I L N U O V 0 CIMENTO
VOL. 79 B, N. 2
11 F e b b r a i o 1984
A Proof of Kolmogorov's Theorem on Invariant Tori Using
Canonical Transformations Defined by the Lie Method.
G. Br;NETTIN
Dipartimento di Fisiea dell' Univevsilh - V~a -F. Marzolo 8, Padova, Italia
(~ruppo ~Vazionale di Struttura della Materia del G.N.R. - Unit~ di Padova, Italia
L. GALGANI
Dipartiraento di Matematica dell'Universqth - Via Scddird 50, Milano, Italia
A . GIORGILLI
Dipartimento di Eisiva dell'U~tiversith - Via Geloria 16, Milano, Italia
g.-3i. S ~ E L C ~
Department de Mathdmatiques, Centre Scienti/ique et -Folytgchnique
Universitd Paris.Nard - 93 Villetaneuse, .France
(ric~v~to il 7 Luglio 1983)
S u m m a r y . - - I n this p a p e r a proof is given of Kolmogorov's theorem
on the existence of invariant tori in nearly integrable Hamiltonian systems.
Tho scheme of proof is t h a t of Kolmogorov, the only difference being in the
way canonical transformations near the identity are defined. Precisely,
use is made of tile Lie method, which avoids any inversion and thus any
use of the implicit-function theorem. This technical fact eliminates a
spurious ingredient and simplifies the establishment of a central estimate.
PAC8. 03.20. - Classical mechanics of discrete systems: general mathematical aspects.
1. - Introduction and formulation of Kolmogorov's theorem.
1"1. - T h e ~ i m of ~,his p a p e r is t o g i v e a p r o o f of K o l m o g o r o v ' s t h e o r e m on t h e
e x i s t e n c e of i n v a r i a n t t o r i in n e a r l y int, e g r a b l e H a m i l t o n i a n s y s t e m s (1.J~) w i t h
(1) A. N. KOL~sOGOrtOV: Dokl. Akad. ~u
$ 8 8 R , 98, 527 (1954) (Math. Rev., 16,
No. 924); English translation in G. CASAXI and J. FORD (Editors): .Decture ~Votes i~
201
202
G. BENETTLN, L. GALGANI, A. GIORGLLLI
and
J.-~.
STRELCYN
mostly pedagogi(:al i n t e n t ; thus we consider the simple.~t ca.~e, namely that
of ,~nalytie and nondegener:~tc lIamiltoni,%ns, but give, however, expli(.itly all
relevant estinmtes. I n parlieular, in deMing whith canonical transfornmtions nenr the id~,ntity we make use of the Lic m e t h o d which avoids
:tny inversion and t~hus any reference to the implicit-function theorem. This
f~ct, which is u purely technical else, eliminates ~ spin'ions ir~gredienL ~nd
simplifies the establishment of ~ central estimate. A p a r t from this technie,~l
fact, we follow here the original scheme of Kohnogorov, which can be considered
to be, e.~peci,dly from a pedagogical point of view, somehow simpler (although
possibly less powerful) t h a n the scheme of Arnold. Moreover, the proof of
convergence of the iteration scheme is given in ,~ r a t h e r simple way which,
in purticul,~r, makes use only of trivial geometric series. Thus it is hoped t h a t
the present versio~ will be u.~eful 'it least in expanding the familiarity with
this iml)ortant theorem, which still seems bo be eonsidcced, especially a m o n g
physicists, :~s an extremely difficult, one.
1"2. - In the theory of pertttrb:ttions one is concerned with an l f a m i l t o n i a n
H ( p , q) = H~
+ HI(p, q) ,
Physics, No. 93 (Berlin, 1979), p. 51.
('-) V. I. ARNOLD: USp. Mat. Nauk, 18, 13 (1963); Russ. Math. Surv., 18, 9 (1963);
Usp. Mal. Nauk., 18, No. 6, 91; Russ. Math. Surv., 18, No. 6, 85 (1963).
(2) V. I. ARNOLD and A. AVEZ: Probl~raes ergodiques de la mdcanique classique,
(Paris, 1967).
(4) J. MOSES: Proc~',.dings el the international Con]erence on Functional Analysis and
Related Tapics (Tokyo, 1969), p. 60; Stable a ~ random motions irt dynamical
systems (Princeton, 1973).
(~) S. STERIUBERG: Celestial Mechanics (New York, N. Y., 1968), Part. 2.
(*) E. A. GREBENNIKOV and Ju. A. RiABOV: New Qualitative Methods in Celestial
Mecbanics (Moskow, 1971), ill Russian.
(~) H. Ri)SSMa.'~N: Nachr. Akad. Wiss. Gall. Math. Phys. Kl., 67 (1970); l, {1972);
Lecture Notes ia Pky.~ies, No. 38 (Berlin, 1975), p. 598; in SeIect~ Mathematics, Vol. 5
(Heidelberg, Berlin, 1979); in G. Hr.LLEMAN(Editors): Ann. N. Y. Acad. Sci., 357, 90
(1980); see also On the con.~truction el invariant tori of nearly integrablc Hamillonian
systems with applieatio~ts, notes of lectures given at the Institut de Statistique, Laboratoire do Dynamique Stellaire, Uaiversit6 Pierre et Marie Curie, Paris.
(s) R. BaRRAR: Celts. Mech., 2, 494 (1970).
(~) E. ZE~r~DER: Co~nmun. Pure Appl. Math, 28, 9l (1975); 29, 49 (1976); scc also
Stability Rind instability in celestial mechanics, lectures given at Ddpartment de PhFsique,
L.~baratoire de Physique Thdorique, Ecole l'olytec.hnique Fdd5rale, Lausanne.
(~o) A. I. NEICHST,t~T: J. Appl. Math. Mech., 45, 766 11981).
(l~) G. Ga-LL~VOTrI: Mecca,ties elementare (Torino, 1980); The Eleme~ts o] 3[eclm~ics
(Berlin, 1983); in J. FR6LmH (Editor): Progress in Physics (Boston, ]982); L. ('HI~:RCmh anti G. GALLaVOTTI: Nuovo Uimento B, 67, 277 (1982).
(12) j. POESCnEL: Commun. Pure Appl. Math., 35, 653 (1982); Celest. Mech., 28, 133
(1982).
A P R O O F OF K O L M O G O R O V ' 8 TIIEORE.',.I ON INWARIA~'T T O R I ETC.
203
where
p = (p~, . . . , p . ) e B c R ~
~nd
q ---- (q~, . . . , q.) ~ T " ,
B being an open b.tll of R" a n d T" t h e n - d i m e n s i o m d torus. The variables p
a n d q a.re called t h e actions a n d the angles, respectively. T h e functions on
the torus T" are n a t u r a l l y identified with t h e functions defined on R" a n d of
period 2.u in q~, . . . , q , . F o r a n y p a B t h e u n p e r t u r b e d a n g u l a r frequencies
a) := (co~, ..., c o , ) - - ( ~ H o / ~ p ~ , . . . , E H ~
~H~
are defined. I n Kohnogor o y ' s t h e o r e m the n o n d e g e n e r a c y condition det(Sco~/~p,)-.~ 0 is a s s u m e d ; as
is well known, t h e extension to d e g e n e r a t e s y s t e m s was accomplished b y
ARNOLD.
