%%% This is a plain TeX file %%%
\magnification=\magstep1
\parskip=10pt
\parindent=0pt
\def\giorno{20/5/98}
\def\a{\alpha}
\def\b{\beta}
\def\g{\gamma}
\def\ga{\gamma}
\def\de{\delta} %% NON ridefinire come \d !!!!
\def\eps{\varepsilon}
\def\phi{\varphi}
\def\la{\lambda}
\def\ka{\kappa}
\def\s{\sigma}
\def\z{\zeta}
\def\om{\omega}
\def\th{\theta}
\def\vth{\vartheta}
\def\B{{\cal B}}
\def\E{{\cal E}}
\def\F{{\cal F}}
\def\G{{\cal G}}
\def\R{{\cal R}}
\def\L{{\cal L}}
\def\M{{\cal M}}
\def\T{{\rm T}}
\def\V{{\cal V}}
\def\W{{\cal W}}
\def\Ga{\Gamma}
\def\De{\Delta}
\def\La{\Lambda}
\def\Om{\Omega}
\def\S{\Sigma}
\def\Th{\Theta}
\def\pa{\partial}
\def\pd{\partial}
\def\d{{\rm d}} %% derivative
\def\x{\times}
\def\xd{{\dot x}}
\def\yd{{\dot y}}
\def\grad{\nabla} %% gradient
\def\lapl{\triangle} %% laplacian
\def\ss{\subset}
\def\sse{\subseteq}
\def\Ker{{\rm Ker}}
\def\Ran{{\rm Ran}}
\def\ker{{\rm Ker}}
\def\ran{{\rm Ran}}
\def\iff{{\rm iff\ }}
\def\all{\forall}
\def\<{\langle}
\def\>{\rangle}
\def\eor{{$\odot$}}
\def\({\left(}
\def\){\right)}
\def\[{\left[}
\def\]{\right]}
\def\=#1{\bar #1}
\def\~#1{\widetilde #1}
\def\.#1{\dot #1}
\def\^#1{\widehat #1}
\def\"#1{\ddot #1}
\def\section#1{\bigskip\bigskip {\bf #1} \bigskip}
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapleft#1{\smash{\mathop{\longleftarrow}\limits^{#1}}}
\def\mapup#1{\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\en#1{\eqno(#1)}
\def\ref#1{[#1]}
\def\Remark#1{{\tt Remark #1.}}
\font \petit = cmr9
\font \pit = cmr9
\font \ptt = cmtt9
\font \pbf = cmbx9
{\nopagenumbers
~~ \vskip 2 truecm
\centerline{\bf Reduction and Equivariant Branching Lemma}
\medskip
\centerline{\bf without finite-dimensional reduction}
\footnote{}{{\tt \giorno }}
\bigskip\bigskip\bigskip
%\centerline{Giampaolo Cicogna}
%\centerline{\it Dipartimento di Fisica,
%Universit\`a di Pisa,}
%\centerline{\it P.za Torricelli 2, 56126 Pisa (Italy)}
%\medskip
%\centerline{\tt cicogna@ipifidpt.difi.unipi.it}
%\bigskip
\centerline{Giuseppe Gaeta}
\centerline{\it Dipartimento di Fisica,}
\centerline{\it Universit\`a di Roma, 00185 Roma (Italy)}
\medskip
\centerline{\tt giuseppe.gaeta@roma1.infn.it}
\bigskip\bigskip
\vfill
{\bf Summary.} {In the bifurcation study of nonlinear
evolution PDEs with symmetry, one usually performs first a
reduction to a finite dimensional critical space, thus
obtaining the bifurcation equation (which inherits
symmetries properties from the original problem), and then
employs the symmetry -- tipically through the reduction
lemma and/or the equivariant branching lemma -- to study
this reduced problem. We argue that one could as well
proceed in the opposite way, i.e. apply bifurcation
analysis on a symmetry-reduced problem; this is done using
some general results of Palais for variational analysis of
$G$-invariant functionals. Such an approach presents some
delicate points, which we discuss in detail.}
\vfill\eject
~~
\vfill\eject}
\parindent=0pt
\pageno=1
\section{Introduction.}
The equivariant branching lemma (EBL in the following) is,
in its various versions \ref{1-4} and together with
the reduction lemma (RL in the following) \ref{3}, at the
very basis of equivariant bifurcation theory \ref{5}; we
stress here that, although the EBL is usually formulated
in the framework of linear group action, it also holds in
the more general context of smooth (Lie-point) nonlinear
group action \ref{6}.
When studying bifurcation in the presence of symmetry, one
follows a standard strategy, common also to the generic (no
symmetry) case: first of all one performs a reduction from
the full space (e.g. an infinite dimensional function
space) in which the original problem is set, to the
(usually, finite dimensional) critical space, i.e. the
space associated to the critical eigenvalues. This
reduction can be performed using different techniques (e.g.
the Lyapounov-Schmidt method, or using a Center Manifold
reduction), depending on the problem at hand; in general,
it is based on an implicit function theorem, and hence
requires the existence of a ``spectral gap'': that is,
there must be a gap separating the projection on the real
axis of the critical eigenvalues and of the rest of the
spectrum (of the appropriate linear operator entering in the
bifurcation analysis).
Such a reduction belongs to ``general'' -- i.e. not
necessarily equivariant -- bifurcation theory \ref{7-15};
the symmetry present in the original problem (if any) plays
a ``passive'' role in it: that is, while on the one side
the symmetry determines the multeplicity of critical
eigenvalues (up to accidental, and therefore nongeneric,
degeneracies) and it is -- in appropriate form -- inherited
by the reduced problem (notice that other symmetries, not
present in the original problem, can be present in the
reduced one \ref{16}), i.e. by the {\it bifurcation
equation}, on the other side in the reduction procedure one
has to take into account the symmetry properties but does
not have any help from them (except for handling the
degeneration of critical eigenvalues enforced by the
symmetry itself).
Once the above mentioned reduction -- typically, to a
finite dimensional problem -- has been obtained, one can
use the symmetry of the reduced problem, i.e. of the
bifurcation equation; this use can go typically
through further reduction into invariant subspaces
determined on the basis of symmetry considerations (RL),
and/or using on this (maybe reduced) equation tools which
are essentially of topological origin (EBL). This procedure
will be assumed to be known by the reader, and we will not
discuss it any further, referring to the vast literature
available (see e.g. \ref{5-25} and references therein).
However -- and this is the main observation we want to
present in this note -- one could also proceed in opposite
order, and operate a symmetry reduction {\it directly} on
the original, and thus possibly infinite dimensional,
problem.
Once this is done, one can study bifurcation problems in the
symmetry reduced setting, either directly by topological
considerations, either by implementing in it the standard
bifurcation analysis procedure (if possible), as discussed
below.
It should be stressed that this direct symmetry reduction
is {\it not } always possible; this is not only in the
obvious sense that not all the problems present a symmetry,
or a symmetry which gives a nontrivial reduction, but also
in the sense that if a symmetry is present, this and the
equation under study have to satisfy some further
conditions, as discussed below. However, when the direct
reduction is possible, it can easily deal with infinite
dimensional problem, and even tackle situations in which the
standard procedure can not even be started.
When we have to deal with $G$-equivariant flows in a function
space or in a space of sections of a fiber bundle, with $G$
a compact Lie group, the idea of direct symmetry reduction is
a rather simple observation, amounting to an extension of the
standard (Golubitsky-Stewart) reduction to suitable
infinite-dimensional settings. However, the situation is
more delicate when we wish to consider a non-compact Lie
group $G$; and even more when we have to study variational
problems defined by a $G$-invariant functional, as it will
be explained below.
It is, of course, well known that this variational setting is
actually the pertinent one in a number of physical
situations. Also, although many groups arising in Physics are
compact -- e.g. the orthogonal, unitary or symplectic ones --
other relevant groups, such as the Euclidean, Poincar\'e or
Lorentz groups, are not compact; indeed, one is not able to
study equivariant bifurcation theory for such groups,
although relevant results can be obtained by assuming
invariance under a subgroup of them (e.g. a lattice
subgroup in the study of convection patterns). Such a
reduction to a discrete function space also eliminates the
continuous spectrum, and thus a major obstacle to
bifurcation analysis.
In the present note we will at first consider equivariant
evolution equations, for which our extension is rather
easy; we will then, for the main part of this note, focus on
the more difficult (and physically relevant) variational
case.
Our direct symmetry reduction approach -- particularly in
the variational case -- will be based on, and closely
follow, the ideas providing the {\it ``Symmetric Criticality
Principle''} (SCP) \ref{26,27} of Palais\footnote{$^1$}{
Actually, we will essentially just formalize the application
of the SCP to bifurcation problems. See also \ref{28,29} for
a previous example of such an extension.}, and thus will
have the same range of validity -- and limitations -- as
this.
We would like to mention that an extension to the setting of
{\it gauge theories} would also be possible, by use of the
Michel's geometric theory\footnote{$^2$}{Michel's theory
aims at obtaining common solutions to all the problems in a
given symmetry class (e.g. points which are critical for any
$G$-invariant potential defined on a $G$-manifold $M$), and
its developement was motivated by the study of
model-independent $SU(3)$ symmetry breaking \ref{28,29};
this has also been extended to study supersymmetry breaking
\ref{32}. See the references in \ref{33} for a partial
bibliography on Michel's theory and related matters.} of
symmetry breaking \ref{34-37}. Such an extension will
not be presented here as it would require the introduction
of a rather heavy formalism, which is maybe not of interest
to most of the readers working in bifurcation theory;
however, this extension would be obtained along the lines,
and using the results, of \ref{33} (see section 6 there)
with very little additional effort. See also \ref{38} for
an application of Michel theory in infinite-dimensional
non-variational problems.
