$
Notice that we can easily define the restriction of $\Phi$ to $\La^H$,
being given in
terms of the $\Phi_H$ with $[H] \le [K]$; this will be called $\Psi_H$.
>From these definitions we have immediately, as a corollary of lemma 1 and
with the
same proof, the
{\bf Lemma 2.} The number $k[H]$ of critical $G$-orbits of $F$ in $M$ with
symmetry
type $[H]$ or higher, is equal to the number of critical points of $\Psi_H$.
We want to consider the situation in which either $M$ is closed, or there
is a closed
submanifold $M_0 \sse M$ which is invariant under both the $G$-action and
the gradient
flow $f = \grad F$, i.e. such that $G(M_0 ) = M_0$ and on $\partial M_0$
the gradient
of $F$ in the outward normal direction has definite sign. For ease of
notation, we
just consider the case $M$ is closed (the other case being exactly the same
provided
we restrict our considerations to $M_0$), and that if $\partial M$ exists,
$f = -
\grad F$ points inward on $\partial M$; we say then that $M$ is contracting
under $F$.
In this case, we are guaranteed that there is at least a critical point of
$F$ in the
interior of $M_0$, and topological considerations can also guarantee there is a
higher number of critical points on $M$.
It should be stressed that now the difference between lemma 1 and lemma 2
is that the
first applies to a potential $\Phi_H$ which could well not have critical
points in
its domain of definition $\S^H$; while in the second case we are guaranteed
$\Psi_H$
has critical points in its domain of definition $\La^H$.
The critical points $x_i$ of $F$ will have a symmetry type $[G_{x_i} ]$,
and we are
interested precisely in this, i.e. we are interested in the {\it symmetry
breakings}
corresponding to critical points of the symmetric potential $F$.
Michel's theorem [1] permits to conclude that if an orbit $\om_0$ is
isolated in its
stratum, then $\om_0$ is a critical point of any $\Phi$, and
correspondingly it is a
critical orbit of any $G$-invariant potential $F$. Here we do not aim at such a
precise identification of critical orbits, but rather at identifying only
the symmetry
types of critical orbits, and the numbers $n[H]$ of critical orbits with
symmetry
type $[H]$; we are also interested in the numbers $\nu [H]$ of critical
orbits with
symmetry type $[H]$ or higher, $\nu [H] = \sum_{[H] \le [K]} n[H]$.
We would like in particular to know what is the minimal possible value of
$n[H]$, i.e.
the number of critical $G$-orbits for $F$ with given symmetry type $[H]$
which is
possible independently\nota{$^9$}{In this context, the original formulation of
Michel's theorem [1] guarantees that if $\S^H$ is the union of $h$ isolated
points
$( \om_1, ..., \om_h )$, as each of these is critical, then $n[H] = h$. The result we
obtain below is a direct generalization of this.} of the actual $F$, just
on the basis
of the symmetry under $G$; we denote this minimal value of $n[H]$ by $m[H]$.
Similarly, we would also like to know the minimal possible value of $\nu
[H]$, which
we call $\mu [H]$.
\medskip
Let us first decompose $\La^H$ into disjoint connected components $\La^H_a
\sse \Om$,
where $a = 1 , ..., c(H)$; we can then consider separately the $\La^H_a$.
We recall that the LS category (or LS index) [12,16] of a connected set $A
\sse X$,
denoted by $\LS (A,X)$, is defined as the minimal number $k$ such that
there exists a
covering of $A$ by closed sets $A_i$,
$$ A = A_1 \cup ... \cup A_k , \eqno(19) $$
with all the $A_i$'s contractible in $X$.
A classical result in variational analysis [12], and more specifically in
LS theory, is
that: {\it a scalar function $p : A \to R$ has at least $\LS (A,X)$
critical points in
$A$.}
Thus, applying this to $\La^H_a$ and $\Phi_H$, we have that if $\La^H_a$ is
contractible, we have at least a critical point of $\Phi_H$ in it, and if
$\La^H_a$ is
non contractible, we have at least as many critical points of $\Phi_H$ as
we need
contractible open sets to cover $\La^H_a$. Therefore, the above result from
LS theory
permits to conclude immediately that:
{\bf Lemma 3.} The number $\mu [H]$ defined above is given by
$$ \mu [H] = \sum_{a=1}^{c(H)} \LS \left( \La^H_a , \Om \right) . \eqno(20) $$
We can therefore consider the lattice of isotropy subgroups of $G$ and use
the above
result to study symmetry breakings under $F$. For the sake of definiteness,
we will
consider the minima of $F$, and assume that if $M$ has a border, then $f =
\grad F$
points outward of $M$ on $\partial M$, i.e. $M$ is contracting under $F$.