I n t h e u n p e e t u r b e d case ( H ~ - - 0 ) , t h e equations of motion reduce to
-- 8H~
- - 0, ~ - - c" H /0c' ~, p - - ~o(p) with solutions p ( t ) - pO, q(t) - - qO
~- ( o ( p ~
(mod2.'z). Tlms the p h a s c space B • T" is foliated in(o toni {p} • T ",
p e B, each of which is inv,~riant for t h e corresponding H a m i l t o n i a n flow
a n d supports quasi-periodic motions characterized b y a f r e q u e n c y ,~ ~ ~o(p).
We recall t h a t 2 is said to be n o n r e s o n a n t if t h e r e does not e ~ s t k e Z ' ,
:
k ~ 0, such t h a t ~t.k - - 0, w h e r e we denote X.k ~ i 2~k~.
t--I
I n K o h n o g o r o v ' s 1~heorem the a t t e n t i o n is restricted to those tori which
s u p p o r t quasi-periodic motions ' w h h a p p r o p r i a t e n o n r e s o n a n t frequencies,
namely frequencies belonging to t h e set -Qv defined b y
(1.1)
for a given positive y, w h e r e [2 --- o~(B) is tile ima.ge of 11 b y t.he m a p oJ ( ~ is
a s s u m e d to be bounded, which can alw.~ys be o b t a b t e d b y possibly reducing
t h e radius of t h e ball B ) .
Here,
as above, )..k = ~ ) . , k , and, m o r e o v e r ,
!k[ - - m a x [k~l; more generally, for v, w E C", we will d e n o t e v . w - - ~ v ~ w ~
and !v[l - - m ~ x Iv~]; aml.logously, t h e uorm :CI] of a m a t r i x {C,j} will be defined
as for a n*-vector C e C ~', n a m e l y b y ]iC:i = m a x ]C,j i. I t is a simple classical
result (see, for e x a m p l e , ref. (2)) t h a t , for a n y tixed y, the c o m p l e m e n t of .(2v
in ~ is dense a n d open and t hat~ its relative Lebcsgue m e a s u r e tends ~o zero
with y. Thus for a l m o s t all frequencies ), E [2 one can find a positive 7 such
t h a t 2 e ~v. Because of t h e relation .Q~, c ~Q~, if 7 < Y', it will be no restriction to ~ake y < 1. One is led to t h e d i o p h a u t i n c condition defining f2v, b y
the p r o b l e m of solving a. p a r t i a l differential equation of the f o r m
.8/v
20~
G. B E N J ~ T T I N , L . OA.LGA.NI, A. G I O R G I L L I
and
J.-M.
8TP~L~Y~*
where t h e unknown function 2' a n d the given function G are defined on t h e
torus T" a n d G has v a n i s h i n g average. An analogous a r i t h m e t i c a l condition in
a different b u t strongly rela.ted p r o b l e m was first i n t r o d u c e d b y SrEOEL (~~).
L e t us fix some m o r e notations. Being i n t e r e s t e d in the analytic case,
we will h a v e to consider complex extensions of subsets (:f R -~" and of real
a n a l y t i c functions defined there. Since we h ' w e fixed p * e B a n d a positive
~ 1 so small t h a t 1;he real closed ball of radius ~ c e n t r e d at p* is c o n t a i n e d
in B, a c e n t r a l role will be p l a y e d b y ~he subsets Do,,. of C ~" defined b y
(1.2)
D o . , . == {(p, q) e C~"; lip -
p * ..I : : ~ ,
IlImql[<~},
w h e r e I m q - - - - ( I m q , , . . . , h n q , ) . F o r w h a t concerns functions, let ~Q.,. be
t h e set of all complex continuous functions defined on De.,., a n a l y t i c in t h e
interior of D o . , , which h a v e period 2~ in q~, ..., q, and arc real for real vMues
of t h e v~riables. I f / e ~/~ ~., its n o r m is t h e n t a k e n to bc [] 'q,~. ---- sup I](x)l;
m
zEDo 1~*
in the case of functions ] = (],, ..., 1,) with values in C', we also write ] ~ .~/,,,.
if / , e ~ , / , ~ . (i ----1, ..., n), and we set [!/iio,,. = m a x IIJ,IIQ.~*. In a g r e e m e n t
,
f;
w i t h t h e convent.ion m a d e above, if C is a n X n m a l r i x whose e l e m e n t s C ,
belong to ~r '
w e set IIC,,q,~. = mGta x liC,~',le,,.; thus, iu p a r t i c u l a r , for a n y
v e C" one has t h e inequality ]ICv[IQ.~.,;nl]Cl]o,,Hv 1. F o r notational simplicity
we also set D o - - D o , o , Mo------do,o, ]].]~ = ]]llo,o. Averaging over angles will
be d e n o t e d b y a b a r : n a m e l y , if J c= ~o.~., t h e n we denote
2~
(1.3)
](p*)
-
9~
]
...
,
0
0
Functions ](p), ](q) of the actions only or of t h e angles only will be t h o u g h t
of as defined in. Dq,~. b y trivial extension.
L e t us fin~flly r e m a r k t h a t , for the given t t a m i l t o n i a n H ~ s/q,,., one can
a s s u m e ]]H]:o,. < .l; indeed this can always be o b t a i n e d b y t h e change of
variables (p, q) ~ (up, q) with a suitable posit~ive a, which leaves the equations
of motion in I-Iamiltonian form with a new ttanfiltonia.n H ' = - a l l .
1 " 3 . - K o l m o g o r o v ' s t h e o r e m c~n t h e n be s t a t e d in tlle following w a y :
Theorem 1. Consider t h e I I a m i l t o n i a n H(p, q)---He(p)
in B X T" and, a f t e r having fixed p * e B, denote
(1.4)
). =
~o(p*) =
P~H ~
~
O/)
(p*),
~e>,
c ~ := ~
(13) C. L. SIEGEL: A~n. Math., 43, 607 (1942).
(p*) -
~,:H ~
H'(p, q) defined
~p, ~p, (p*) .
A P R O O F OF K O L M O G O R O V ' 3 T H E O R E M ON IN'VARIANT T O R I "F=TC.
205
Assume t h e r e exist positive n u m b e r s 77 ~, d ~ 1 such t h a t
(.1.5)
i)
). e ~ y ,
(1.6)
ii)
H o, H ' e d e , , . ,
(1.7)
iii)
dliv ~ < ilC*vJi <d-liiv][
for a n y v ~ C"
and, m o r e o v e r , I!H[:l.p. <1. T h e n t h e r e exist positive m m i b e r s E a n d ~' with
Q'<~ such t h a t , if the n o r m ][H 1 Io.o* of t h e p e r t u r b a t i o n H 1 satisfies
(1.s)
[n'll,.,. <
~,
one can c o n s t r u c t a canonical a n a l y t i c change of variables (p, q) = ~(P, Q),
,p: D ~ , - ~ D o , , . , ~ ~ ~ , ,
which brings t h e t t a m i l t o n i a n H into t h e f o r m
H': Ho~ given by
(1.9)
H'(}', Q) = (HoyJ)(P, Q) = a + ; t . P 4-/~(/>, Q);
here a e R a n d the r e m a i n d e r R e x / d
]P]l t. I n particular, one can t a k e
is, as a f u n c t i o n of P , of t h e order
E -~ c, Tl d * ~s(.+l~
(1.10)
where
(1.H)
c.
