Reduction and Integrability

10 Pages·0.16 MB·

by  Nguyen Tien Zung

REDUCTION AND INTEGRABILITY
2
0 NGUYEN TIEN ZUNG
0
2
Abstract. Wediscusstherelationshipbetweentheintegrabilityofadynam-

n ical system invariant under a Lie group action and its reduced integrability,
a i.e. integrabilityofthecorrespondingreducedsystem
J

1
1
1. Introduction
]
S
The aim of this note is to address the following question (Q):
D
GivenamanifoldM (eventuallywithsingularities),aLiegroupGwhich
.
h acts on M (in such a way that the quotient M/G is a manifold with
t singularities), and a vector field X on M which is invariant under the
a
m actionofG. Denote by X/Gthe projectionof X onM/G. What is the
relationshipbetweentheintegrabilityof(M,X)andthe integrabilityof
[
(M/G,X/G) (a.k.a. the reduced integrability of (M,X,G)) ?

2
Theabovequestionisverynatural,sincedynamicalsystemsoftenadmitnatural
v
symmetrygroups,andbyintegrabilityofaprobleminclassicalmechanicsoneoften
7
8 means its reduced integrability. It seems to me, however, that Question (Q) has
0 not been formally addressed anywhere in the literature, and that’s why this note.
1 We will consider two different cases: Hamiltonian and non-Hamiltonian. For
0
simplicity, we will assume that G is a compact group. We will show that, when
2
the action of G is Hamiltonian, Hamiltonian integrability is the same as reduced
0
/ Hamiltonian integrability. In the non-Hamiltonian case, integrability still implies
h
reduced integrability, though the inverse needs not be true. The proof of these
t
a facts is elementary : we simply play with the dimensions of various spaces and
m
their intersections, quotients, etc.
:
v 2. Hamiltonian integrability
i
X
Let (M,Π) be a real Poisson manifold, with Π being the Poisson structure.
r
Let H be a (smooth or analytic) function on M, and X be the corresponding
a H
Hamiltonian vector field. Assume that we have found a set F of first integrals of
X , i.e. each F ∈ F is a function on M which is preserved by X (equivalently,
H H
{F,H}=0, where {.,.} denotes the Poisson bracket as usual). Denote by ddim F
thefunctionaldimensionofF,i.e. themaximalnumberoffunctionallyindependent
functions in F. To avoid pathologies, we will always assume that the functional
dimensionoftherestrictionofF toanyopensubsetofM isequaltoddim F. (This
is automatic in the analytic case).
We will associate to F the space X =XF of Hamiltonian vector fields XF such
that X (G) = 0 for all G ∈ F and F is functionally dependent of F (i.e. the
F
Date:firstversion,11/Jan/2002.
1991 Mathematics Subject Classification. 37JXX,37C80,70HXX.
Key words and phrases. symmetry,reduction,integrability,reducedintegrability.
1
2 NGUYEN TIEN ZUNG
functional dimension of the union of F with the function F is the same as the
functional dimension of F). Clearly, X belongs to X, and the vector fields in
H
X commute pairwise. Denote by ddim X the functional dimension of X, i.e. the
maximal number of vector fields in X whose exterior (wedge) product does not
vanish.
With the above notations, we have the following definition, due essentially to
Nekhoroshev [8] and Mischenko and Fomenko [7] :
Definition 2.1. AHamiltonianvectorfieldX onanm-dimensionalPoissonman-
H
ifold(M,Π) iscalledHamiltonianly integrablewiththe aidofasetoffirstintegrals
F, if m = ddim F +ddim XF. It is called properly Hamiltonianly integrable if F
satisfies the following additional properness condition :
There are q functions F ,...,F in F, where q =ddim F, which are functionally
1 q
independent, and whose joint “moment map” (F ,...,F ) : M → Rq (here we
1 q
considerthe realcase)isa propermapfromM to its image,andtherearep vector
fields X ,...,X in X, where p=ddim X, such that the image of their singular set
1 p
{x∈M,X ∧X ∧...∧X (x)=0}underthemap(F ,...,F ):M →Rq isnowhere
1 2 p 1 q
dense in Rq.
Remarks
1. The above notion of integrability is often called generalized Liouville integra-
bility,oralsonon-commutativeintegrabilitybyMischenko-Fomenko,duetothefact
thatthe functions in F do not Poisson-commuteingeneral,and inmany cases one
may choose F to be a finite-dimensional non-commutativeLie algebraof functions
(under the Poisson bracket). When the functions in F Poisson-commute, we get
back to the classical integrability `a la Liouville.