In the following we will assume -- as we have somewhat
already done in this introduction -- the reader is familiar
with bifurcation and equivariant bifurcation theory
\ref{5}, and freely refer to it; readers not being in
this condition, however, should not have problems in
following the main line of our discussion, which is
reasonably self-contained.
The plan of the paper is as follows.
As already mentioned, we will consider theories defined by a
$G$-invariant (with $G$ a Lie group) to be minimized. In
section 1 below we will give an informal (and overly
non-rigorous) discussion of the main idea behind our work.
In the following section 2 we will recall the SCP of Palais;
in section 3 we apply it to bifurcation problems, in which
it can be seen as a ``direct'' RL, in order to obtain a
``direct version'' of the EBL; at this point we pause to
mention how our discussion and result easily applies also to
the non-variational setting. Finally, section 4 is devoted
to the illustration of some examples, concerning both the
variational and the non-variational setting.
\section{1. An informal discussion.}
As mentioned in the introduction, our results will require
to tackle some relatively delicate matters concerning
non-compact group actions and variational analysis.
However, we believe that illustrating the rough idea to be
implemented should help to grasp the point and to follow our
discussion; thus, we will at first give an informal -- and
not rigorous at all -- idea of the approach we want to
follow.
Consider an infinite dimensional $\V$ in a function space,
and an interval $\La \sse R$; on this we have an action of a
Lie group $G$. Let us now consider an equivariant flow on
$\V$, say $$ {\pa u \over \pa t} \ = \ \Phi [u;\la ]
\eqno(1.1) $$ (the square brackets denote that $\Phi$ can
depend on spatial derivatives of $u$ as well); the
equivariance means that
$$ \Phi \, [ gu ; \la ] \ = \ D_g \, \Phi \, [u; \la ]
\eqno(1.2) $$ where $D_g$ is the $G$-action on the tangent
space to $\V$. Now, if $gu = g$ for all $g$ in a subgroup $H
\sse G$, we have that $D_g \Phi [u; \la] = \Phi [u; \la]$
for all $g \in H$; thus, considering the space $\V^H $ of
points in $\V$ which are invariant under $H$, this is
invariant under $\Phi$, and our system can be reduced to
$\V^H$. As the reader will have noticed, this is just the
idea behind the reduction procedure of Golubitsky and
Stewart.
Let us now consider another function space $\F$, on which
$G$-action is defined; we give a smooth functional $L : \F
\x \La \to R$ and consider its functional gradient (also
called variational derivative) $\de L$. The idea is that if
$L$ is {\it invariant} under a Lie group $G$ (acting in
$\F$), then $(\de L)$ will be {\it equivariant} under $G$;
this means that, with $L[g \phi , \la ] = L[\phi , \la ]$
for all $g \in G$, $\la \in \La$ and $\phi \in \F$, then $$
(\de L) [g \phi , \la ] = \( ( \de g ) (\de L) \) [\phi ,
\la ] \ ; \eqno(1.3) $$ thus, if we compute $\de \phi$ at
a $\phi \in \F$ which is invariant under a subgroup $H \sse
G$, we get $$ \( ( \de g ) (\de L) \) [\phi , \la ] \ = \
(\de L) [g \phi , \la ] \ = \ (\de L) [\phi , \la ]
\eqno(1.4) $$ for all $g \in H$. In other words, we should
have that $(\de L) [\phi , \la ]$ is directed along the
space $\F^H$ of functions which admit $H$ as (part of) their
isotropy subgroup; thus the equation $(\de L) [\phi , \la ]
= 0$ can be reduced to $\F^H$, uniformly in $\la \in \La$.
Clearly, this discussion -- which has the advantage of
depicting a direct generalization of the one applying in
finite dimension \ref{3,5,34} -- is too cavalier to be
correct in this form, i.e. in full generality; however, it
is also clear that when it is indeed (under additional
hypotheses and with a real proof) correct, one can
reduce the variational problem, i.e. consider the
restriction $L_H$ of $L$ to the space $\F^H$, claiming that
critical points of $\L_H$ are automatically also critical
points of $L$; this is a direct analogue of the
RL \ref{3,5}.
Moreover, when the reduced space $\F^H$ has some favourable
property, i.e. low dimension or suitable topology, one
could use tools and results -- essentially of topological
nature -- of the kind considered in the usual EBL in order
to guarantee the existence of solutions, and smooth
branches of solutions once the dependence on $\la$ is taken
into account, with given behaviour and located precisely in
$\F^H$; this would be a direct analogue of the EBL
\ref{1-6,39}.
It should also be remarked that our argument above does not
really use the variational structure, but is just based on
the equivariance of $(\de L)$; thus, it is to be expected
that the same kind of results that can be obtained for a
general (i.e. not necessarily stemming from a variational
problem) $G$-equivariant application $f : \F \x \La \to \T
\F$ also hold in the $G$-invariant variational case,
provided $\de L$ is indeed equivariant.
Actually, the variational structure should be apriori
expected to make the reduction more difficult: in the
reduced setting we are, by definition, not able to consider
the variation of $L$ in directions orthogonal to $\F^H \ss
\F$; indeed, the above discussion used precisely the idea
that the $G$-invariance of $L$ guarantees that $(\de L) [\phi
, \la ]$ would be automatically zero in directions
transversal to $\F^H$ \ref{26,27}.
In the following of the paper, we will make the above
discussion rigorous; this will obviously require to impose
suitable conditions on the groups, functions, spaces etc.
to be considered. We will not attempt to give optimal
conditions, but be satisfied with ``reasonable'' ones.
In order to show clearly that such conditions are indeed
necessary, the following example (which is example 3.2 of
\ref{26}) should be quite convincing, and also showing how
one should be careful even in the simpler finite dimensional
setting.
Consider the space $R^2$ with coordinates $(x,y)$, and in
this the action of the group $G=R$ defined by $g_t : (x,y)
\to (x+ty,y)$. Obviously, a potential $V(x,y)$ is
$G$-invariant if and only if it is actually a function of
$y$ alone, $V = V(y)$, e.g. $V(x,y) = y^k$; we will
consider in particular $V(x,y) = y$. The set $\F^G$ of
points having the full group $G$ as isotropy subgroup
corresponds to the $x$ axis, $\F^G = \{ (x,0) \}$. Notice
that the restriction $V_0$ of a $G$-invariant $V$ to the $x$
axis is simply $V_0 (x) = c = V(0,0)$, and thus all points
of $\F^G$ are critical for $V_0$; however, it is clear that
for $V(x,y) = y$ there is no critical point, so that the
Symmetric Criticality Principle -- and therefore the
reduction procedure sketched above -- does not apply.
Actually, the situation is even worse: indeed, the gradient
$(\nabla V) (x,y) = (0,1)$ of the $G$-invariant potential
$V(x,y) = y$ is {\it not equivariant}. As shown later on in
the examples section, the case in which we deal with an
equivariant flow leads to a reduction. Notice that the group
acting in this case is not compact.
\section{2. Equivariant flow on $G$-manifolds and
reduction.}
Let us consider a Lie group $G$ and a smooth
manifold\footnote{$^3$}{Throughout this paper, a manifold is
possibly an infinite dimensional manifold in a function
space; by {\it smooth} we mean ${\cal C}^\infty$.} $M$ on
which a $G$ action is defined.
Recall that a chart at $\mu \in M$ is a diffeomorphism
$\Phi$ of an open set $U \ss M$ (with $\mu \in U$) onto an
open set $\Phi (U)$ in a Banach space $V$, such that $\Phi
(\mu) = \{ 0 \}$.
For any point $\mu \in M$ and any subgroup $H \sse G$ we
can consider
$$ G_x \ := \ \{ g \in G \ : \ g x = x \} \eqno(2.1) $$
$$ M^H \ := \ \{ x \in G \ : \ g x = x \
\forall g \in H \sse G \} \eqno(2.2) $$
The subgroup $G_x$ is called the isotropy subgroup of $x$,
and the set $M^H$ is the fixed points set under $H \sse G$.
Notice that $G_x$ is obviously a subgroup of $G$, but $M^H$
is not guaranteed to be a submanifold of $M$.
Let us consider a point $\mu \in M$, such that $g \mu =
\mu$ for all $g \in H \sse G$, i.e. $H \sse G_\mu$ and $\mu
\in M^H$. Then to $\mu$ is associated an action of $G$ in
$T_\mu M$, denoted by $(Dg_\mu)$.
We say that $G$ is {\it linearizable about $\mu$} if there
is such a $\Phi$ that for each $g \in G$ the map
$$ \Psi \ = \ \Phi \cdot g \cdot \Phi^{-1} \ : \ \Phi (U)
\, \to \, V \eqno(2.3) $$
is the restriction to $\Phi (U)$ of a linear map $A_g : V
\to V$. These $A_g$ provide a linear representation of $G$
in $V$, and $(D \Phi_\mu ) : T_\mu M \to V$ sets up an
equivalence of this with $(D g_\mu)$, which is therefore
called the linearization of $G$ at $\mu$.
As the $A_g$ are linear maps, the set
$$ W \ = \ \{ v \in V \ : \ A_g v = v \ \forall g \in H \}
\ \equiv \ V^H \eqno(2.4) $$
is a closed linear subspace of $V$; as $\Phi : U \cap M^G
\to \Phi (U ) \cap W$, $M^H$ is locally a smooth
submanifold of $M$ at $\mu$. Therefore:
{\bf Lemma 1.} {\it If $M$ is a smooth $G$-manifold, and
$H$ a subgroup of $G$ such that the $G$-action of $H$ is
linearizable about each point $ x \in M^H$, then
the set $M^H \sse M$ of symmetric points is a smooth
submanifold of $M$.}
We stress that the converse of this lemma is not
necessarily true, as shown by example 3.2 of \ref{26}.