\5~/
\4n+l/\n+1]
"
T h e change of variables is n e a r t h e identity, in t h e sense t h a t
i!~ - - i d e n t i t y l l d - + O,
as I]H~i!~,,. -~ O .
1"4. - L e t us m)w add some c o m m e n t s on t h e i n t e r p r e t a t i o n of K o l m c g o roves theorem. I n classical p e r t u r b a t i o n theory one a t t e m p t e d a t c o n s t r u c t i n g
a canonical change of variables with t h e a i m of eliminating t h e angles Q in
t h e new H a m i l t o n i a n H ' ; in such a way t h e p h a s e space would t u r n out to
be foliat,ed into i n v a r i a n t tori P -- const, which is possible only for t h e exceptional class of integrable systems. As one sees, in t h e m e t h o d of K o h n o g o r o v
one looks inste~d~ in correspondence with a n y good frequency ), (i.e. a n y
). ~ f2~ for some 7), for ~ related change of variables which eliminates t h e
angles Q only at first order in D. I n such a way, one renounces to h a v e t h e
general solution P(t) -~ p 0 Q(t) ~ Qo _~ co(1:,o)t (mod2~) corresponding to all
14 - I I N a c r e Cim~nto B .
20~
G. BRN]gTTIN, L. GALG&NI, A. OIOROTT.TJ a n d J.*M. STRELCYN
initial data (po, QO) ~ Dq., b u t one gels, however, for a n y good f r e q u e n c y 2,
the particular solutions P(t) ~-- po, Q(t) ~ Q ~ 2t (mod2g) for all initial d a t a
(/~, Qo), with / ~ - 0 ,
Q~ T"; this is immediately seen by writing down t h e
equations of motion for H', namely /5 ~ --~R/~Q, Q ~ ). + &R/~p and using
the fact t h a t R ( P , Q) i~. of order I[PI]z. Thus with the H ~ m i t o n i a n in the
new form H ' , a d a p t e d to the chosen frequency 2, one is not g u a r a n t e e d
to have a foliation into i n v a r i a n t tort, b u t one just ~ees b y inspection
the invariance of one torus supporting quasi-periodic motions with angular
f r e q u e n c y 2. I n terms of the original variables (p, q) canonically conjugated
to (P, Q) by the mapping yJ, this torus is described b y t h e p a r a m e t r i c equation~ (p, q ) ~ yJ(O,Q), Q e T ~, and is invariant for the H a m i l t o n i a n flow
induced b y H. Thi~ t~rus is a small p e r t u r b a t i o n of t h e torus p = p*, q E T ",
which is invariant for the Hamiltonian H 0, supporting quasi-periodic motions
with the same angtflar frequency.
In such a way, having fixed a positive y < 1, to ~ny p * e B with a good
frequency )~ ~- w(p*) e Y2v (and also satislying the two /urther conditions o] theorem 1), one can associate ~ t o m s which i~ invariant for the original Hamiltonian H suppol%ing quasi-periodic motions with f r e q u e n c y ~. :Now, as in
v i r t u e of the nondegeneracy condition t h e mapping r
co(B) is a
diffeomorphism, one has t h a t t h e ~et of points {p* e B ; co(p*)Ctgv} has a
Lebesgue measure which tends to zero as Y --~ 0. This r e m a r k suggests tha.t
t h e 2n-dimensional Lcbesgue measure of the set of tort whose existence is
g u a r a n t e e d b y Kolmogorov's theorem is positive and t h a t the measure of its
complement in B • T ~ tends to zero as t h e size of the p e r t u r b a t i o n tends to
zero. Actually this fact, which was ah'eady stated in t h e original paper of
Kolmogmov, was p r o v e d b y A~NOLD (see also (~o) for a r e c e n t i m p r o v e m e n t
and (~)) using a v a r i a n t of Kohnogorov's method, but will not be p r o v e d in
t h c present paper.
2. - R e f o r m u l a t i o n o f t h e t h e o r e m .
2"1. - The proof of the t,heorem starts with a trivial r e a r r a n g e m e n t of the
t t a m i l t o u i a n which reduces it to the form actually considered b y KOL~O~0ROV.
Indeed, after a translation of p* to t h e origin, by u Taylor expansion in p
one can write the Hamiltonian H in t h e form
(2.1)
H(p, q) = a -~- A(q) 4- [). ~- B(q)J.p + ~ ~ C , ( q ) p , p ~ - R(p, q),
where a e R is a constant which is uniquely defined by the condition ~ ---- 0,
while A, B,, C,, R E d s and 1~ is, as a function of p, of t h e order lip[j3.
207
A PROOF OF KOL.~IOOOROV'$ TIIEORE.~I ON INVARIAI~T TORI ]ETC.
One has clearly
a = Ho(O) + i i ' ( 0 ) = • ( 0 ) ,
A(q) = ~'(0, q ) - ~7,(o) = H(o, q) -- a ,
B,(q) = 8H1
ep--~ (0, q) ---- ~~H (0, q ) - - :t,,
(2.2)
(0, q).
Co(q) - - 8p, -~P4
This Hamiltonian H has t h e n already the w a n t e d form (1.9) of t h e o r e m 1,
a p~rt from t h e disturbing t e r m A(q) + B(q).p, which thus constitutes t h e
actual pertm'bation in t h e problem at hand and will be elinfinated in the following b y a sequence of canonical transformations.
2'2. - One is thus interested in giving estimates on A, B----{B~} and
C -- {Cu}. To this end we will make use of t h e familiar Cauchy's inequality,
which will also be used repeatedly in the f u t u r e sections. This inequality
will be used in our f r a m e w o r k in two forms: given ] e ~r a positive (~ <
and nonnegative int;egers kt, l, (i--~ 1, ..., n), then one has
(2.3)
[
~k,+.. +k~+z,+...+s,
k, ! ... k, ! l, ! ... I, !
i@~' ... @ 2 ~qi' ... ~.q':/(v, q)]<.~ ~,+ .~..,,+...,.
I:/1'.~
for all points (p, q ) e D ~ _ e , and
(2.4)
~.r,+. +~,
k~ ! ... k~ !
bp~, :.. @2 I(o, q) < ~ , ~ + ~
I:11!~
for all points (0, q) with IlIm(q)l! :;Q. The proof cart be given exactly along
the same lines as in t h e case of a p o l y d i s k (see, for example, ref. (u)).
I t is easily seen t h a t one has
(2.5)
m~x(l[Aiio , hBli~) < 2E/q
and t h a t t h e r e exists a positive n u m b e r m < 1 such t h a t one has
(2.6)
mllvll < 1:By',,
IlCvll,.<m-'llvll
for an v e C=;
precisely one can take
(2.7)
m -~ d / 2 .
(a4) B. A. Fuel,s: Introduction to the Theory of Analytic F~tnctions o/ Several Complex
Variables, Translations o] mathematical monographs, Vol. 8. (Providence, R. I., 1963).
G. BENETTINt L. GA.LGA-NI~ A. GIORGILLI and J.-M. ~TRELCYN
208
For what concerns (2.5), let us r e m a r k t h a t one has
"(2.8)
IIaHo<2fn'll~,
IIBII0<~ H n ' l , ,
the first one following immediately from the very definition (2.2) of A, while
the second one folJows from Cauchy's inequality (2.4) applied to
]/~ll, = max ~up
@ , (0, q) .