2. We always have m ≤ ddim F +ddim X (even for non-integrable systems),
becausethevectorfieldsinX aretangenttothe commonlevelsetsofthefunctions
in F. Thus the integrability condition m = ddim F +ddim X is a maximality, or
fullness, condition on F.
3. Casimir functions of (M,Π), i.e. functions whose Hamiltonian vector fields
vanish, must be functionally dependent of F in the integrable case - otherwise we
could add them to F to increase the functional dimension of F, which contradicts
the above remark about m≤ddim F +ddim X.
4. It follows directly from the fact that if X ∈ X then F is functionally
F
dependent of F that we have ddim X ≤ ddim F. We didn’t mention the rank of
thePoissonstructureΠintheabovedefinition,butofcourseddim X ≤1/2rank Π.
When m = ddim F +ddim X and ddim X < 1/2 rank Π, some authors also say
thatthesystemissuper-integrable(becauseinthiscaseonefindsmorefirstintegrals
than necessary for integrability).
5. We prefer to use the term Hamiltonian integrability, to contrast it with the
non-Hamiltonian integrability discussed in the next section.
6. Under the additional properness condition, one get a natural generalization
of the classical Liouville theorem [8, 7]: the manifold M is foliated by invariant
isotropictorionwhichtheflowofX isquasi-periodic(thusthe behaviorofX is
H H
veryregular,justifyingtheword“integrable”),andtherealsoexistlocalgeneralized
action-angle coordinates. The existence of action-angle coordinates for Liouville-
integrable systems is often referred to as Arnold-Liouville theorem, though it was
probably first proved by Mineur [6].
REDUCTION AND INTEGRABILITY 3
Assume that there is a Hamiltonian action of a Lie group G on (M,Π), given
by an equivariant moment map π :M →g∗, where g denotes the Lie algebra of G,
such that the following conditions are satisfied :
1)The actionofGonM isproper,sothatthequotientspaceM/Gisasingular
manifold whose ring of functions may be identified with the ring of G-invariant
functions on M.
2) Recall that the image π(M) of M under the moment map π : M → g∗
is saturated by symplectic leaves (i.e. coadjoint orbits) of g∗. Denote by s the
minimalcodimensioning∗ ofacoadjointorbitwhichliesinπ(M). Thenweassume
that there exist s functions f ,...,f on g∗, which are invariant on the coadjoint
1 s
orbits which lie in π(M), and such that for almost every point x ∈ M we have
df ∧...∧df (π(x))6=0.
1 s
Forexample,whenGiscompactandM isconnected,thentheaboveconditions
are satisfied automatically.
If π(M) contains a generic point of g∗, then s=ind g, where ind g denotes the
index of g, i.e. the corank of the corresponding linear Poissonstructure on g∗, and
if there are (ind g) global functionally independent Casimir functions on g∗ they
may play the role of requiredfunctions f ,...,f in the above assumption. If π(M)
1 s
lies in the singular part of g∗ then s may be greater than the index of g.
Since Π is preserved by G, it can be projected to a Poisson structure, denoted
by Π/G, on M/G. In the case when M is a symplectic manifold, the symplectic
leaves of (M/G,Π/G) are known as Marsden-Weinstein reductions.
Let H be a function on M which is invariant by G. Then H may be viewed as
thepull-backofafunctionhonM/Gviatheprojectionp:M →M/G,H =p∗(h).
Denote by X (resp., X ) the Hamiltonian vector field of H (resp., h) on (M,Π)
H h
(resp., (M/G,Π/G)). Of course, X is G-invariant, and its projection to M/G is
H
X .
h
With the above notations and assumptions, we have :
Theorem 2.2. If the system (M/G,X ) is Hamiltonianly integrable, then the sys-
h
tem (M,X ) also is Hamiltonianly integrable. If G is compact and (M/G,X )
H h
is properly Hamiltonianly integrable, then (M,X ) also is properly Hamiltonianly
H
integrable.
Proof. Denote by F′ a set of first integrals of X on M/G which provides
h
the integrability of Xh, and by X′ = XF′ the corresponding space of commuting
HamiltonianvectorfieldsonM/G. WehavedimM/G=p′+q′wherep′ =ddim X′
and q′ =ddim F′.