Notice that if $M$ is Riemannian and $G$ acts
isometrically, tyhe action is linearized about $\mu \in
M^G$ by just passing to geodesic coordinates.
Thus, we are naturally led to ask when it is that the $G$
action can be linearized about any point of $M^G$ for $M$ a
$G$-manifold. It is known that when $M$ is a Banach manifold
and $G$ is compact with ${\cal C}^1$ action on $M$, then the
$G$ action can be linearized about any point $\mu \in M^G$
\ref{26,40}; however, the situation is much less clear for
$G$ non compact, even for $G$ a connected and semisimple Lie
group. It was conjectured by Palais and Smale in 1965 (see
\ref{26//PS}) that in this latter case the action is indeed
linearizable about any $\mu \in M^H$, but such conjecture is,
as far as I know, not proven; on the other side, it is known that the
conjecture is true when $G$ acts by a finite dimensional
linear representation \ref{41}, and also when $M$ is a real
analytic manifold and the $G$ action is real analytic
\ref{42,43}.
This brief discussion shows how difficult it is to obtain
general results for non-compact groups. On the other side,
when we deal with a specific $G$-action we only have to check
explicitely if the $G$-action can be linearized about
points in $M^H$, which is a much easier task. Needless to
say, when we have to deal with linear group actions -- as
it is often the case in physically relevant situations --
we do not have to worry about this.
The condition of linearizability of the $G$-action is
obviously needed to define an equivariance of $\Phi : M \to
\T M$; however, it is {\it not} sufficient to infer that $\Phi :
M^H \to \T M^H$.
In order to discuss this, we have to
introduce the concept of admissible Banach space.
Consider a Banach linear $G$-space $\V$, i.e. a Banach
space on which $G$ acts linearly. Let $\V^*$ be the Banach
space dual to $\V$; this is also a linear $G$-space, with
$G$ action given by $(g^* \phi ) (v) = \phi (g^{-1} v)$,
where $\phi \in \V^*$ and $v \in \V$.
Denote by $\W \sse \V$ the linear subspace on which $G$ acts
as the identity, i.e. $$ \W \ = \ \{ v \in \V \ : \ g v = v
\ \forall g \in G \} \ . \eqno(2.5) $$ Correspondingly,
denote by $\W^* \sse \V^*$ the linear subspace of $\V^*$ on
which $G$ acts as the identity,
$$ \W^* \ = \ \{ \phi \in \V^* \ : \ g \phi = \phi
\ \forall g \in G \} \ ; \eqno(2.6) $$ this is obviously the
space of linear $G$-invariant functionals on $\V$.
To any linear subspace $V \ss \V$ is associated its
annihilator ${\rm Ann}(V)$, also denoted as $V_0$, i.e. the
linear subspace of $\V^*$ given by functionals which vanish
identically on $V$,
$$ {\rm Ann}(V) \ \equiv V_0 \ := \ \{ \phi \in \V^* \ : \
\phi (v) = 0 \ \forall v \in V \sse \V \} \ = \ \{ \phi \ :
\ V \sse \Ker (\phi ) \} \ . \eqno(2.7) $$
Thus we can in particular consider $\W_0$, the annihilator
of $\W$; the condition $\W^* \cap W_0 = \{ 0 \}$ means that
a linear $G$-invariant functional which vanishes
identically on $\W$ does actually vanish identically on
$\V$.
When this is the case, i.e. when
$$ \W^* \cap W_0 = \{ 0 \} \ \ {\rm for} \ \ \W = \{ v \in
V \ : \ gv = v \ \forall g \in G \} \, \eqno(2.8) $$
we say that the Banach linear $G$-space $\V$ is {\it
admissible}.
Notice that if $G$ is compact, then any Banach linear
$G$-space $\V$ is admissible. For general $G$, there is a
simple criterion to check if a Banach linear $G$-space is
admissible:
{\bf Lemma (Palais)} {\it A Banach linear $G$-space $\V$ is
admissible provided that for each nonzero $G$-invariant
linear functional $\phi :\in \V^*$ the invariant hyperplane
$\phi^0$ has an invariant complementary space.}
{\bf Corollary.} {\it If $\V$ is completely reducible, so that
every closed invariant subspace has a complementary closed
invariant subspace, then $\V$ is admissible.}
This discussion is immediately applied to Banach $G$
manifolds via the:
{\bf Definition.} {\it A smooth $G$-manifold $M$ is
admissible if the $G$ action is linearizable about each
symmetric point $ \mu \in M^G$ and the tangent space $V_\mu
= \T_\mu M$ equipped the linearization $Dg_\mu$ is an
admissible Banach linear $G$-space.}
This will be the necessary (and sufficient) condition to
be able to consider a reduction of equivariant flows on
$M$ to $M^G$ (and {\it a fortiori} to $M^H$ for all $H \sse G$);
see the discussion below in section 4.
\section{3. The symmetric criticality principle}
In this section we will briefly recall the ``symmetric
criticality principle'' (SCP) of Palais; our discussion
will closely follow \ref{26}. We refer to \ref{27} for
further detail; see \ref{40,44} for variational analysis in
the presence of symmetries.
In short, the SCP says that if we have a functional $L$
defined on a $G$-space $\F$, and we consider the
restriction $L_0$ of this to the subspace $\F^G \sse \F$ of
$G$-invariant functions, critical points of $L_0$ are also
crirical points of $L$; one says also that ``critical points
on symmetric space are symmetric critical points'' \ref{26}.
The SCP is {\bf not} valid in general, but one knows
conditions which guarantee it applies (as well as a number
of counterexamples when these are not satisfied)
\ref{26,27}; thus, what we really have to discuss are
conditions which can guarantee the validity of the SCP, and
be satisfied in physically meaningful situations.
Let us first of all consider the case, standard in
Physics (although not always explicitely
formulated in this way) in which $\F$ is some space of
sections of a fiber bundle \ref{45-51}. Notice that this
bundle could be trivial, e.g. $\R^n \times U$ if we consider
a functional over functions $f : \R^n \to U$: in this case we
can also see the functional as defined over the space $\F$
of sections of this bundle; imposing some condition on $f$,
i.e; choosing a function space, is equivalent to selecting
a subspace of sections of the bundle: e.g., considering
$L^2$ functions corresponds to considering $L^2$ sections,
or a finite energy condition will lead to considering a
Sobolev space of sections.
We will thus consider a fiber bundle $(E,\pi , B)$ over a
smooth manifold $B$ with total space $E$ and fiber $\pi^{-1}
(x) \simeq F$; we will then consider a space $\S$ of
sections of $E$, and we will need this to be complete and
equipped with a scalar product; thus, it is natural to
require that $\S$ is a Sobolev space of sections (in
physical applications, these appear naturally by finite
energy conditions).
We will then consider a smooth functional $L : \F \x \La
\to R$, where $\la \in \La \sse R^n$ will be parameters on
which $L$ (and $\L$ to be introduced below) can depend; we
will be mainly interested in the case $\La \sse R^1$.
We will, for definiteness, suppose $L$ to be written
in terms of a density $\L : J^p \F \x B \x \La \to R$, with
$J^p F$ representing a jet over $\F$ of suitable (finite)
order $p$, as
$$ L [\phi , \la ] \ = \ \int_B \L [ \phi (x)
, \la ; x ] \, \d \mu (x) \ \ ; \eqno(3.1) $$
here $\d \mu (x)$ is a measure over the base space $B$, and
the square brackets indicate that $L$ and $\L$ depend on
derivatives of $\phi$ up to some finite order. In (3.1) we
have included a possible explicit dependence of $\L$ on the
point $x \in B$.
As the parameters will not play any role in this section, we
will drop them from the notation for ease of writing; the
results obtained here will be uniform in $\la \in \La$. In
next section we will reintroduce $\la$ in our discussion.
We will consider the case where there is a group $\Ga$ of
operators $\ga : \S \to \S$ acting
regularly in $\S$ and such that $$ L [\ga (\s ), \la ] \ = \
L [ \s , \la ] ~~~~ \all \ga \in \Ga ,~ \all \la \in \La \ ;
\eqno(3.2) $$ in this case we say that $\Ga$ is a symmetry
group of $L$.
Recall that the regularity of the group action means that
the topologies of $\Ga$ and of $\S$ are compatible, i.e. that
there are no sections which are near in the $\S$ topology
but at a ``large'' distance along a $\Ga$ orbit. A simple
example of non-regular action in finite dimension (here the
role of sections is, of course, played by points), with
$G=R$, is provided by the irrational flow on the torus
$T^2$: now any neighbourhood, no matter how small, of a
point $x_0$ contains points which are on the $G$-orbit of
$x_0$, at large distance from $x_0$ along the orbit (but
not, of course, in the metric on $T^2$).