Inequality (2.5) then follows from Q,~I and from (1.8), which reads now
Coming now to (2.6) ~nd (2.7)~ let us remark t h a t , by definitions (2.2) of
C and (.1.4) of C*, one has
(2.9)
C ( q ) - C* ~--- / ~2H~
}
Lop, ~p, (o, q) ,
so t h a t from C~uehy's i n e q u a h t y (2.4) with our definition of the norms one
deduces
(2..10)
If(C- c * ) ~ i l ~ < ~ IIB'~ Ilvl.
By [IH~I!~~ E and the estimates (1.10) and (1..11) for E one has t h e n surely
also
(2.11)
II (c - c*) v I1~ < (a12)I!v II,
which guarantees
(2..12)
I:C*v] - ] ( c - c*)~ I1> (a/z):l~ll ~ o ,
as is seen by using J[.-~.[]< :J...l'-~ and (1.7). Using now the i d e n t i t y
(2.13)
C -~ C* ~ (C -- C*),
or also C .... C* ~- (C--C*)~ by (2.12) one gets
(2.~ 4)
I;Vv I; > Jli c* ~ II -
ll(C -
namely t,he first of (2.6) with m .-d/2.
(2.15)
c*) v Ill >
(d/2)ll v II,
Analogously, from (2.13) one gets
ljcvlt~ < IlC*~ll § II(C - c*)vl!~,
209
& P R O O F O F KOL~KOGOROV'S THEOR.E.~I ON I N ~ r A R I A N T T O R I ]~TC.
or with (1.7) a n d (2.11)
(2.16)
ll~vll~<(~-~+ d!2)'v, I < 2d-~llvll
2 " 3 . - I t is t h e n i m m e d i a t e to deduce t h e o r e m i from t h e following
Theorem 2. F o r g i v e n positive n u m b e r s y, Q, m < 1, consider t h e t t a m i l tonian 11(p~ q) ~ He(p, q) ~-11~(p, q) defined in t h e donmin De b y
(2.17)
11O(p, q) : a -- )..p ~- 89~_, C,j(q)p,p,~- R(p, q),
(2.18)
11~(p, q ) = A(q) + N' B,(q)p,
with ]1//11o<1; h e r e a e R , ]t~oQ:,, A, B~, 6'~j, R ~ ; ~ o a n d A -----0, while R
is, as a function of p, of t h e order ilpl]~. :For t h e m a t r i x C(q) = {C,(q)}, a s s u m e
(2.19)
mllvll < IlCvll,
:lCv:l~ < m-~l!vll
for a n y v e C " .
T h e n t h e r e exist positive n u m b e r s s a n d ~' w i t h o ' < ~ such that~ if
(2.20)
max(llAJIe, IIBII~) < ~,
one can c o n s t r u c t a canonical a n a l y t i c a l change of variables ~o:DQ,--+Dr,
~0 e ~r
which brings t h e t t a m i l t o n i a n 11 into t h e f o r m
(2.21)
tt'(t', Q) : (11o~)(P, Q) : a' --{- ~ . P + R'(P, Q),
where a ' e R, while t h e function R ' E dQ, is, as a function of P, of t h e order
I]PII~. T h e change of variables is n e a r t h e identity, in t h e sense t h a t
I1 - identitYl]~
o as [)H,t)o o.
I n particular, one can t a k e
(2.22)
m.o (
=
26A2U31
,
where
{711
(2.23)
A = 2(an + 1)' ~ ,
(2.24)
= 2,.+, ( n - ~ )
"+' .
T h e scheme of proof of t h e o r e m 2 is as follows. One p e r f o r m s a sequence
of canonical a n a l y t i c a l changes of variables such t h a t t h e disturbing t e r m H 1
210
G. B E N E T T I N ~
L . GALGA_NI~ A , G I O R G L L L I a n d
J.-M.
STRELCYN
1
of the tt~miltonian H at step k decreases with k, its norm 'i H~llq~
being essentially of Lhe order of H t '~
while the other parameters Q~ and m~ are kept.
controlled. The convergence of the scheme with H kie~-~0~
~'
~}~-+~>0
and m~-> m~ > 0, as k--> c% is then established. The iterative lemma and
the proof of convergence are given in sect. 4 and 5, respectively, while two
~uxiliary lemmas (giving suitable estimates in connection with ca.nonical
changes of variables defined by the Lie m e t h o d ~nd in connection with the use
of the diophantine condition ). e ~2v in solving a differential equation of the
form ~ ~.(~E/Oq~) - G with (~ ---- 0) will be recalled ia Lhe next section.
3. - Canonlcal transformations and small denominators, lemma.
3"1. For a given function )~(p, q), Z e .~%, let ;/~ be defined by
(3.1}
z ; = max
~ ,
~ .
and let. {., .} denote the Poisson bracket
{3.~)
{1, 9} = ~ ~ ; ~q, -~q, ~p, 9
Then, for 7., / e d ~ and a n y positive 5 < o, the inequalities
(3.3)
(3.4)
I1{~., I}!1.-, ....- 2n(z./~) Illl!.o,
$
2
II{ z, {z, I} }11o-. ::~,~(~,~ + ~)(z./~) !11,:
9
i:
are deduced from Cauchy~s inequality and from an enumcra'Aon of terms in
Poisson brackel s.
We come now to the definition of canonical transformations, or changes
of variables, near the identity by means of the Lie method, which has, with
respect to the standard method, the advantages of avoiding a n y use of the
implicit-function theorem and of providing the relevant estimates in a simple
way. In such a method a canonical trausformaiion (p, q) = q(/}, Q) is defined
through the solution at time one of a system of canonical differential equations
with Hamiltonian (here called ~generating function ~>) X; then t h e functions ]
defined in phase space change correspondingly by ] F-~ ~'] = ]o9~ and it is
well known t h a t one has formally ~ ] --= / -~- {Z, ]} + 89{ Z, {X,/} } + - . . ; i n d e c d
this is no~hing but Taylor's expansion of /, using d]/dt = {Z, ]}, i.e.
d~ (p(t), q(t)) = {Z, f} (p(t), q(t)).
A PROOle OF KOLMOGOROV'S T H E O R E M ON INVA/~IA_~IT TORI ETC.
9.1].
The existence of such a canonical transformation ~ and operator ~ and the
relevant estimates are then afforded by the foUowing
Zemma 1. Take an analytic function :~(p, q) defined in Dr whose derivatives
belong to ~r for a given positive 0, and consider the corresponding system of
canonical differential equations
(3.5)
p=
ez
8q '
4-ez
~p "
With Z: defined by (3.1), for a positive ~ < ~ assume
(3.6)
Z:<8/2.
Then for all initial data (P, Q) DQ_~ the solution (p, q) of (3.5) at t : 1 exists
in Do, thus defining a canonical transformation w:Da-~ -*Do, T e ~q-a. The
operator ~:~Q--~ ~r
with
'~11:1o~
(3.7)
is then well defined and one has the estimates
(3.8)
II~0-identitT II
<
and
I1 /11o- < II111o,
$
(3.9)
ll z/- llo-o<4n
lllllo,
Iiq,l--/--{Z,/}~o-o<i6n('2n
+ 1) (:~)'
t!fl[..