Recall that, by our assumptions, there exist s functions f ,...,f on g∗, which
1 s
are functionally independent almost everywhere in π(M), and which are invariant
on the coadjoint orbits which lie in π(M). Here s is the minimal codimension in
g∗ of the coadjoint orbits which lie in π(M). We can complete (f ,...,f ) to a set
1 s
of d functions f ,...,f ,f ,...,f on g∗, where d = dimG = dimg denotes the
1 s s+1 d
dimension of g , which are functionally independent almost everywhere in π(M).
Denote by F the pull-back of F′ under the projection p : M → M/G, and by
F ,...,F the pull-backoff ,...,f under the momentmapπ :M →g∗. Note that,
1 d 1 d
since H is G-invariant,the functions F arefirst integralsofX . And of course,F
i H
isalsoasetoffirstintegralsofX . DenotebyF theunionofF with(F ,...,F ).
H s+1 d
(It is not necessary to include F ,...,F in this union, because these functions are
1 s
G-invariantand projectto Casimir functions on M/G, which implies that they are
4 NGUYEN TIEN ZUNG
functionally dependent of F). We will show that X is Hamiltonianly integrable
H
with the aid of F.
Notice that, by assumptions, the coadjoint orbits of g∗ which lie in π(M) are of
genericdimensiond−s,andthefunctionsf ,...,f maybeviewedasacoordinate
s+1 d
system on a symplectic leaf of π(M) at a generic point. In particular, we have
<df ∧...∧df ,X ∧...X >=6 0,
s+1 d fs+1 fd
which implies, by equivariance :
<dF ∧...∧dF ,X ∧...X >=6 0.
s+1 d Fs+1 Fd
SincethevectorfieldsX ,...,X aretangenttotheorbitsofGonM,andthe
Fs+1 Fd
functions inF areinvariantonthe orbitsofG, itimplies thatthe set(F ,...,F )
s+1 d
is “totally” functionally independent of F. In particular, we have :
′ ′
(2.1) ddim F =ddim F +ddim (F ,...,F )=q +d−s,
s+1 d
where q′ =ddim F′. On the other hand, we have
′ ′
dimM =dimM/G+(d−k)=p +q +d−k,
where p′ = ddim XF′, and k is the dimension of a minimal isotropic group of the
action of G on M. Thus, in order to show the integrability condition
dimM =ddim F +ddim XF,
it remains to show that
(2.2) ddim XF =ddim XF′ +(s−k).
Consider the vectorfields Y =X ,...,Y =X on M. They span the tangent
1 F1 d Fd
space to the orbit of G on M at a generic point. The dimension of such a generic
tangent space is d−k. It implies that, among the first s vector fields, there are at
leasts−k vectorfieldswhicharelinearlyindependentatagenericpoints: wemay
assume that Y1∧...∧Ys−k 6=0.
Let X ,...,X be p′ linearly independent (at a generic point) vector fields
h1 hp′
which belong to XF′, where p′ =ddim XF′. Then we have
Xp∗(h1),...,Xp∗(hp′),Y1,...,Ys−k ∈XF,
andthesep′+s−k vectorfieldsarelinearlyindependentatagenericpoint. (Recall
that,ateachpointx∈M,the vectorsY1(x),...,Ys−k(x)aretangenttothe orbitof
Gwhichcontainsx,whilethelinearspacespannedbyXp∗(h1),...,Xp∗(hp′) contains
no tangent direction to this orbit).
Thus we have ddim XF ≥ p′+s−k, which means that ddim XF = p′+s−k
(because, as discussed earlier, we always have ddim F +ddim XF ≤ dimM). We
have proved that if (M/G,X ) is Hamiltonianly integrable then (M,X ) also is.