Let us first consider $G$ itself, and correspondingly $\La^G = \S^G \sse
\Om$ if it
exists. In this case, we are sure there is at least a critical point $\om
\in \La^G$
(indeed $\S^G$ is a maximally singular stratum, if it exists), and actually
we know
that there are at least $\mu [G] = \LS (\S^H , \Om )$ critical points of
$\Phi$ with
symmetry type $[G]$ (i.e. invariant under $G$, as $H \in [G]
\Leftrightarrow H=G$
since $G$ is the full group).
Assume now that the critical points in $\La^G$ are known to be maxima, i.e.
that the
symmetry $G$ is broken. Let us consider a maximal isotropy subgroup $H$ of
$G$. We
can now consider $\La^H$ and claim that $\mu [H] = \LS ( \La^H , \Om )$, but the
information that the symmetry $G$ is broken allows also to consider $\S^H =
\La^H
\backslash \La^G$ being guaranteed that $\S^H$ is contracting under $\Phi$.
Thus, we
can also affirm that there are at least $m[H] = \LS \left( \S^H , \Om \right)$
critical points of $\Phi$ with symmetry type $H$.
It is clear that the procedure can be iterated along any chain of
subgroups. In this
way, we arrive at the following conclusion:
{\bf Lemma 4.} Let $H = H_0 \ss H_1 \ss H_2 \ss ... \ss H_n \sse G$ be a
complete
chain of isotropy subgroups, i.e. all the $H_i$ are isotropy subgroups of
$G$ for the
$G$-action on $M$, and $H_i$ is a maximal isotropy subgroup of $H_{i+1}$
($H_n$ a
maximal isotropy subgroup of $G$ if there is no $x \in M$ for which $G_x = G$).
Assume that the symmetries $H_1,...,H_n$ are broken, i.e. that the critical
points of
$\Phi$ with symmetry types $[H_1 ] , ... , [H_n ]$ are known to be
unstable. Then,
there are at least
$$ m [H] = \LS \left( \S^H , \Om \right) \eqno(21) $$
critical points of $\Phi$ with symmetry type $[H]$.
{\bf Example.} As a simple example, let us consider $R^2$, on which we take
coordinates $(x,y)$; and $G = Z_2 \times Z_2 = Z_2^{(x)} \times Z_2^{(y)}$, with
$Z_2^{(x)}$ generated by $h: (x,y) \to (-x,y)$ and $Z_2^{(y)}$ generated by
$k: (x,y)
\to (x,-y)$. As for the orbit space $\Om$, this can be identified with the first
quadrant of $R^2$,
$$ \Om \simeq R_{++} = \{ (x , y ) ~:~ x \ge 0 ,~ y \ge 0 \} . $$
We have the following lattice of isotropy subgroups, where an arrow
indicates a group-subgroup relation:
$$ {\tt diagram} $$
$$ \{ e \} \ss \left\{ \matrix{ Z_2^{(x)} \cr Z_2^{(y)} \cr} \right\} \ss G .$$
It is immediate to check that there are four strata in $M$ (we denote by $\xi =
(x,y)$ points of $R^2$):
$$ \matrix{
\s_0 &=& \{ (0,0 ) \} & &~~ G_\xi =& G ~~ \cr
\s_1 &=& \{ (0,y ) \} &~(y \not= 0) &~~ G_\xi =& Z_2^{(x)} \cr
\s_2 &=& \{ (x,0 ) \} &~(x \not= 0) &~~ G_\xi =& Z_2^{(y)} \cr
\s_3 &=& \{ (x,y ) \} &~(xy \not= 0) &~~ G_\xi =& \{ e \} \cr } $$
The strata $\S_i$ in $\Om$ are immediately obtained from these by
considering $\s_i
\cap R_{++}$. Notice that $Z_2^{(x)}$ and $Z_2^{(y)}$, although isomorphic,
are {\it
not} conjugated in $G$, so that we have different strata corresponding to these.
The $G$-invariant potentials $V(x,y)$ are of the form $V = V(x^2 ,
y^2 )$, and the requirement that there is an invariant $M \sse R^2$ is
satisfied if
e.g. $\lim_{x,y \to \infty} V = \infty$, i.e. if $V$ is convex at
infinity, which we
assume to be the case; equivalently, we choose a rectangle $M = I_x \times
I_y$ such
that $\grad V$ points outward on $\partial M$.
Clearly, any $V = V(x^2 , y^2)$ has a critical point at the origin. It is
also clear
that on the $x$ and on the $y$ axis, the gradient $\grad V$ is directed
along the
axis itself, so that if the origin is a maximum, there has to be (at least)
a minimum
on the $x$ axis for positive $y$ and one for negative $y$, which are
related by $k$;
and similarly on the $y$ axis we have (at least) a minimum for positive $x$
and one for
negative $x$, related by $h$.
Indeed, applying lemma 4 we get exactly this result: considering any
compact $M \ss
R^2$, the LS indexes of $\S_i$ are all equal to one; then, if the origin is an
unstable critical point, we have one critical orbit (at least) in each of
$\s_1$ and
$\s_2$; and if these are also unstable, we get one critical orbit (at
least) in the
generic stratum $\s_4$.