Proo]. First of all, using Cauchy's inequality (2.3), one guarantees that
in the s u b d o m a i n Dq_~/2 tile second members of system (3.5) have finite derivatives, so that the Lipschitz constant K of the system is finite in Dq_~/2 (precisely
one has K < 4nx*q/O). Moreover, as max(I]/i!l, 1141i)<~Z~< 6/2, if one takes
initial data (P, Q) in Dq_~, the standard existence and valiqueness theorem
guarantees that the corresponding solutions (p(t), q(t)) with (p(0), q(0))--~
= (P, Q) exist in Dq..6/~ for any t with 0 < t ~ l . Thus, for any such t, one has
a mapping ~t :Dq_~ --> DQ_~/~r D e with cf'(P, Q) = (p(t), q(t)) and, in particular,
the mapping ~ = ~ is also defined. By the standard existence theorem such
a mapping is analytic; furthermore, the periodicity in q~, ..., q, is immediately
checked. Finally, the mapping ~0 is well known to be canonical, being the
, time-one solution, of a canonical system.
212
G. B]~N~.TTIN~ L. GA.LG/tNI~ A. GIORGI:LLI
L e t us now come to t h e e s t i m a t e s (3.8)
an i m m e d i a t e consequence of t h e m e a n - v a l u e
for a n y t in 0 < t < l , one has ]~0'-- identity]q,
trivial. The second a n d the t h i r d ones follow
first and second order, respectively, n a m e l y
={z'l}l'''at
J . - M . STR~'LCYN
and (3.9). E s t i m a t e
t h e o r e m , according to
~:Z:, while the first of
from T a y l o r ' s f o r m u l a
d!=]
dt
and
_]
2 dff :t"
2
(3.8) is
which,
(3.9) is
for ] of
{z, {z, 1}}1,.
with 0 < t', t " < 1. R e m a r k i n g t h a t , as recalled above, one has T(P, Q) E Dq_x,
if 0 < t < l
and ( P , Q ) ~ D Q _ ~ , one h a s t h e n to e s t i m a t e II{Z,/}II~x~ a n 4
I]{;~, {;6/}}[[Q-~/2. E s t i m a t e s (3.9) t h e n follow b y (3.3) a n d (3.4).
3"2. I n t h e course of t h e proof of t h e o r e m 2 a f u n d a m e n t a l role is p l a y e d
by the possibility of solving a differenti'd equation of t h e f o r m ~ 2 d S F / ~ q , ) : G
for functions F a n d G defined on t h e torus 7" if 2 is a ((good f r e q u e n c y ,>.
Zemma
(3.10)
2. Consider the e q u a t i o n
~F
~ ~,x-- = 6I
where Ir~ and G are functions defined on t h e torus T", a n d a s s m n e 2 ---- (21, ...
..., 2,) ~ ~2~ for some y > 0 a n d G e zCe for some positive ~ with G = 0. T h e n ,
for a n y positive 5 < 9, eq. (3.10) a d m i t s a unique solution F e ~/o-~ with
P = 0, and one has t h e e s t i m a t e s
(3j1)
(3j2)
!I-FII <
<r
IlOI ,
I1 11 ,
wheJe a = a(n) is defined as in (2.24).
This is a well-established result. F i r s t of all, a f o r m a l solution in t e r m s
of F o u r i e r eoeff.cients ]~ a n d g~ (0 =/= k e Z ' ) of F a n d G, respectively, is imm e d i a t e l y obtained, b y ],. = ig~(~.k) -1 if one has go = 0, i.e. G = O. T h e
convergence is t h e n easily established, as t h e coefficients g~ decay exponentially
with k in v i r t u e of t h e a n a l y t i c i t y of G, while ]2-k[ -1 grows at m e s t as a power
b y t h e d i o p h a u t i n e condition (1.1) imposed on ).. T h e details of t h e proof are
deferred to the appendix. O p t i m a l e s t i m a t e s were g i v e n m o r e r e c e n t l y b y
Rtiss~th.~-(*) with b-2. a n d (~-(2,+t~ in {3.11)-(3..12) replaced b y (~-" a n d
5-t"+~), respectively, and a replaced b y a a = 24"+~n!(2 ~ - 1) -1.
A PROOF OF
KOLMOGOROV'S TII]~OREM
213
ON I N ' V A R I A N T T O R I ETC.
4. - T h e i t e r a t i v e l e m m a .
4 " 1 . - As a n t i c i p a t e d in t h e introduction, one applies now a sequence of
canonic'd t r a n s f o r m a t i o n s in order lo eliminate t h e disturbing t e r m H ~ in
t h e original H a m i l t o n i ~ n H considered in t h e o r e m 2. To this end we will
r e p e a t e d l y a p p l y t h e following
Lemma 3. :For ,given p o s i t i v e n u m b e r s y, o, m, e < 1 a n d o. < 0, m . < m
consider t.he H a m i l t o n i a n H(p, q ) - - H ~
q)
H~(p, q) defined in Do b y
ttO(p, q) -- a + ,t.p .f- 89~ C,(q)p,p,-f- R(p, q),
(4.])
~'J
B ' ( p , q) = A(q) ~- ~ B,(q)p,
with ] ! H ! ] o < l , where a e R , ~e~2~,, A, B~, C;~, l ~ e , ~ ' ~ ,
of order ':P'i'- Assume, f u r t h e r m o r e ,
(4.2)
m',vll < [](~v] ,
(4.3)
max(ll A
ilo,
A=0
,Cv.:e< m4.1v][
a n d R is
for any v e C " ,
,Bile) < e .
F o r a n y positive h so small Chat O - - 3,5 "> C~,, let t h e q u a n t i t y ~7 be detined
by
(4.4)
rl - A m '~d~'
~"
z=4n+3,
where A ---- A(n, 7) is the c o n s t ' m t defined b y (2.23), na.mely
(4..~)
A = '_,(an + l ) ~ - ~ ,
7"
a n d a s s u m e t h a t ~: is so small t h a t
m -- n~ilQ2.> m. .
(4.6)
T h e n one can find an analytical canonical cha.nge of variables, ~o:De_ao --> De,
~0 ~ de_3~ , such t h a t t h e t r a n s f o r m e d H a m i l t o n i a n H ' --: q/H = H o ~ can be
decomposed in a way analogous to / / with corresponding p r i m e d quantities
a', A', B', C' ~nd R', b u t with t h e s,n.me ~, a n d satisfies analogous conditions
w i t h positive p a r a m e t e I s 0', m', e ' < 1 given b y
Q' -= 0 - - 3 ~ >
(4.7)
n
e'
=
~'2!Q,, '
~.,
"*
214
O. B:EN-ETTIN, L. O A L G t ~ I I , A. G I O R O I L L I a n d
with, m o r e o v e r , HH'][~.< 1 .