h H
Now assume that G is compact and (M/G,X ) is properly Hamiltonianly inte-
h
grable : there are q′ functionally independent functions g1,...,gq′ ∈ F′ such that
(g1,...,gq′) : M/G → Rq′ is a proper map from M/G to its image, and p′ Hamil-
tonian vector fields X ,...,X in X′ such that on a generic common level set
h1 hp′
REDUCTION AND INTEGRABILITY 5
of (g1,...,gq′) we have that Xh1 ∧...∧Xhp′ does not vanish anywhere. Then it is
straightforwardthat
∗ ∗
p (g1),...,p (gq′),Fs+1,...,Fd ∈F
and the map
(p∗(g1),...,p∗(gq′),Fs+1,...,Fd):M →Rq′+d−s
isapropermapfromM toitsimage. Moreimportantly,onagenericlevelsetofthis
mapwehavethatthe (q′+s−k)-vectorXp∗(h1)∧...∧Xp∗(hp′)∧Y1∧...∧Ys−k does
notvanishanywhere. Toprovethislastfact,noticethatXp∗(h1)∧...∧Xp∗(hp′)∧Y1∧
...∧Ys−k(x)6=0 for a point x∈M if andonly if Xp∗(h1)∧...∧Xp∗(hp′)(x)6=0 and
Y1∧...∧Ys−k(x)6=0(oneofthesetwomulti-vectorsistransversaltotheG-orbitof
xwhilethe otherone“liesonit”),andthattheseinequalitiesareG×Rp′-invariant
properties, where the action of Rp′ is generated by Xp∗(h1),...,Xp∗(hp′). ♦
Remark. RecallfromEquation(2.2)abovethatwehaveddim XF−ddim XF′ =
s−k, where k is the dimension of a generic isotropic group of the G-action on M,
andsis the (minimal)coranking∗ ofacoadjointorbitwhichlies inπ(M). Onthe
otherhand, the difference between the rank ofthe Poissonstructure onM andthe
reduced Poissonstructure on M/G can be calculated as follows :
(2.3) rank Π−rank Π/G=(d−k)+(s−k)
Here (d−k) is the difference between dimM and dimM/G, and (s−k) is the
differencebetweenthecorankofΠ/GinM/GandthecorankofΠinM. Itfollows
that
(2.4) rank Π−2ddim XF =rank Π/G−2ddim XF′ +(d−s)
Inparticular,ifd−s>0(typicalsituationwhenGisnon-Abelian),thenwealways
haverank Π−2ddim XF >0 (because we alwayshaverank Π/G−2ddim XF′ ≥0
duetointegrability),i.e. theoriginalsystemisalwayssuper-integrablewiththeaid
of F. When G is Abelian (implying d = s), and the reduced system is Liouville-
integrablewiththeaidofF′(i.e. rank Π/G=2ddim XF′),thentheoriginalsystem
is also Liouville-integrable with the aid of F.
Remark. Following Mischenko-Fomenko [7], we will say that a hamiltonian sys-
tem (M,Π,X ) is non-commutatively integrable in the restricted sense with the
H
aid of F , if F is a finite-dimensional Lie algebra under the Poisson bracket and
(M,Π,X )is Hamiltonianly integrablewiththe aidofF. Inother words,we have
H
an equivariant moment maps (M,Π) → f∗, where f is some finite-dimensional Lie
algebra, and if we denote by f ,...,f the components of this moment map, then
1 n
they are first integrals of X , and X is Hamiltonianly integrable with the aid of
H H
thissetoffirstintegrals. Theorem2.2remainstrue,anditsproofremainsthesame
ifnoteasier,ifwereplaceHamiltonianintegrabilitybynon-commutativeintegrabil-
ityintherestrictedsense. Indeed,ifM →g∗ istheequivariantmomentmapofthe
symmetry group G, and if M/G → h∗ is an equivariant moment map which pro-
vides non-commutative integrability in the restrictedsense onM/G, then the map
M → h∗ (which is the composition M → M/G → h∗) is an equivariant moment
map which commutes with M →g∗, and the direct sum of this two maps, M →f∗
where f=gLh, willprovide non-commutativeintegrabilityin the restrictedsense
on M.
6 NGUYEN TIEN ZUNG
The above remarks show that the notion of Hamiltonian integrability (or non-
commutative integrability if you prefer), rather than integrability a` la Liouville, is
the most natural one when dealing with systems admitting (non-Abelian) symmetry
groups.
Examples. 1) The simplest example which shows an evident relationship be-
tween reduction and integrability is the classical Euler top : it can be written as
a Hamiltonian system on T∗SO(3), invariant under a natural Hamiltonian action
of SO(3), is integrable with the aid of a set of four first integrals, and with 2-
dimensional isotropic invariant tori. 2) The geodesic flow of a bi-invariant metric
on a compact Lie group is also properly Hamiltonianly integrable : in fact, the
corresponding reduced system is trivial (identically zero).
It is known that most Hamiltonianly integrable systems are also integrable `a la
Liouville,i.e. thereexistsasetoffirstintegralswhichcommutepairwiseandwhich
make the system integrable - see e.g. [5] for a very long discussion on this subject.