\Remark{1} That the regularity of the action plays an
essential role can be easily seen by considering the finite
dimensional example just mentioned, i.e. the action of $G=R$
on $T^2$ given by an irrational flow; this situation can be
compared with the one obtained by considering a rational
flow (in which case only $G/Z \simeq SO(2)$ acts
effectively, so that we are actually dealing with a compact
group action). \eor
\Remark{2} If for all elements of $H \sse G$ the action of
$H$ is linearizable about each point of $\S^H$, then $\S^H$
is a smooth submanifold of $\S$ \ref{26}. Notice also
that, as is seen by the example mentioned in the
previous remark 1, this requires however the assumption
of a regular action. \eor
If $E$ is a $G$ fiber bundle (i.e. a fiber bundle having
$G$ as structure group), it is natural to consider groups
of the form $\Ga = \Ga_b \otimes_\to \Ga_f$, where $\Ga_b$
is a group of transformations of the base space $B$,
$\Ga_f \sse G$ acts on the fiber, and $\otimes_\to$
denotes semidirect product; it is implicit in this that
$B$ is a $\Ga_b$-manifold (i.e. a manifold on which it is
defined an action of $\Ga_b$). More generally, assuming that
$B$ is a $\Ga$-manifold, we can assume that $\ga \in \Ga$
acts in $\S$ as $$ (\ga \s) (x) \ = \ \ga [ \s ( \ga^{-1}
(x) ) ] \ . \eqno(3.3) $$
\Remark{3} As stressed in \ref{26}, the assumption that $B$
is a $\Ga$-manifold is to be understood to include the
assumption that the map $(\ga , x) \to \ga (x)$ is a smooth
map from $\Ga \x B$ to $B$, and not just a smooth map $B \to
B$ for each $\ga \in \Ga$. \eor
We can then consider subgroups $H \sse \Ga$, and to any
such subgroup we can associate the subset $\S^H$ of
sections which are invariant under $H$, i.e.
$$ \S^H \ = \ \{ \s \in \S \, : \ \ga (\s ) = \s \ \all
\ga \in H \} \ . \eqno(3.4) $$
When $\Ga$ is a {\bf compact} Lie group, the $\S^H$ are
smooth submanifolds of $\S$ (union, possibly disjoint, of
linear subspaces if $\Ga$ acts linearly), and we can
consider the restriction $L_H$ of $L$ to $\S^H \x \La$.
\Remark{4} Notice that assuming $\Ga$ compact does
automatically guarantee the regularity of its action on
$\S$. \eor
Under the same condition of compactness of $\Ga$, we have
that: {\it The critical points of $L_H$ are also critical
points of $L$.} This statement constitutes indeed the
``Symmetric Criticality Principle'' of Palais \ref{26,27},
so that we are stating that it applies under the present
conditions, and in particular for $\Ga$ compact.
We summarize the discussion so far in the following
{\bf Theorem (Palais).} \ref{26} {\it Let $G$ be a compact
Lie group, $B$ a smooth $G$-manifold, $(\E,\pi , B)$ a
smooth $G$-fiber bundle over $B$ with fiber $\pi^{-1} (x)
\simeq F$, and $M$ a Banach manifold of sections of $\E$.
Let $G$ act on $M$ by $(g \mu ) (x) = g [ \mu (g^{-1} x)]$,
and let $L : M \to R$ be a smooth $G$-invariant function on
$M$. Then the set $\S \sse M$ of $G$-equivariant sections is
a smooth submanifold of $M$, and if $\a$ is a critical point
for $L |_\S$ it is in fact a critical point for
$L$.}
It should be stressed again that, as mentioned above, the
SCP is not valid in full generality, i.e. if we remove the
assumptions considered above. The example of Palais
mentioned at the end of section 1 is perhaps the best
illustration of this.
Unfortunately, these assumptions can be too restrictive in
a number of physically relevant cases; this applies in
particular to the condition of having a compact group
$\Ga$. We have thus to consider if it is possible to have
less restrictive conditions which still guarantee the
validity of the SCP.
Indeed the assumptions we are considering here can be
somewhat weakened; let us shortly mention how this
generalization goes, again following closely \ref{26}. For
the sake of generality we will also consider Banach spaces
(as already done in section 2 above) instead than
restricting only to Hilbert ones. The discussion of section
2 is actually nearly completely transported in this frame;
this is due to the fact that already there we had to focus
on $G$-invariant functionals on a Banach space.
We consider now $\F$ a Banach $G$-manifold, and let $\F^G$
be the set of points $f \in \F$ such that $g f = f $ for
all $g \in G$. As already mentioned in sect.2, if the
$G$-action is linearizable about each point $f \in \F^G$,
then $\F^G$ is a smooth submanifold of $\F$. When this
condition is verified, we can enquire if $\F$ is an
admissible Banach $G$-manifold, i.e. if $\T_f \F$ is an
admissible Banach linear $G$-space for all $f \in \F^G$,
see again the definitions and the discussion is section 2.
This is indeed a relevant feature for the SCP: in fact, we
have the
{\bf Lemma (Palais)} \ref{26} {\it The symmetric
criticality principle is valid for $G$-invariant
functionals $L : \F \to \R$ defined on an admissible Banach
$G$-manifold $\F$.}
By recalling the discussion in section 2 on admissible
Banach $G$-manifolds, we also have immediately:
{\bf Corollary} {\it If $G$ is a compact Lie group, then any
smooth $G$-manifold $\F$ is admissible and hence the
symmetric criticality principle applies for any
$G$-invariant functional $L : \F \to \R$.}
{\bf Corollary} {\it If the Banach $G$-manifold $\F$ is
real analytic and $G$ is a connected semisimple Lie group
with a real analytic action on $\F$, then the smooth
$G$-manifold $\F$ is admissible and hence the
symmetric criticality principle applies for any
$G$-invariant functional $L : \F \to \R$.}
\Remark{5} Notice that in the above discussion, $G$ is
required to act linearly when we deal with a Banach space,
but it can act nonlinearly -- provided the action is
separately linearizable around each point -- when we deal
with a smooth $G$-manifold. Needless to say, when the $G$
action on a manifold is linear, the whole discussion is
much easier. \eor
\Remark{6} Notice that this discussion could be
extended to the case where instead than having $L [ \ga
\phi , \la ] = L [\phi , \la ]$, we have $L [ \ga \phi , \la
] = A [\phi , \la ] \cdot L[\phi , \la ]$, with $A$ a scalar
function: in this case the gradient $(\de L)$ at $\phi \in
\S^H$ is also tangent to $\S^H$ and zero in transverse
directions. This is similar to what happens when discussing
the relation between symmetries of the lagrangian and
conserved quantities (Noether theorem): when the lagrangian
is not invariant under a group $G$, but gets multiplied by a
scalar factor (as is the case e.g. for scaling applied to
harmonic oscillators systems), we have invariance of the
Euler-Lagrange equations, and useful results follow as well
(see e.g. \ref{52} for detail). \eor
\bigskip
\Remark{7}
It is maybe also of some interest to briefly mention the
relation of the topics discussed here (and in the
following) with the geometrical theory of symmetry breaking
developed by physicists in the context of fundamental
interactions.
The SCP can be seen, especially from a physicist's
point of view, as an extension of results obtained (and
methods employed) by Michel \ref{34} in the study of the
geometry of spontaneous symmetry breaking for fundamental
interactions; see also the works of Michel and Radicati
\ref{30,31,35,36} for physical applications of Michel
analysis, and \ref{32,33,37} for later extensions of his
approach.
As discussed in \ref{53}, it is possible {\it a
posteriori} to see the ``usual'' RL and EBL as deriving
from the approach and results of Michel; however, most of
bifurcation theorists were unaware of physicists' work (a
notable exception being provided by Sattinger \ref{19}),
and the physicists were unaware of the possible relevance
of this work in the context of Nonlinear Dynamics and
bifurcation theory (an exception being provided by Cicogna
\ref{1,39}). Notice that the present author is in the
unconfortable situation of having being aware of Michel's
results and working in bifurcation theory, for a long time
without realizing the connection among the two; thus this
remark should not be seen as critical toward anybody else.
Again from a physicist's point of view, it is
remarkable that at the same time of Michel and Radicati
work, Cabibbo and Maiani \ref{54} developed a related
approach; this approach constitutes in many ways an
anticipation of the SCP.
As it should be clear by the previous discussion, the
$G$-invariant variational case presents greater difficulties
than the study of $G$-equivariant flows; it is thus maybe
surprising to the more mathematically oriented reader that
these early works attacked directly (and successfully !)
this more difficult case. However, it should be recalled
that on physical grounds, fields configurations which are
related by a symmetry transformation should be seen as
exactly the same: in this way, it is natural -- and
necessary -- to discuss the theory in $\F / G$ rather than in
$\F$. It is remarkable that obeying to the physical needs
led (so to say, nearly intuitively) Michel, Radicati,
Cabibbo and Maiani to a deep understanding of a
mathematical theory which would not have been needed if
they had chosen to work with gradients of the invariant
functionals. \eor
\section{4. The SCP in the bifurcation framework.}
In the discussion of the previous section, the possible
dependence on a parameter $\la \in \La$ had no role.
Indeed, such $\la$ was confined to the role of a
parameter, and a little thinking shows moreover that if
the dependence of $\L$ on $\la$ is smooth, and assuming
that the symmetry holds for any value of $\la$, is
completely independent of $\la$, and does not act on $\La$,
then also all the objects considered above -- and in
particular the fixed spaces $\S^H$ as embedded submanifolds
of $\S$, or the infinite dimensional generalization $\V_*^H
\ss \V_*$ of these -- also depend smoothly on $\la$ {\it for
regular values of $\la$}, i.e. at the exception of
bifurcation points.
In this case, away from bifurcation points, everything
depends smoothly on $\la$, and in particular the solutions
to $(\delta \L) (\phi ) = 0$, among which those
determined via the SCP, will depend smoothly on $\la$.
Thus the fundamental fact discussed above,
i.e. that we can consider the restriction $L_H$ and be
guaranteed that critical points of $L_H$ are also critical
points of $L$, holds true also in the case $L$ depends on a
parameter.