J . - M . STRELCYN
One has, f m ' t h e r m o r e , for a n y ] e zCQ
(4.8)
II~1-
IHo, < ~
II111~ 9
4"2. - Proof. a) I f one p e r f o r m s a canonical change of variables with
, g e n e r a t i n g f u n c t i o n , Z in t h e sense of l e m m a 1, one obtains in place of H
t h e new H a m i l t o n i a n H ' = r
a n d one can m a k e a decomposition H'---= 8 '~ q - H '~ in a w a y analogous to t h e decomposition H = 8 ~ q - H ~ ; precisely, using again t h e s y m b o l (p, q) i n s t e a d of (P, Q) for a p o i n t of t h e new
d o m a i n , one has
H 'o-- a'-[- ),.p + ~e ~_, C~(q)P,P, ~ R'(p, q),
H ' ~ = A'(q) ~, B'(q).p ,
where
a'
A'(q)
(4.9)
=//'(0),
H'(O, q) - - a'
B~(q) = OH'
~p-- (0, q) - - 2~,
~2H'
c;,(q) = ~p~-~p-. (o, q)
and R ' is of order '!pill I n t h e spirit of p e r t u r b a t i o n theory, one thinks t h a t
b o t h H ~ a n d the g e n e r a t i n g function Z are of first order a n d one chooses Z
in order to eliminate t h e undesired t e r m s of t h e s a m e order in t h e new H a m i l tonian H ' . To this end one first writes t h e i d e n t i t y
(4.10)
H' - ~ H = 8 ~ + H~ + {z, Ro} + [{z, m } + ~ -
n - {z, ~ } ] ,
where in t h e expression [...] are isolated t h e t e r m s t h a t h a v e to be considered
of second order, in a g r e e m e n t w i t h t h e e s t i m a t e given in t h e t h i r d of (3.9).
T h e n one tries to choose g in such a way t h a t t h e first-order t e r m s in H',
n a m e l y H 1 "- {Z,//o}, do not c o n t r i b u t e to H ' I ; this is obtained b y imposing
H ' ~- {;~, H ~ = c - - 0(l!p!!*), where c is a constant.
b) Following KOLbIOGOR0V, We show t h a t this condition is m e t b y a
g e n e r a t i n g function Z of t h e f o r m
(4.11)
Z = ~'q -}- X(q) -4- ~, :Y,(q)p, ,
t
where the c o n s t a n t ~ e R" a n d t h e functions X(q), :Y~(q) (of period 2rt in ql, ..., q.)
215
A P R O O F OF K O L M O G O R O V ' S TH:EOR~M ON II~-VARI/~IT T O R I ETC.
have to be suitably determined. A straightforward calculation gives, b y
recalling the definitions (2.17)-(2.18) of H ~ and H ~,
(4.12)
H'4- {Z,H ~ =--~,2,4-a(q)--~2,
,
~q-~+
~X
~ Y,]
Thus it is sufficient to impose
.
~X
(4.13)
~ Z, ~-~ :
(4.14)
Z
. ~Y,.
A(q),
(
-
~q,]
,
i =
i, ..., n .
B y the smalJ denominators, lemma 2, eq. (4.13) in t h e unknown X ' can be
solved, as A = 0. Then one has to d e t e r m i n e the unknown constant ~ in
such a way t h a t t h e mean value of the r.h.s, of (4.14) vanish. This leads to
a linear equation for ~, which in compact notation can be written as
~X
and such an equation can bc solved by (4.2), which g u a r a n | e e s det C ~ 0.
:Equation (4.14) in the unknown l 7 can then be solved too.
c) The existence of the wanted generating function Z is thus ascertained
and one remains with the problem of specific estimates. By a twofold application of l e m m a 2, one obtains by easy calculations the following
Main estimate. Equations (4.13)-(4.15) in the unknowns }, X(q) and Yj(q),
which define by (4.11) t h e generating function Z, can be solved with X,
Y~e ~ , 5 = - - 2(~ for any positive b < @/2, and, for the q u a n t i t y Z; defined
b y (3.1), one gets
0.2
(4.16)
8
Z~,<(4n + 1) 7~ m, 5,,+ ,
or equivalently
(4.1.7)
Z~<
7
1
2(4n -}- 1) ~/"
The details are deferred to t h e end of the present section.
Remarking now t h a t condition (4.6) evidently implies ~ < 1, from (4.17)
one gets Z~ < 0/2, so t h a t lemma 1 can be applied and Z generates a canonical
transformation with domain D~,, where ~ ' ~ 6 - - 6 ~ ~ - 3&
216
o.
BI~N-ETTIN, L . ( } A L G A N I , A. G I O R G I L L I
and
j.-M.
8TRELCYN
d) We come now to estimates (4.7)-(4.8) and ][H'][Q,<I. The inequality ]]H'[o, :/[[H[]q<I is just the first of (3.9). L e t us t h e n come to #.
For wha~ concerns A', by definition (4.9), using expression (4.10) for H '
and rec~flling thai, in virtue of the choice of Z, only the last term contributes to A', one gets i!A' o,<2]]{Z,m} +
H- r-fz, n}l]o,. Using now
(3.3) and the thb'd of (3.9), together with llml',~<Kn+l)~, I:Ri~<l,
9' ---- 6 - - 3, :~nd (4.17), one gets
9
~ 35"
e X~
< ( 4 n + l ) y a ~m
3 q._ 16n(2n q-- 1)
< [ 1 6 n ( 2 n q - 1 ) q - 1 ] 2(4
-1)
K
<~'12,
where (4.17) was used together with
0-2
~n(n+ 3)< (4~ + 1)-~m'~7
and
16n(2n q- 1) -}- I ~ 2(4n q- 1 )2.
1
2 . Concerning B', for the same reason, from (4.9) and
Thus one has [A t [~,~
(4.10) and recalling (~cHo/~:p,)(O, q ) : = 2~, using Cauchy's inequality (2.4) one
has
ilB'l[.o'< i[{z,H'} + ~ - - ~ - - { Z ,
~7~
~..
Rki!o. < ~ < 2~,"
In conclusion, one has max(iA'][o,, [B'[i~,) < ~' with 0'----o--33 and
(4.18)
s ' = ~/*/~, .
For what concerns inequality (4.8), from the second of (3.9), with ~ in
place of Q, one has
(4.19)
il~] -IIIo' <
llllI~,
from which (4.8) follows by using (4.17) and [[]~$ < H/[16.
L e t us finally come to the estimate for m'. To this end, we recall first t h a t ,
by thc definition of C and C', one has
•92
G,(q) -- Gdq) - ~p, ~p, [~zE-- lr](0, q).
217
A PROOF OF KOLMOGOROV~S THEOREM ON INVARIAN~T TORI :ETC.
ThtL% from Cauchy's inequality (2.4) and (4.8), one gets
(4.20)
IlC',-
c,,~o, < 2 I~.H
HIIo,<~,
-
which gives
(4.21.)
I[(c'-
c) ,, II~, < (n,#e:)!j v II 9
Thus from (4.2) and (4.21), using a.lso II.-:.ll <:1...1~ ,, one has
IIC~v~ -
',l(C'- C)vll >(m-
n,~/d)Hvll > o
in virtue of condition (4.6) a nd~ moreover, from the i d e n t i t y C ' = C -{-- (C'-- C),
(4.22)
IIC'v", > (,~ -
Furthermore, one has
~v/Q~.)llv',i 9
[[C'vil,<lICvllQ,-~ ji(C'--C)vil,,
or, with ( 4 . 2 1 ) a n d
!ICv',l~, < IICvll~ < m-'l',v:!,
(4.23)
iiC'v:;Q, < (m -t- nv/e~)-'[vll
.