A question of similar kind, which is directly related to the inverse of Theorem 2.2,
is the following :
IfX isHamiltonianlyintegrablewiththeaidofF ,andifF isanother
H 1 2
set of first integrals of X which contains F , then is it true that X
H 1 H
is also Hamiltonianly integrable with the aid of F ? In particular, let
2
F denotes the set of all first integrals of X . If X is Hamiltonianly
H H H
integrable, then is it true that it is integrable with the aid of F ?
H
A related question is the following :
Suppose that F = (F ,...,F ) is a set of independent first integrals of
1 q
X such that regular common level sets of these functions (F ,...,F )
H 1 q
are isotropic submanifolds of (M,Π). Is it true that X is integrable
H
with the aid of F ?
Remark that the inverse to the later question is always true. And if we can say
YES to the later question, then we can also say YES to the former one, because
adding first integrals has the affect of minimizing invariant submanifolds, and a
submanifold of an isotropic submanifold is again an isotropic submanifold. It is
easy to see that, at least in the smooth proper case, the answer to the above two
questionsisYES:smoothproperHamiltonianintegrabilityofaHamiltoniansystem
is equivalentto the singularfoliationofthe Poissonmanifoldby invariantisotropic
tori (This fact is similar to the classical Liouville theorem).
For the following theorem, we use the same notations and preliminary assump-
tions as in Theorem 2.2 :
Theorem 2.3. If G is compact, and if the Hamiltonian system (M,X ) is Hamil-
H
tonianlyintegrablewiththeaid ofF (thesetofall firstintegrals), thenthereduced
H
Hamiltonian system (M/G,X ) is also Hamiltonianly integrable. The same thing
h
holds in the smooth proper case.
Proof. By assumptions, we have dimM = p+q, where q = ddim F and p =
H
ddim XFH, andwe canfindp firstintegralsH1,...,Hp ofH suchthat XH1,...,XHp
are linearly independent (at a generic point) and belong to XF . In particular, we
H
have X (F)=0 for any F ∈F and 1≤i≤p.
Hi
REDUCTION AND INTEGRABILITY 7
An important observation is that the functions H ,...,H are G-invariant. In
1 p
deed, if we denote by F ,...,F the components of the equivariant moment map
1 d
π :M →g∗(viaanidentificationofg∗withRd),thensinceH isG-invariantwehave
{H,F }=0,i.e. F ∈F ,whichimpliesthat{F ,H }=0 ∀1≤i≤d, 1≤j ≤p,
j j H j i
which means that H are G-invariant.
i
Denote by h the projection of H on M/G (recall that the projection of H on
i i
M/G is denoted by h). Then the Hamiltonian vector fields Xhi belong to XFh :
Indeed, if f ∈ F then p∗(f) is a first integral of H, implying {H ,p∗(f)} = 0, or
h i
{h ,f}=0, where p denotes the projection M →M/G.
i
To prove the integrability of X , it is sufficient to show that
h
(2.5) dimM/G≤ddim F +ddim (X ,...,X )
h h1 hq
Butwe denote by r the genericdimensionofthe intersectionofa commonlevelset
ofp independent first integralsof X with an orbitof G in M, then one cancheck
H
that
p−ddim (Xh1,...,Xhq)=ddim XFH −ddim (Xh1,...,Xhq)=r
and
q−ddim F =ddim F −ddim F ≤(d−k)−r
h H h
where (d−k) is the dimension of a generic orbit of G in M. To prove the last
inequality, notice that functions in F can be obtained from functions in F by
h H
averaging with respect to the G-action. Also, G acts on the (separated) space of
commonlevelsets ofthe functions inF , andisotropicgroupsofthisG-actionare
H
of (generic) codimension (d−k)−r.
The above two formulas, together with p+q = dimM = dimM/G+(d−k),
implies Inequality (2.5) (it is in fact an equality).