When we consider such restriction $L_H : \S^H \x \La \to
R$, the critical points of this will depend on $\la$, i.e.
we will have families of critical points $\phi_\la$,
depending smoothly on $\la$.
If we consider suitable but standard hypotheses, we can in
this way recover the setting of bifurcation theory, and in
particular guarantee that some given critical point
bifurcates into a branch of critical points with given
residual symmetry; although this will just follow the
standard equivariant bifurcation theory \ref{17-25}, we will
briefly recall what these ``suitable assumptions'' could be
(referring to \ref{17-25} for more detailed discussion and
possible weaker sets of conditions).
First of all, as implicit in the fact we are using the SCP,
we have to require that the space $\S$ on which the
functional $L$ of interest is defined is such that the SCP
holds; this means we have to require the:
{\tt Condition A.} {\it $\S$ is an admissible Banach
manifold.}
(Obviously here the ``A'' stands for admissibility).
This is the condition peculiar to the case of infinite
dimensional spaces and possibly non-compact groups; the
other conditions will be standard for bifurcation theory.
We denote by $\B_{c_0}$, or briefly $\B_0$, the ball of
radius $c_0 > 0$ in $\S$. The $c_0$ entering in this has
been assumed, for the sake of simplicity, to be independent
of $\la$, but some suitable dependence could also be allowed.
We will then also require that:
{\tt Condition B.} {\it There is some $\s_0 \in \B_0
\ss \S$ which is a critical point of $L[\s , \la ]$ for all
$\la \in \La$, and that there is a $\la_0$ in the interior
of $\La$ such that this $\s_0$ is a linearly stable critical point
for $\la < \la_0$, and becomes linearly unstable for $\la >
\la_0$, with $d {\rm Re}(\s_0 ) /d \la > 0$ at $\la = \la_0$.}
Here the ``B'' stands for bifurcation; indeed, when $\S$ is
a complete space and condition C is satisfied, condition B
guarantees the existence of a branch of solutions
bifurcating from $\s_0$ at $\la_0$.
Denoting by $\< .,. \>$ the scalar product in $\S$, in the
following we will assume as a blanket hypothesis that:
{\tt Condition C.} {\it $\L$ and therefore $L$ are convex
for $\< \s , \s \> > c_0$, uniformly in $\la \in \La$.}
The condition just introduced will be used to
guarantee existence of solutions inside a ball of radius
$c_0$, and rule out ``explosive'' behaviour at bifurcation,
but will play no role in the other parts of our discussion.
Obviously, the ``C'' stands for convexity.
Now, thanks to condition A, we can consider the
directions in $\T_{\s_0} \S$ along which the instability
arises; we denote this linear subspace as $\T^+_{\s_0} \S$
(this is tangent to the unstable manifold through $\s_0$ for
the -- in general, infinite dimensional -- dynamical system
defined by ${\dot \s} = (\de L) (\s )$ \ref{8,12}).
We denote by $\Ga_0 \sse \Ga$ the symmetry group of $\s_0$,
i.e. $\Ga_0 = \{ \ga \in \Ga \, : \, \ga (\s_0 ) = \s_0
\}$. Let us now consider the subgroups of $\Ga_0$; to each
of these, say $\Ga_H$, we associate the corresponding fixed
space $\S^H$ defined as in (4). Obviously, for each $\Ga_H
\ss \Ga_0$, $\s_0 \in \S^H$.
It is a simple but relevant observation that if a direction
$\ga \in \T_{\s_0} \S^H$ is unstable, i.e. $\ga \in
\T^+_{\s_0} \S$, then all the directions in $\T_{\s_0}
\S^H$ are unstable: in other words, if there is a $\ga \in
\[ \T_{\s_0} \S^H \, \cap \T^+_{\s_0} \S \]$, then $\T_{\s_0}
\S^H \sse \T^+_{\s_0} \S$. Indeed, by the equivariance of
$(\de L) (\s )$ -- defined in $\T_{\s_0} \S$ -- the action
of $\Ga$ takes unstable directions into unstable directions
(and similarly for stable ones) \ref{55}.
Thus, it makes sense to speak about subgroups $\Ga_H$ whose
fixed space $\S^H$ is tangent in $\s_0$ to the unstable
tangent space $T^+_{\s_0} \S$; for the sake of brevity
these will be called unstable subgroups at $\s_0$. An
unstable subgroup at $\s_0$ will be called maximal if it is
not contained in any other unstable subgroup at $\s_0$.
We have thus proven the following result.
{\bf Proposition 1.} {\it Let us consider the variational problem
corresponding to the $\Ga$-invariant functional $L [\s , \la ]$,
and let the above condition A, B and C be satisfied; then for each
$\Ga_H$ which is a maximal unstable subgroup at $\s_0$, the set
$\[ \( \S^H \backslash \s_0 \) \cap \B_0 \]$ is an invariant set
under the dynamics of ${\dot \s} = - (\de L) (\s )$.}
\Remark{9} This means that we can restrict our variational problem
to each of these sets, and the critical points of such restricted
problems will automatically be also critical points of the
original problem. This statement represents a {\bf Reduction Lemma}
\ref{3,5} for the bifurcation problem in the considered
(infinite dimensional) setting. \eor
Moreover, by construction -- and due to the completeness of
$\S$ -- we are guaranteed of the existence of a stable critical
point $\s_\la$ (different from $\s_0$ for $\la > \la_0$) in
$\S^H$, and the smoothness of $L$ in $\la$ guarantees, as remarked
above, that $\s_\la$ depends smoothly on $\la$. Thus we obtain
also the following proposition 2, and from this and
proposition 1 the corollary below.
{\bf Proposition 2.} {\it In the same case and under the same
hypotheses as in Proposition 1 (in particular that conditions
A,B and C are satisfied), let $H$ be any maximal unstable
subgroup of $\Ga_0$ at $\s_0$. Then, the restricted variational
problem given by $L_H [\s , \la]$ has a local branch of stable
critical points $\s_\la$ for $\la$ in a right neighbourhood of
$\la_0$, which represent a bifurcating branch of critical points all
invariants under $\Ga_H$.}
{\bf Corollary} {\it Under the same hypotheses as in
Proposition 1, and for any $H$ a maximal unstable
subgroup of $\Ga_0$ at $\s_0$, the full variational
problem given by $L [\s , \la]$ has a local branch of stable
critical points $\s^H_\la \in \S^H$ for $\la$ in a right
neighbourhood of $\la_0$.}
\Remark{10} This statement represents an {\bf Equivariant
Branching Lemma} \ref{1-6} for the variational problem
in the considered (infinite dimensional) setting. \eor
\Remark{11} It should be stressed that in the above discussion, not
only the ambient space $\S$ was admitted to be infinite
dimensional, but also the manifolds $\S^H$ to which the
reduction operates could be infinite dimensional. \eor
\bigskip
Let us mention that these results could be rather easily
generalized in several ways; some of the possible extensions
are sketched in the following remarks.
\Remark{12} When the topology of $\[ \( \S^H \backslash \s_0 \)
\cap \B_0 \]$ is such to guarantee -- e.g. by Morse theory
\ref{56,57;35,40,49} -- the existence of several stable
critical points or of critical points of different
stability, one can similarly conclude the existence of local
branches of critical points with corresponding invariance
properties. \eor
\Remark{13} Similar considerations could be applied to subsequent
bifurcations, i.e. to study the case in which the
bifurcating stable branches of solutions become themselves
unstable for $\la > \la_1$; in this case the relevant sets
obtained by intersection of the unstable manifolds with the
convexity ball $\B_0$ and excluding the set which is already
known to be unstable (the point $\s_0$ in the case
considered above, the branch $\s_\la$ in more complex cases)
have a richer and richer topology: they are thus on the one
side more difficult to study, and on the other side more
prone to give information by application of topological
tools. \eor
\bigskip
Finally, we notice that in the present section we dealt
directly with the variational case. However, it is clear
that we could proceed in the same way if, instead than
considering a variational problem defined by a functional $L
: \S \to R$ invariant under $\Ga : \S \to \S$, we were
considering a problem defined by a dynamical system $$
{\dot \s} = \Phi (\s ) \ , \eqno(4.1) $$ where $\Phi : \S
\to \T \S$ is a $\Ga$-equivariant (not necessarily gradient)
smooth vector field on the infinite dimensional manifold
$\S$, provided $\S$ is complete (otherways we cannot
conclude the existence of fixed points), and provided we can
define for any point $\s \in \S$ a scalar product in $\T_\s
\S$, e.g. if $\S$ is locally modelled on a Hilbert space
(or, in other words, is a Hilbert manifold) \ref{40}.
Both these conditions are automatically satisfied when $\S$
is a Sobolev space and, in particular, a Sobolev space of
sections of a fiber bundle; we can restrict to this setting for
the sake of concreteness (and for its relevance in
applications).
If we are guaranteed that for such $\Phi$ -- with the same
notation as above for the space of $H$-invariant sections,
with $H \sse \Ga$ -- we have
$$ \Phi : \S^H \to \T \S^H \eqno(4.2) $$
(i.e. in particular if $\S^H$ is a smooth submanifold of
$\S$), then we can at once repeat the discussion and
conclusion of the variational case. The condition (2) is
guaranteed by the condition of $\Ga$ acting regularly in
$\S$, and in particular, when $\S$ is a Sobolev space of
sections of a $G$-fiber bundle $P$ and $\Ga$ the
corresponding Sobolev space (of matching order) of
${\rm ad} P$, is verified automatically provided $G$ is a
compact group.