\bigskip\bigskip
{\bf 5. Discussion and generalizations}
\bigskip
Notice that the above lemma 4 has a ``global'' formulation, in the sense that we
require {\it all} the critical points with higher symmetry $[H_1 ] , ... ,
[H_n ]$
are unstable. However, it is quite clear that we could equally well make
``local''
equivalent considerations, i.e. restrict our attention to a neighbourhood
$U_0$ of a
given critical point $\om_0$ of symmetry type $[G_0 ]$ when $\om_0$ is
unstable, or to
a neighbourhood $U_1 \sse U_0$ of a given critical point $\om_1 \in U_0$ if both
$\om_0 $ and $\om_1$ are unstable, and so forth. (Here each $G_j$ plays the
role of the
$H_{n-j}$ in the previous formulation).
One could also consider the case $F$ -- and therefore $\Phi$ -- depends on
a control
parameter $\lambda$ in such a way that for varying $\lambda$ the stability
of critical
points are changed. We could use the approach sketched above to follow a
chain of
symmetry breakings, or in the mathematical language a chain of
bifurcations; such an
extension would just repeat the lemma 4 in a ``local'' formulation,
requiring moreover
the existence of a suitable chain of $\lambda$-dependent neighbourhoods.
In this way, one could count -- in terms of the LS index -- the generic
number of
bifurcating branches with a given symmetry type at the $k$-th symmetry
breaking in the
chain. This would be a generalization of the ``Equivariant Branching
Lemma'' (EBL) of
Cicogna and Vanderbauwhede [20,21,25] in the frame of variational (stationary)
bifurcation problems.
It should be mentioned that the EBL was also generalized to study the case
of Hopf
bifurcation (of periodic solutions) and of quaternionic bifurcation
[22-24,32,33]. It
is then natural to ask if the approach developed in the present paper could
also be
generalized to such a setting. I have no answer to such a question at this
stage,
although a natural way of attempting such an extension would combine the present
approach, the $S^1$ index of Benci [13-15], and the ``splitting principle''
recently
proposed by the present author [41].
The $S^1$ symmetry of the Benci index would be a dynamical one, i.e.
correspond to
motion along the periodic solutions rather than being a symmetry in the sense
considered here. Clearly, the two notions can coincide, i.e. one can have a
periodic
orbit which lie in one $G$-orbit; in this case one has a {\it relative
equilibrium}.
It appears that combining the $S^1$ index and the splitting principle would be
particularly efficent in considering bifurcation of\nota{$^{10}$}{This
should not
be confused with the bifurcation {\it from} relative equilibria, for which
see the
paper by Krupa [42].} relative equilibria.
In many cases, one is interested in variational problems in infinite dimensional
spaces, e.g. spaces of section of a fiber bundle (as in gauge theories).
Although the
extension of Michel theory to such a setting is very difficult -- and a full
extension probably impossible [18,19] -- as the stratification is not
properly defined
in this case, the equivalent of (16) does still hold, and yields the
Palais' {\it
Symmetric Criticality Principle}; also, a partial extension of Michel theory to
this context (essentially, for maximal isotropy subgroups) is possible
[18,19]. Thus,
the approach proposed here can also be extended to infinite-dimensional
variational
problems, essentially in the cases -- and with the limitations --
considered by Palais
[10,11].
As a final remark, already anticipated in the introduction, we notice that
if we want
to apply Morse theory, by using the same construction we can apply it to
$\Phi$, i.e.
in $\Om$, rather than to $F$, i.e. in $M$. Similarly, if we want to
specifically look
for periodic solutions (in a non-gradient system) we could apply the $S^1$-index
[13-16] to $\phi$, i.e. in $\Om$, rather than to $f$, i.e. in $M$.
\bigskip
In concluding, more than stressing the strict content of the previous
lemmata, we would
like to emphasize the main idea -- and result -- of the present approach:
namely, that
we can use the {\it ordinary} LS theory [12,16] (as opposed to the
equivariant one) to
study symmetric variational problems and symmetry breaking solutions,
provided we work
in $\Om$ rather than in $M$, and we utilize Michel's theory.
\bigskip\bigskip\bigskip\bigskip
{\bf Acknowledgements}
\bigskip
I would like to thank the Centre de Physique Th\'eorique of the Ecole
Polytechnique
(Palaiseau), where this work was started and -- at a later time --
concluded, and
where I enjoyed the usual warm hospitality. I would also like to thank N.
Tchou for
bibliographical help with the LS category and the LS equivariant index, and
C. Bachas
for recently reviving my interest in the topic, which allowed me to
complete this
work.
\vfill\eject
{\bf References}
\bigskip
\parindent=10pt
\parskip=0pt
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}
\bye