Thus with the trivial inequality a - * _ b < ( a - - b )
obtains
(4.24)
IIC"vll~.<
(m -
-~ for 0 < b < a < ]
one
nv/e,~)-':]'~ll,
so t h a t the estimatc for m' is proven.
e) To conclude the proof, we give now the main estimate (4.]6). Firsl~
by lemma 2 applied to eq. (4.13) with ]A!io< ~, one finds
(~t.~5)
IlX I1.~-,,<~,
8X
1
fie
-~
as
~_~<76T.+1 9
Coming to eq. (4.15) we first establish
(4.26)
IB -- G ~X
< 2 7m ~+
' ---~o '
making use of
lIB il,-, <~,
and of a/(ym~ ~'+1) > 11. Thus,
<
II -a- 11o-,
by I!.ZII < II...;I,-~, from (4.15) and
Ilf,I <
21~
o. BEN'ETTIN, L. GKLGANI, A. GIORGrLLI &nd J.-M. STRELC'~N
< m - ' l [ ~ l ! , i.e. t h e definition of m, one gets
ll~ll
(4.27)
<~
G8
ym ~ ~,~+~ 9
Coming t h e n to eq. (4.14], in order to estimate the r.h.s, we r e m a r k t h a t
I, o~~,1~, ~-II- ~~I~ ..+o~
so t h a t , b y (4.26), (4.27) with m , ~ 1, one has
,428,
]l~-~~- ~
I
~
- e~ . - . < 4 ~ m - s - ~ i 9
A direct application of lenmla 2 t h e n gives
II~II,_,~<4~,2 m ~'~
s ~,.+I,
I~~'y ,_~<4pm ~'~
s 54.+~ '
As a consequence, recalling definition (3.1) of Z; (with ~ - ~ Q - - 2 6 ) , b y
definition (4.11) of X which gives
1~1 ,,~,,, {~,
I~
i~,1 ~,
and using ~, y, ~, m < 1 with 3a -* ~ 1, one gets
O'z~
(4.29)
Z ~ < ( 4 n + 1) 7 , m , 6 , . , ' .
The main estimate has thus been checked and the proof of ]emma 3 completed.
5.
-
Conclusion of the proof.
5"1. - One has now to apply repeatedly t h e iterative lemma 3 in order to
eliminate t h e p e r t u r b a t i o n H I ( p , q) ---- A(q) --k B ( q ) . p . Thus, starting from
given positis'e numbers ~, m, E < 1, which we denote now b y eo, too, ~0, b y
assigning ~ sequence {3k}k~ one can define recttrsivcly Q~+I, m~+~, ~.~ by relations analogous to (4.7), which we write now in the form
r
~,+1 = ~ k - - 3 6 , ,
(5.1)
~ ~:+1
--
,:
e, \ m s ~64
,
m.§ = m , - (riM,)~L,,
219
A P R O O F OF K O L ~ O G O R O Y ' S TI~EORE3I O.N" I N Y A R I A N ~ T O R I ZTO.
where definition (4.4) of ~ was used and, moreover, in t h e rel.ttion between
m' and m, ~ was expressed through d. One has, however, to satisfy the
two consistency conditions ~,+~ > 0 , , m**~ > m . (k = 0, 1, ...), where ~. < ~
and m . < m o are a r b i t r a r y positive numbers. I n v i r t u e of
0|
=
~ o - - ~ 3~,
m|
and
==
mo-- (n/Qt,) ~ ~lt+l
-u*
__
/e,=o
k,,,o
these consistency conditions are satisfied if one guarantees
(5.2)
3 ~ ~<
~o-- ~*,
k--O
(5.3)
/r
We notice, however, ttiat t h e relation connecting e,+l with m,~ 5, a n d t ,
can also b~ r e a d as defining 6, if e,+~ is given; precisely one has
A~
(5A)
e~
~ ' = q,m~~+,"
~oLice t h a t m~ is defined in terms of e~ and m0 b y t h e last of (5.1). Thus one
can t h i n k of defining arbitrarily the sequence {e~}, instead of the sequence
{5~}. Makhag also She position
(5.5)
k
e~-= e~eo,
=
0r 19 21 ... (Co--= 1) ,
one has thus to define ~ sequence of positive numbers v~ with eo = 1 so t h a t
t h e two series
(5.6)
s~=
~
k,.-0
k..,o
k--O
k--O
/ - 2 :--
xl]2V
t%1%+1)
---- s ,
converge. T h e n the two consistency conditions (5.2} and (5.3) determine eo;
indeed (5.3) gives
(5.8)
~o<:
e~(mo-- m,)~
while (5.2), b y using m ~ > rn., gives
(5.9)
t, n2
,
220
G. B E N E T T I N , L. GA.LGANI~ A. G I O R G I L L I a n d
J . - M . STRELCYN
A rough e s t i m a t e for eo is o b t a i n e d as follows. Choose c~ : 2-~'~; this
gives Sk --- 2"2 -~ a n d ta = 2 -'t~+~, so t h a t s = 4 a n d t = 1 / ( 2 " - - 1) < 1. F o r
the two up to now a r b i t r a r y n u m b e r s 0, and m , m a k e the simple choice
0, = 00/13 xnd m"* = m~o/2. This ens~zre~ t h a t the r.h.s, of (5.8) is larger t h a n
t h a t of (5.9)~ so thai; we conclude t h a t t h e ~,wo consistency conditions (5.2)
~nd (5.3) are su~;isfied if ore} assumes, for e x a m p l e ,
_ eomg leA"
(~.]0)
~~ ~6-;i; ~]~)
"
5"2. - We finally c o m e to t h e c o n v e r g e n c e of t h e sequence of canonical
t r a n s f o r m a t i o n s , thus giving t h e
P r o p / o/ theorem 2. S t a r t i n g f r o m t h e H a m i l t o n i a n He : H defined in De,
with IIHo/l~,.-.-], characlerized b y positive p a r a m e t e r s y~ ~ , me "< 1, consider
t h e q u a n t i t y ~:o satisfying (5.10) a n d a s s u m e lnaX(IA]i.%, ][B][~.)< Co. T h e n
one can a p p l y recursively t h e i t e r a t i v e lemm'~ 3, defining at each step k > l
a canonica.1 tr,~nsformation q~:Do~--> Dq~_,, with t h e corresponding o p e r a t o r
q/~:Jo,--* .~.,;
f u r t h e r m o r e , frmn (4.7) a n d (4.8) one has t h e e s t i m a t e
(5.11)
II~d
-
t[:~, < *7~-1,lf I~_,
with ~/~_,-~. ~, so that, in particular, the series
~?k i.,: known to be converk=l
gent. We e~n now detine 1;he composite ca,nonical transformation ~ k : D ~ --* Do~
by ~ = ~Lo...oq~ "rod the corresponding eomposit.e operator ~ ' a / o . -> aa/0k,
defined by ~
: to93k, or equivalently by ~ : q / , , o ~ _ ~ ,
~---idenlAty.
Clearly, in order Lo p r o v e tim convergence of t h e sequence {~} of ca~nonical
t r a n s f o r m a t i o n s restricted to De , it is sufficient to p r o v e t h e convergence
of the corresponding sequmm(, {~k} of operators for e v e r y / e ago,. This in
t u r n is seen by r e m a r k i n g t h a t , f r o m (5.1.1) and t h e first of (3.9), one has
a n d so also, for a n y / > 1 ,
k+l--1
I(~ ,-
(5.12)
Thus, as the series
~z~.)/llo,, ~:: :]liloo ~:
~,.
~ converges, one deduces t h a t the sequence ~%] conk--1
verges uniformly, for a n y / e ~Jo.; b y Wcierstras.~' t h e o r e m one has t h e n
A PROOF
OF KOLMOGOROV~S THEOREM
ON I N V A R I A N T
TORI ETC.