\section{5. Examples.}
{\tt Example 1.} We start by considering again the finite
dimensional situation dealt with at the end of section 1 in
the variational case; we do now consider a nonvariational
case, in order to point out how the symmetry reduction has
a wider range of vailidity when we deal with equivariant
flows (see also the next example 2 for the infinite
dimensional case).
Thus, we consider the action of the group
$G = R$ on $M = R^2$, in which we take coordinates $(x,y)$,
given by $$ g_a \ : \ (x,y) \ \to \ (x+ay , y) \ ;
\eqno(5.1) $$ thus, the action of $G$ is given by the matrices
$$ T_a \ = \ \pmatrix{1 & a \cr 0 & 1 \cr} \ . \eqno(5.2) $$
It is readily seen that there is a submanifold $M_0$
invariant under the full group $G$, and this is precisely
the $x$-axis: indeed, $g_a (x,0) = (x,0)$.
Obviously $G$ is a maximal isotropy subgroup, and $M_0$ is
a maximal invariant subspace; thus $M_0$ must be invariant
under any equivariant flow in $M$. We write such a flow in
coordinates as
$$ \eqalign{ {\dot x} \ = & \ X (x,y) \cr {\dot y} \ = & \ Y
(x,y) \ , \cr} \eqno(5.3) $$
and the equivariance condition
$$ \eqalign{
& X (x,y) + a Y (x,y) \ = \ X (x + ay , y) \cr
& Y (x,y) \ = \ Y (x + ay , y) \cr} \eqno(5.4) $$
requires that
$$ a Y \, = \, ay X_x \ \ ; \ \ 0 = a y Y_x \ ; \eqno(5.5) $$
this implies that $X_{xx} = 0$ and hence
$$ \eqalign{
X (x,y) \ = & \ A(y) \, + \, B(y) \, x \cr
Y (x,y) \ = & \ B(y) \, y \ . \cr} \eqno(5.6) $$
On $M_0$, i.e. for $y=0$, we have ${\dot y} = 0$, so that
indeed the flow leaves the $x$ axis invariant; the equation
describing the restriction of the flow to this invariant
manifold is simply
$$ {\dot x} \ = \ \a + \b x \eqno(5.7) $$
with the real constants $\a , \b$ given by $\a = A(0)$, $\b
= B(0)$.
Thus, the symmetry reduction to $\F^G = M_0$ is valid in
this nonvariational case, although -- as we have seen in
section 1 -- it would not generally apply to a variational
problem in the same setting.
Notice that here we have not considered any dependence on a
control parameter $\la$, for ease of notation, but it would
not be difficult to include this.
\bigskip
{\tt Example 2.}
We will now consider an infinite dimensional version of the
case considered in example 1 above.
Let us consider functions $u (x,y)$ defined on $R^2$, and
let now $M$ be a suitable\footnote{$^4$}{ Suitable means
that we have the required structure of an Hilbert manifold,
and that the $G$ action to be considered is smooth and
regular; thus the considerations to follow would also
apply if we were considering, e.g., Sobolev spaces on $M$.}
space of such functions, say $L^2 (R^2, R)$ for
definiteness. We consider $G=R$, acting in $M$ by the
unitary representation $$ g_a \ : \ u(x,y) \ \to \ u (x+ay ,
y) \ . \eqno(5.8) $$
There is again a subspace $M_0$ invariant under the full
group $G$: this is given simply by the functions which are
independent of $x$,
$$ M_0 \ = \ \{ u \in M \ : \ u_x = 0 ,\ u (x,y) = u(y) \}
\ . \eqno(5.9) $$
Again $G_0 = G$ is a maximal isotropy subgroup and $M_0$ a
maximal invariant subspace, and $M_0$ must be invariant
under any $G$-equivariant flow in $M$, as we are going to
check explicitely.
In doing this, we can deal with the Lie generator $\ga$ for
the group $G$, given simply by $\ga = y \, \pa / \pa_x$.
We want now to consider PDEs of the form $u_t = F[u; \la ]$
(square brackets denote, as earlier in this paper,
dependence on $u$ and its derivatives up to a finite order
$p$), which will describe a flow in $M$.
In order to determine explicitely the PDEs invariant under
$G$ (that is, under $\ga$), we have first to fix the order
$p$ of the equation and then apply standard computations
\ref{52,58,59}. Let us, for the sake of simplicity,
consider $p=1$, so that we write the equation $\E$ as $$ u_t
\ = \ F (u, u_x , u_y ; \la ) \ . \eqno(5.10) $$
We have then to consider the first prolongation
\ref{52,58,59} of $\ga$, given by
$$ \ga^{(1)} \ = \ y \, {\pa \over \pa x} \ - \ u_x \, {\pa
\over \pa u_y} \ \ ; \eqno(5.11) $$
the symmetry condition is $\ga^{(1)} [\E ] = 0$, i.e.
$$ \ga^{(1)} \ \[ u_t - F(u, u_x , u_y ; \la ) \] \ = \ 0 \ ,
\eqno(5.12) $$
which in view of (11) is simply $u_x \pa F / \pa u_y = 0$,
so that the required invariant PDEs are those of the form
$$ u_t \ = \ F(u , u_x ; \la ) \ . \eqno(5.13) $$
It is clear that $M_0$ is invariant under these.
Moreover, when we restrict to $M_0$, i.e. to $u(x,y;t) = w
(y;t)$, we have that (13) reduces to an ordinary differential
equation $$ {d w \over dt} \ = \ \F (w, \la ) \ , \eqno(5.14)
$$ with $\F (x,\la ) = F (x,0; \la )$.
Notice that the group $G$ considered here is not
compact, but is locally compact.
If we consider variational problems for a functional $L : M
\to R$, say $L = \int \L [u] \, dx \, dy $ for
definiteness, we could not apply the symmetry reduction.
Indeed, it suffices to consider e.g. the case
$$ L \ = \ \int \ {1 \over 2} \, \[ u_y^2 - u^2 \] \ + \
u_x \ dx \, dy \eqno(5.15) $$
which on $M_0$ gives the restricted functional
$$ L_0 \ = \ \int \ {1 \over 2} \, \[ u_y^2 - u^2 \] \ dy
\eqno(5.16) $$ leading to the Euler-Lagrange equations $u_{yy} = -
u$; the $u$ solution of this are not critical points for the full
$L$. We stress again $G$ is not compact.
\bigskip
{\tt Example 3.} We will now consider the group $G =
SO(2)$. First of all we notice that the analogue of the
finite dimensional case considered in example 1, with $M =
R^2$, and $M_0 = \{ 0 \}$, would be trivial: the origin is
invariant under any $SO(2)$-equivariant dynamics, as well
known, so that the symmetry reduction for equivariant flow
is a triviality; as for the variational case, invariant
functions are of the form $V = V(r)$, and the gradient of
these is of course equivariant. Let us thus consider
directly the infinite dimensional setting.
We consider again $M = L^2 (R^2,R)$, and coordinates
$(x,y)$ or $(r , \phi)$ in $R^2$, with the $G$-action given
now by $$ g_\th \ : \ u(x,y) \ \to \ u \( \cos (\th ) x -
\sin (\th ) y , \sin (\th ) x + \cos (\th ) y \) \eqno(5.17)
$$ in cartesian coordinates, or
$$ g_\th \ : \ u ( r , \phi ) \ \to \ u (r , \phi + \th )
\eqno(5.18) $$ in polar coordinates. The $G$-invariant
functions are obviously those independent of the angular
coordinate, $$ M_0 \ = \ \{ u (r, \phi) = w (r) \} \eqno(5.19)
$$
Repeating the check of the reduction procedure for first
order equations as in example 2 would be immediate, leading
again to an ODE for $ w(r)$ (the same holds for higher
order equations, with a more complicate expression for the
prolongation $\ga^{(p)}$ of the Lie generator of $G$).
Similarly, if we consider e.g. the $SO(2)$ symmetric
equation
$$ \Delta u \ - \ f(u,\la ) \ = \ 0 \eqno(5.20) $$
this is immediately restricted to $M_0$, yielding
$$ {1 \over r} \, {d \over d r} \, \( r \, {d
\over d r} \) \, w \ = \ f (w , \la ) \ . \eqno(5.21) $$
Notice that in this case $M_0$ is infinite dimensional, and
we are not guaranteed of the existence of a solution to
(21).
Let us now consider an example of variational problem in the same
setting; we choose the simple functional on $M$ given by
$$ L \ = \ \int \int \[ {1 \over 2} \( u_x^2 + u_y^2 \) \, - \, V
(u;\la ) \] \, \d x \d y \ ; \eqno(5.22) $$
in polar coordinates this reads
$$ L \ = \ \int \int \[ {1 \over 2} \( u_r^2 + {1 \over
r^2} u_\theta^2 \) \, - \, V (u;\la ) \] \, r \d r \d \theta \ ,
\eqno(5.23) $$
and thus the restriction of $L$ to $M_0$ is simply
$$ L \ = \ 2 \pi \, \int \[ {1 \over 2} u_r^2 \, - \, V (u;\la )
\] \, r \d r \ . \eqno(5.24) $$
The Euler-Lagrange equations for (22),(23) are
$$ \lapl u \ = \ - {\pa V \over \pa u} \ \equiv \ f (u ; \la )
\eqno(5.25) $$
or, recalling the expression of the Laplacian in polar
coordinates,
$$ u_{rr} \, + \, {1 \over r} \, u_r \, + \, {1 \over r^2 } \,
u_{\theta \theta} \ = \ f (u ; \la ) \eqno(5.26) $$
while the Euler-Lagrange equations for (24) are
$$ u_{rr} \, + \, {1 \over r} \, u_r \ = \ f (u ; \la ) \
; \eqno(5.27) $$
This shows that critical points $u_0 \in M_0 $ of $L_0$ are also
critical points of $L$.