We will leave the proper case to the reader as an exercise. ♦
3. Non-Hamiltonian integrability
The interest in non-Hamiltonian integrability comes partly from the fact that
there are many non-Hamiltonian (e.g. non-holonomic) systems whose behaviors
are very similar to that of integrable Hamiltonian systems, see e.g. [1, 3]. In
particular, in the proper case, the manifold is foliated by invariant tori on each of
which the system is quasi-periodic. Another common point between (integrable)
Hamiltonian andnon-Hamiltonian systems is that their localnormal formtheories
are very similar and are related to local torus actions, see e.g. [9, 10]. The notion
of non-Hamiltonian integrability was probably first introduced by Bogoyavlenskij
[2], who calls it broad integrability, in his study of tensor invariants of dynamical
systems. Let us give here a definition of non-Hamiltonian integrability, which is
similar to the ones found in [1, 2, 3, 9, 10] :
Definition 3.1. Avectorfieldona(eventuallysingular)manifoldM iscallednon-
Hamiltonianly integrable with the aid of (F,X), where F is a set of funtions on M
and X is a set of vector fields on M, if the following conditions are satisfied :
a) Functions in F are first integrals of X: X(F)=0 ∀F ∈F.
b) Vector fields in X commute pairwise and commute with X : [Y,Z] = [Y,X] =
0 ∀ Y,Z ∈X.
c)FunctionsinF arecommonfirstintegralsofvectorfieldsinX: Y(F)=0 ∀Y ∈
X,F ∈F.
d) dimM =ddim F +ddim X.
8 NGUYEN TIEN ZUNG
If,moreover,thereexistpvectorfieldsY ,...,Y ∈X andqfunctionallyindependent
1 p
functions F ,...,F ∈ F, where p = ddim X and q = ddim F, such that the map
1 q
(F ,...,F ) : M → Rq is a proper map from M to its image, and for almost any
1 q
levelsetof this mapthe vectorfields Y ,...,Y arelinearlyindependent everywhere
1 p
on the level set, then we say that X is properly non-Hamiltonianly integrable with
the aid of (F,X).
Remarks.
1. It is straightforward that, in the proper case, the manifold is a (singular)
foliation by invariant tori (common level sets of some first integrals) on each of
whichthe vectorfieldX is quasi-periodic. (This is similar to the classicalLiouville
theorem).
2. If a Hamiltonian system is (properly) Hamiltonianly integrable, then it is
also (properly) non-Hamiltonianly integrable, though the inverse is not true : it
may happen that the invariant tori are not isotropic, see e.g. [2, 4] for a detailed
discussion about this question.
Oneofthemaindifferencesbetweenthenon-HamiltoniancaseandtheHamilton-
iancaseisthatreducednon-Hamiltonianintegrabilitydoesnotimply integrability.
Infact,intheHamiltoniancase,wecanliftHamiltonianvectorfieldsfromM/Gto
M via the lifting ofcorrespondingfunctions. Inthe non-Hamiltoniancase,no such
canonical lifting exists, therefore commuting vector fields on M/G do not provide
commuting vector fields on M. For example, consider a vector field of the type
X =a ∂/∂x +a ∂/∂x +b(x ,x )∂/∂x on the standard torus T3 with periodic
1 1 2 2 1 2 3
coordinates (x ,x ,x ), where a and a are two incommensurable real numbers
1 2 3 1 2
(a /a ∈/ Q), and b(x ,x ) is a smooth function of two variables. Then clearly X
1 2 1 2
is invariant under the S1-action generated by ∂/∂x , and the reduced system is
3
integrable. On the other hand, for X to be integrable, we must be able to find
a function c(x ,x ) such that [X,∂/∂x +c(x ,x )∂/∂x ] = 0. This last equa-
1 2 1 1 2 3
tion does not always have a solution (it is a small divisor problem, and depends
on a /a and the behavior of the coefficients of b(x ,x ) in its Fourier expansion),
1 2 1 2
i.e. therearechoicesofa ,a ,b(x ,x )forwhichthevectorfieldX isnotintegrable.
1 2 1 2
However,non-Hamiltonianintegrabilitystillimpliesreducedintegrability. Before
formulating a precise result, let us mention a question similar to the one already
mentioned in the previous section :
For a vector field X on a manifold M, denote by F the set of all
X
first integrals of X, and by X the set of vector fields which preserve
X
each function in F and commute with X. Suppose that X is non-
Hamiltonianly integrable. Is it then non-Hamiltonianly integrable with
the aid of (F ,X ) ? In other words, is it true that vector fields in F
X X
commute pairwise and ddim X +ddim F =dimM ?
X X
Itiseasytoseethatthe answertothe abovequestionisYESinthepropernon-
Hamiltonianly integrable case, under the additional assumption that the orbits of
X are dense (i.e. its frequencies are incommensurable) on almost every invariant
torus(i.e. commonlevelofagivensetoffirstintegralsF). InthiscaseX consists
X
of the vector fields which are quasi-periodic on each invariant torus. Another case
where the answer is also YES arises in the study of local normal forms of analytic
integrable vector fields, see e.g. [10].
