A proof of Kolmogorov's theorem - Dipartimento

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[, 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)] (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~ 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~, 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, 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~