Similar considerations would apply for $M = L^2 (R^3 , R)$
and $G = SO(3)$; a more specific application concerning
$SO(3)$ will be presented in the next section.
\bigskip
{\tt Example 4.} We will now consider again functions
$u:R^2 \to R$, but $M = L^2 (B,R)$ ($B$ the square of side
$2 \pi$ in $R^2$), with now $G = Z_2 \times Z_2$ generated by
two elements $g_x$, $g_y$ which act in $M$ by $$ \eqalign{
g_x \ : & \ u(x,y) \ \to \ u(-x,y) \cr
g_y \ : & \ u(x,y) \ \to \ u(x,-y) \ . \cr} \eqno(5.28) $$
We wish to consider the equation
$$ u_t \ = \ \la u \, + \, a \, u_{xx} \,
+ \, b u_{yy} \, - \, u ^3 \eqno(5.29) $$
with boundary conditions on the square $B = [- \pi ,\pi]
\x [-\pi ,\pi]$ corresponding to zero normal derivative; here
$a,b$ are real constants, $0 < a < b$.
Looking for stationary solutions to (29) is the same as
looking for extremals for the functional
$$ L \ = \ \int \ \[ {\la \over 2} \, u^2 - {1 \over 4} \,
u^4 \] \ - \ {1 \over 2} \, \[ a u_x^2 \, + \, b u_y^2 \] ÷
dx \, dy \eqno(5.30) $$
It is readily seen that $G$ has four subgroups, listed here
with their corresponding invariant subspaces (we give
generators of $H$ for the subgroup $H$): $$ \matrix{H \sse G
& \S^H \cr ~& ~\cr \{ g_x , g_y \} & M_0 = \{ u(x,y) = \a
~~~~\} \cr g_x & M_y = \{ u(x,y) = w(y) \} \cr g_y & M_x
= \{ u(x,y) = z(x) \} \cr \{ e \} & M \cr} $$
Here $\a$ are constants, and $w,z$ functions of their
argument alone.
The reductions of (29) to the corresponding fixed spaces
$\S^H$ are readily obtained by cancelling the terms
involving $x$ and/or $y$ derivatives.
The functions compatible with the required boundary
conditions will be written as
$$ u(x,y ) \ = \ \sum_{k,\ell = 0}^\infty \, \b_{k
\ell} \, \cos (k x) \, \cos (\ell y) \eqno(5.31) $$
(we will also write $\a = \b_{00}$); with this the linear
part of (29) is simply
$$ {\dot \b}_{k \ell} \ = \ (\la - a k^2 - b \ell^2 ) \,
\b_{k \ell} \ . \eqno(5.32) $$
Notice that $u=0$ is a solution for all values of $\la$;
for $\la < 0$ this is stable, and becomes unstable as $\la$
crosses zero. For $\la \in [0,a )$ the space $M_0 = \S^G$
(corresponding to $\b_{k \ell} \equiv 0$) is not only
invariant but also hyperbolically attractive, so that the
stable solutions bifurcating from $u=0$ are described by
solutions to $\a_t = \la \a - \a^3$, i.e. are given by
$u(x,y) = \pm \a$.
At $\la = a/4$, these two branches of constant solutions
become first unstable in a direction tangent to $M_x$; the
factor $1/4$ is due to the fact that we have to consider the
linearization around $u = \a$, i.e. $u = \a + v$: the
linearized equation for $v$ is then $v_t = 4 \la v + a
v_{xx}$, using $\a^2 = \la$.
Our results guarantee that in analyzing this secondary
bifurcation we can limit to consider functions in $M_x$,
i.e. set all the $\b_{k \ell}$ with $\ell \not= 0$ to zero.
Notice that by a standard bifurcation argument we can then
obtain the bifurcation equation involving $\a$ and
$\b_{01}$ alone, but this further reduction does not follow
from symmetry considerations.
The new branches of solutions lying in $M_x$ will then become
themself unstable in a direction tangent to $M_y$, at some
higher value of $\la$.
\bigskip
{\tt Example 5.} We consider now $M=L^2 (R^2,R)$ and $G=Z_2
\x Z_2$ with generators $g_x , g_y$ as in the previous
example. We consider now the equation
$$ u_t \ = \ - \la u_xx - (\la - c) u_{yy} - u^3 \eqno(5.33) $$
on which we do not impose boundary conditions. Passing to a
spatial Fourier transform
$$ u(x,y;t) \ = \ {1 \over 2 \pi} \, \int \int \Phi (k ,
\ell ; t) \, e^{i (k x + \ell y) } \, dk \, d \ell
\eqno(5.34) $$ the linear part of (33) reads
$$ {\dot \Phi} (k , \ell ; t) \ = \ \[ \la k^2 + (\la - c)
\ell^2 \] \Phi (k , \ell ; t) \eqno(5.35) $$
so that at $\la=0$ the zero solution becomes unstable in
the directions of the whole $M_x$ space; the corresponding
symmetry reduced equation is obviously
$$ u_t \ = \ - \la u_{xx} - u^3 \ ; \eqno(5.36) $$
notice that now we have a continuous critical spectrum, and
we cannot apply bifurcation theory.
\bigskip
{\tt Example 6.}
Let us consider the complex Ginzburg-Landau equation,
$$ \pa_t u \ = \ \pa_x^2 u + u - u \, |u|^2 \eqno(5.37) $$
which we consider as defining a flow in the space $M = L^2
(R,C)$ of complex functions $u(x)$ on $R$, $u: R \to C$.
Passing to consider the Fourier transform of $u$,
$$ u(x;t) \ = \ {1 \over \sqrt{2 \pi} } \, \int \phi (k,t)
\, e^{ikx} \, dk \eqno(5.38) $$
and writing $|| \phi ||^2 = (1 / 2 \pi ) \int | \phi
(k,t) |^2 \, dk $, eq. (37) reads
$$ {d \phi (k,t) \over dt} \ = \ (1 - k^2 - || \phi ||^2 )
\phi \eqno(5.39) $$
We consider here the group $G = R$ of translations in $x$,
$g_a : u(x) \to u (x-a)$; this acts in $M$ as giving an
extra phase $\th = ka$ to the coefficient of the base
function $e^{ikx}$, that is
$$ g_\tau : \phi (k) \to e^{ik \tau } \phi (k) \ ; \eqno(5.40)
$$ thus we have the maximal invariant subspace $M_0$,
invariant under the full group $G$, which corresponds to
constant functions; other invariant subspaces $M_a$, with
isotropy subgroup $Z$ corresponding to translations of
multiples of $2 \pi / a$, are given by functions for which
(38) reduces to a Fourier series
$$ u(x;t) \ = \ \sum_{k=-\infty}^{+ \infty} \, f_k (t) \,
e^{i k x} \ . \eqno(5.41) $$
It is immediate to check that these subspaces are indeed
invariants under (37). Notice that one is not able to
perform a bifurcation analysis for (37), due to the
presence of continuous spectrum, but the bifurcation
analysis is completely standard (and trivial) for each of
the symmetry reduction corresponding to these isotropy
subgroups.
\bigskip
{\tt Example 7.}
In the same vein, we can consider the Swift-Hohemberg
equation for $u : R \x R_+ \to R$,
$$ \pa_t u \ = \ \la u - (1 + \pa_x^2 )^2 u - u^3 \ ,
\eqno(5.42) $$ which we see again as a flow on $L^2 (R,R)$.
Here the condition of real $u$ requires $\phi (k;t) =
\phi^* (-k ; t)$, and $g_\tau$ acts as a rotation in the
two dimensional spaces spanned by $\cos (k x)$ and $\sin (k
x)$ (except of course for the one dimensional invariant
subspace given by $k=0$). We obtain in this case
essentially the same picture as in example 6 above, i.e.
for each subgroup $Z$ of $R$ corresponding to $\tau $
multiple of $2 \pi / a$ we get an infinite dimensional
invariant subspace given by the $k$ multiples of $a$.
Once again, this equation is not amenable to standard
bifurcation analysis due to continuous spectrum \ref{60},
but the bifurcation analysis is standard and pretty trivial
in each of the invariant subspaces determined in this way.
%\section{5. Discussion and conclusions.}
\vfill\eject
\section{References}
\bigskip\parskip=6pt
\ref{1} G. Cicogna, ``Symmetry breakdown from
bifurcation''; {\it Lett. N. Cim.} {\bf 31} (1981), 600
\ref{2} A. Vanderbauwhede, {\it Local bifurcation and
symmetry}, Pitman (Boston) 1982
\ref{3} M. Golubitsky and I. Stewart, ``Hopf bifurcation
in the presence of symmetry''; {\it Arch. Rat. Mech. Anal.}
{\bf 87} (1985), 107
\ref{4} P. Chossat and M. Koenig; ``Characterization of
bifurcations for vector fields which are equivariant
under the action of a compact Lie group'', {\it C. R. Acad.