REDUCTION AND INTEGRABILITY 9
Theorem 3.2. Let X be a smooth properly non-Hamiltonianly integrable system
on a manifold M with the aid of (F ,X ), and G be a compact Lie group acting
X X
on M which preserves X. Then the reduced system on M/G is also properly non-
Hamiltnianly integrable.
Proof. Let XG denote the setofvector fields which belong to X andwhichare
X X
invariant under the action of G. Note that the elements of XG can be obtained
X
from the elements of X by averaging with respect to the G-action.
X
A key ingredient of the proof is the fact ddim XG =ddim X (To see this fact,
X X
noticethatneareachregularinvarianttorusofthesystemthereisaneffectivetorus
action (of the same dimension) which preserves the system, and this torus action
mustnecessarilycommutewiththeactionofG. Thegeneratorsofthistorusaction
arelinearlyindependentvectorfieldswhichbelongtoXG -infact,theyaredefined
X
locally near the union of G-orbits which by an invariant torus, but then we can
extend them to global vector fields which lie in XG)
X
Therefore, we can project the pairwise commuting vector fields in XG from M
X
toM/Gtogetpairwisecommutingvectorfields onM/G. To getthe firstintegrals
for the reduced system, we can also take the first integralsof X on M and average
themwithrespecttotheG-actiontomakethemG-invariant. Therestoftheproof
of Theorem 3.2 is similar to that of Theorem 2.3. ♦
Acknowledgements. Apartofthisnote,Theorem2.2,datesbackto1999-2000
when I was preparing some lectures on the subject and was surprised by the lack
of such a theorem in the standard literature. I would like to thank A.T. Fomenko
for encouraging me to write up this note.
References
1. L.Bates,R.Cushman,What isacompletely integrablenonholonomic dynamical system?
Proceedings oftheXXXSymposiumonMathematical Physics(Torun´,1998). Rep.Math.
Phys.44(1999), no.1-2,29–35.
2. O.I. Bogoyavlenskij, A concept of integrability of dynamical systems, C. R. Math. Rep.
Acad. Sci. Canada, 18 (1996), 163-168; Theory of tensor invariants of integrable Hamil-
tonian systems. II,Comm.Math.Phys.184(1997), no.2,301–365; Extended integrability
and bi-Hamiltonian systems,Comm.Math.Phys.196(1998), no.1,19–51.
3. R. Cushman, J.J. Duistermaat, Non-Hamiltonian monodromy, J. Differential Equations
172(2001), no.1,42–58.
4. F.Fasso`,Quasi–periodicity ofmotions andcomplete integrabilityof Hamiltonian systems,
ErgodicTheoryandDynamicalSystems18(1998), 1349-1362.
5. A.T. Fomenko, Integrability and nonintegrability in geometry and mechanics, Kluwer, Dor-
drecht, 1988.
6. H. Mineur, Sur les syst`emes m´ecaniques admettant n int´egrales premi`eres uniformes et
l’extension `a ces syst`emes de la m´ethode de quantification de Sommerfeld, C. R. Acad.
Sci., Paris 200, 1571-1573 (1935); R´eduction des syst`emes m´ecaniques `a n d´egr´e de lib-
ert´e admettant n int´egrales premi`eres uniformes en involution aux syst`emes `a variables
s´epar´ees, J.Math.PuresAppl.,IX.S´er.15,385-389(1936).
7. A.S. Mischenko, A.T. Fomenko, A generalized method for Liouville integration of Hamil-
tonian sytems,Funct. Anal.andAppl.,12(1978), 46-56.
8. N.N.Nekhoroshev,Action-anglevariablesandtheirgeneralizations,Trans.MoscowMath.
Soc.,26(1972), 180-198.
9. L.Stolovitch, Singular complete integrability,PublicationsIHES,91(2000), 134-210.
10. Nguyen Tien Zung, Convergence versus integrability in Birkhoff normal form, preprint
math.DS/0104279,andConvergenceofPoincare-Dulacnormalformforcommutingvector
fields,preprintmath.DS/0105193.
10 NGUYEN TIEN ZUNG
GTA, UMR5030CNRS, D´epartementdeMath´ematiques,Universit´e MontpellierII
E-mailaddress: [email protected] URL: www.math.univ-montp2.fr/~tienzung