221
f u r t h e r m o r e one immediately gets t h e estimate
(5.13)
li~. !
-
III~<
(e.~ ?o 0 li/ilo,,
where t is t h e constant defined b y (5.7). I n particular, for t h e g a m f l t o n i a n
H| --~ lim
one has H ~ = H~ - ~ - / / ~ , where b y construction H ~ ( p , q) ~k---*~a ~ , H ,
-~ A ~ ( q ) ~ B ~ ( q ) ' p := O. Finally the mapping qS~ = ~im q~ turns out to be
canonical again in virtue of Weierstrass ~ theorem, as a uniform limit of canonical mappings.
APPENDIX
Proof o f l e m m a 2.
As we are here interested in functions of the angles q only, in the deiinitions of Dq and dQ the actions p will be now considered as parameters and
completely disregarded. Moreover, in addition to our standard norm []k!] =
= m a x [k,I, we will also introduce the norm [k[---- ~ [kd. W e recall first two
i
i
e l e m e n t a r y properties concerning analytic functions on the torus T", and an
e l e m e n t a r y inequality, precisely we prove what follows:
i) if F ~ . ~ q ,
F(q)=
(A.1)
~ : / ~ e x p [ i k . q ] , t h e n for every k e Z ~ one has
Is < liFll~ exp [-- I~"le];
ii) suppose t h a t for some positive constants C and ~2 with 0 ~ 1 and
e v e r y k ~ Z" one has ]f~]< C exp [--]k]a], and consider t h e function 2~(q)=
-~ ~ f, exp [ik.q]; then for any positive 8 < ~ one has F e d e - t and
kr
iii) for ~ny K, s, 3 > 0 one has the inequality
(8).
(A.3)
.K'< ~
_Proof. i) B y definition one has
1
7-n
15 -
I ! N u o v o C,tmento 13.
exp [ K S ] .
222
G. BENETTII~-, L. GALGANI, A. GIORGILLI a n d
As F e ~r
]
1~ - (2a)"
J.-M. STRELCYN
one can suitably shift the integration paths and write
fF(q_ilr~ l q ' " " q " - i T ~k.l ~
, ~ exp
[_ikl(q~_ikg )]
~]0
dq,.., d q , ,
T"
0
where, by conve~tion, we m a y assume ~-~ = 0. One has t h e n
]
p*
ii) Let [ I m q l l < o - - ~, where Imq---- ( I r a q i , . . . , I m q . ) .
sup
Then one has
<
< c y. e~,p [ - Ikl~] e~p [Ikl(~ - 6 ) ]
1:~2's
C2 ~
~
exp
5
kj
=
= c y. e~p [ - IklS] <
keZ~
C2"
exp[--Sk]
=
kl~Os...,kn~'O
= (/9- (1 -- exp [-- 5])- < C
,
because 1 / ( 1 - e r p [--6])-< 2/6, for a n y positive 5 < 1, as is immediately
seen by comparing 1 - 6/2 and exp [--d].
iii) The considered inequality is equivalent to K J ) < c x p [ K S / s - - 1 ] ,
i.e. to x < e x p I x - - 1 ] , which is evidently true even for a n y real x.
We come then to the
Proof of the lemma. In terms of the Fourier coefficients fk an4 g~ of _~ and G,
respectively, one formally satisfies eq. (3.10) by
]~-- )..k '
(A.4)
wherc, by the condition 2 e -Qv, all denominators are, in particular, nonvanishing.
By G a-~1~, property i) and 2 e-Qv, one has then
I/k] < y-~]IGJI_~Ikl" exp [-- Ik[~],
(A.5)
where the inequality ][ktJ<Ikl has been used. F r o m inequality (A.3) with
K - ~ Ikl and s = n, for a n y positive 5 this gives
(~,)
n
Ihl <r-'llal]~
~
exp [ - Ikl(t~-- (~)] = C exp [--
where
c =
r-'l]Gll:
9
I~l(e -
6)],
A PROOF OF KOL~OGOROV'8 THEOR.~M" ON INVARIANT TORI ETC.
223
NOW, f o r a n y ~ < ~, one c a n m a k e u s e of p r o p e r t y ii) ( w i t h ~ - - ~ in p l a c e of o)
a n d one o b t a i n s for / V ( q ) _ - - ~ ] ,
e x p [ik.q] t h a t F e ~/~_~ a n d t h a t
14n\"
T h u s (3.11) is p r o v e d b y t.%king ~/2 in p l a c e of 0, b e c a u s e a > (16n/e)".
Le~ us f i n z l l y p r o v e (3.12). T h e F o u r i e r coefiicients of ~F/~q~ are of t h e
f o r m h r . ~ k~gJ(),.k) -~, ~ = ~, ..., n, so t h a t as for (A.5) o n e g e t s
B y iii) w i t h K----]k,]
a n d s == n q - 1 , t h i s g i v e s
(.
iI,h..,<~-~
e~ /
IIOll~ cxp [ - tkl(e" ~)],
so t h a t o n e o b t a i n s
40_
B y t a k i n g a ~ a i n 6/2 in p l a c e of ~, one t h u s g e t s (3.12) w i t h
9
RIASSUNTO
Nel prosonte lavoro si dimostra il teorema di Kolmogorov sull'esistenza di tori invarianti
in sistomi 1-familtoaiani quasi intcgrabili. Si usa Io schema di dimostrazionc di Kolmogorov, con la sola variante del modo in eui si definiscono le trasformazioni canoniche
prossimo all'identitY. Si usa infatti il metodo di Lie, ehe climina la necessits d'invcrsioni o quindi doIl'impiogo del teoroma delle funzioni implicite. Questo fatto tecnico
evita an iagrodiente spurio e somplifiea il modo in cui si ottiene una delle stime principali.
~]~OI~I3aTeJIbCTBO TeOpeMhl KOJ~Moroposa Ha m m a p s a n T m ~ x T o p a x , gCHO,]b3y~
K a a o m ~ e c K ~ c a p e o 6 p a 3 o B a H n ~ , o H p e ~ e ~ e n m , le c h o m e r o o m MeTO~Ia JIH.
Pe3mMe (*). - - B 3TO~ pa6oTe n p e ~ a r a e T c ~ ~oKa3aTen~TnO TeOpeM~I K o n M o r o p o B a o
cymeCTBOBalIIIH HHBapHaHTHLIX TOpOB B I~a3H-HHTC~'pHpyeMbIX FaMHYIIJTOHOBBIXCHCTCMaX.
Hcrionh3yeTcfl cxeMa lloKa3aTenbcTaa, npe~rIO~CH~da~ KOHMOFOpOBbtM, e~2HHC'rBCHHOe
OTHHqHe c o c r o r r r B cHoco6e, XOTOpbIM onpe~eaamTC~ KaHO~eCKHe I]peo6pa3oBaH~.
B 3TO~ pa6oTe riCHOYlb3yeTclI MeTOjI Jill, KOTOpbl~ HClCJIIOqagT HeO6XO/IJaMOCTb mmepcn~
H, Cae2tOBaTe~qbHO, HCnOIIb3OBah~e TeOpCMbI ~UIJI I~e$1BHOI~ ~yHKHHI4. ~TOT TeXHHqC~KI4~
npaeM llcKamqaeT ~o~rmn~ mtrp:ieHenT H yupomaeT noayqemae FJIaBHOI~ OI~CRKH.
(*) HepeeedeHo pe,gamlue(t.
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