Sci. (Paris)} {\bf 318-A} (1994), 31
\ref{5} M. Golubitsky, D. Schaeffer and I. Stewart, {\it
Singularities and groups in bifurcation theory -- vol. II},
Springer (Berlin) 1988
\ref{6} G. Cicogna, ``A nonlinear version of the
equivariant bifurcation lemma''; {\it J. Phys. A} {\bf 23}
(1990), L1339
\ref{7} S.N. Chow and J.K. Hale, {\it Methods of bifurcation
theory}, Springer (Berlin) 1982
\ref{8} J. Guckenheimer and P. Holmes, {\it Nonlinear oscillations,
dynamical systems, and bifurcations of vector fields}, Springer (New
York) 1983
\ref{9} G. Prodi and A. Ambrosetti, {\it A primer of
nonlinear analysis}, Cambridge 1993
\ref{10} D.H. Sattinger, {\it Topics in stability and bifurcation
theory}, Lecture Notes in Mathematics {\bf 309}, Springer
(Berlin) 1973
\ref{11} V.I. Arnold, {\it Geometrical methods in the
theory of ordinary differential equations}
\ref{12} D. Ruelle, {\it Elements of differentiable
dynamics and bifurcation theory}, Academic Press (London)
1989
\ref{13} G. Iooss ad D.D. Joseph, {\it Elementary stability and
bifurcation theory}, Springer (Berlin) 1990$^2$
\ref{14} J.D. Crawford, ``Introduction to bifurcation theory'',
{\it Rev. Mod. Phys.} {\bf 63} (1991), 991-1037
\ref{15} V.I; Arnol'd, V.S. Afrajmovich, Yu.S.
Il'yashenko and L.P. Shil'nikov, ``Bifurcation
Theory''; in {\it Dynamical Systems V}, V.I. Arnold ed.,
vol. 5 of the {\it Encyclopaedia of Mathematical Sciences},
Springer (Berlin) 1994
\ref{16} M. Golubitsky, J. Marsden and D. Schaeffer,
``Bifurcation problems with hidden symmmetry''; in W..
Fitzgibbon ed., {\it Partial differential equations and
dynamical systems}, Research Notes in Mathematics {\bf
101}, Pitman (Boston) 1984
\ref{17} D. Ruelle, ``Bifurcations in the presence of a symmetry
group'', {\it Arch. Rat. Mech. Anal.} {\bf 51} (1973), 136
\ref{18} D.H. Sattinger, {\it Group theoretic methods in
bifurcation theory}, Lecture Notes in Mathematics {\bf 762},
Springer (Berlin) 1979
\ref{19} D.H. Sattinger, {\it Branching in the presence of
symmetry}, S.I.A.M. (Philadelphia) 1983
\ref{20} M. Field, ``Local structure of equivariant dynamics''; in
M. Roberts and I. Stewart eds., {\it Singularity theory and
applications -- II}, Lecture Notes in Mathematics {\bf ***}, Springer
(Berlin) 1991
\ref{21} M.J. Field and R.W. Richardson, ``Symmetry
breaking and branching patterns in equivariant bifurcation
theory -- I \& II'', {\it Arch. Rat. Mech. Anal.} {\bf 118}
(1992), 297 \& {\bf 120} (1992), 147
\ref{22} M. Field, {\it Lectures on bifurcations,
dynamics and symmetry}, Pitman Research Notes in Mathematics
{\bf 356}, Longman (Harlow) 1996
\ref{23} I. Stewart, ``Bifurcations with symmetry''; in T.
Bedford and J. Swift eds., {\it New directions in dynamical
systems}, Cambridge University Press (Cambridge) 1988
\ref{24} G. Gaeta, ``Bifurcation and symmetry breaking'',
{\it Phys. Rep.} {\bf 189} (1990), 1
\ref{25} J.D. Crawford and E. Knobloch, ``Symmetry and symmetry
breaking bifurcations in fluid dynamics'', {\it Ann. Rev. Fluid
Mech.} {\bf 23} (1991), 341-387
\ref{26} R.S. Palais, ``The principle of symmetric
criticality'', {\it Comm. Math. Phys.} {\bf 69} (1979), 19
\ref{27} R.S. Palais, ``Applications of the symmetric
criticality principle in mathematical physics and
differential geometry''; in: Gu Chachao ed., {\it
Proceedings of the 1981 Shangai symposium on differential
geometry and differential equations}, Science Press
(Beijing) 1984
\ref{28} G. Gaeta: ``Reduction and equivariant branching
lemma: dynamical systems, evolution PDEs, and gauge
theories''; {\it Acta Appl. Math.} {\bf 28} (1992), 43
\ref{29} G. Gaeta: ``Michel's theorem and critical section
of gauge functionals''; {\it Helv. Phys. Acta} {\bf 65}
(1992), 922; and ``Critical sections of gauge functionals: a
symmetry approach''; {\it Lett. Math. Phys.} {\bf 28} (1993),
1
\ref{30} L. Michel and L. Radicati, ``Properties of the
breaking of hadronic internal symmetry'', {\it Ann. Phys.
(N.Y.)} {\bf 66} (1971), 758
\ref{31} L. Michel and L. Radicati, ``The geometry of the
octet'', {\it Ann. I.H.P.} {\bf 18} (1973), 185
\ref{32} G. Sartori, ``Geometric invariant theory. A
model-independent approach to spontaneous symmetry and/or
supersymmetry breaking'', {\it Riv. N. Cim.} {\bf 14}
(1991), no.11
\ref{33} G. Gaeta and P. Morando, ``Michel theory of
symmetry breaking and gauge theories'', {\it Ann. Phys.
(N.Y.)} {\bf 260} (1997), 149
\ref{34} L. Michel, ``Points critiques de fonctions
invariantes sur une G-vari\'et\'e'', {\it Comptes Rendus
Acad. Sci. Paris} {\bf 272-A} (1971), 433
\ref{35} L. Michel, ``Symmetry defects and broken symmetry.
Configurations. Hidden symmetry'', {\it Rev. Mod. Phys.}
{\bf 52} (1980), 617
\ref{36} L. Michel, ``Nonlinear group action. Smooth
action of compact Lie group on manifolds'', in R.N. Sen and
C. Weil eds., {\it Statistical Mechanics and Field Theory},
Israel University Press (Jerusalem) 1971
\ref{37} M. Abud and G. Sartori, ``The geometry of
spontaneous symmetry breaking'', {\it Ann. Phys. (N.Y.)}
{\bf 150} (1983), 307
\ref{38} G. Gaeta and P. Morando, ``Commuting-flow
symmetries and common solutions to differential equations
with common symmetries'', {\it J. Phys. A: Math. Gen.} {\bf
31} (1998), 337
\ref{39} G. Cicogna, ``Bifurcation from symmetry and
topological arguments'', {\it Boll. U.M.I.} {\bf
3A} (1984), 131
\ref{40} R.S. Palais and C.L. Terng, {\it
Critical point theory and submanifold geometry}, Lecture Notes in
Mathematics {\bf 1353}, Springer (New York) 1988
\ref{41} N. Jacobson, {\it Lie Algebras}; Interscience (New
York) 1962; reprinted by Dover
\ref{42} R. Hermann, ``The formal linearization of a semisimple
Lie algebra of vector fields about a singular point'', {\it
Trans. A.M.S.} {\bf 130} (1968), 105-109
\ref{43} V. Guillemin and S. Sternberg, ``Remarks on a paper by
Hermann'', {\it Trans. A.M.S.} {\bf 130} (1968), 110-116
\ref{44} J. Frohlich and M. Struwe, ``Variational problems on
fiber bundles'', {\it Comm. Math. Phys.} {\bf 131} (1990),
431-464
\ref{45} M. Spivak, {\it A Comprehensive
Introduction to Differential Geometry}, Publish or
Perish, Berkeley 1979
\ref{46} B.A. Dubrovin, S.P. Novikov and A.T. Fomenko,
{\it Modern Differential Geometry}, Springer,
Berlin 1992
\ref{47} T. Eguchi, P.B. Gilkey and A.J. Hanson, ``Gravitation, gauge
theories and differential geometry'', {\it Phys. Rep.} {\bf 66} (1980),
213-393
\ref{48} C.J. Isham, {\it Modern differential geometry for physicists},
World Scientific 1989
\ref{49} C. Nash and S. Sen, {\it Topology and
Geometry for physicists}, Academic Press, London 1983
\ref{50} M. Nakahara, {\it Geometry, Topology and
Physics}, Adam Hilgher -- IOP, Bristol 1990
\ref{51} J.E. Marsden, {\it Lectures on Mechanics}, L.M.S.
Lecture Notes Series {\bf 174}, Cambridge 1992
\ref{52} P.J. Olver, {\it Application of Lie groups to differential
equations}, Springer (New York) 1986
\ref{53} Gaeta, ``A splitting lemma for equivariant
dynamics'', {\it Lett. Math. Phys.} {\bf 33} (1995), 313;
and ``Splitting equivariant dynamics'', {\it Nuovo Cimento
B} {\bf 110} (1995), 1213
\ref{54} N. Cabibbo and L. Maiani, ``Weak interactions and
the breaking of hadronic symmetry''; in M. Conversi ed.,
{\it Evolution of particle Physics (E. Amaldi
Festschrift)}, Academic Press (London) 1970; compare also
the contribution by L. Michel and L. Radicati in the same
volume.
\ref{55} G. Cicogna and G. Gaeta,
``Symmetry invariance and center manifolds for dynamical systems'',
{\it Nuovo Cimento} {\bf B 109} (1994), 59
\ref{56} J. Milnor, {\it Morse theory}, Princeton University Press 1963
\ref{57} M. Morse, {\it The calculus of variations in the large}, A.M.S.
(Providence) 1964
\ref{58} G.W. Bluman and S. Kumei, {\it Symmetries and
differential equations}, Springer (New York) 1989
\ref{59} G. Gaeta, {\it Nonlinear symmetries and nonlinear
equations}, Kluwer (Dordrecht) 1994
\ref{60} P. Collet and J.P. Eckmann, {\it Instabilities and
fronts in extended systems}, Princeton 1990
\bye