H.K. Moffatt- KAM-Theory
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8/3/2019 H.K. Moffatt- KAM-Theory
OBITUARY 719. V. STRASSEN,Algebraische Berechnungskomplexitat,Perspectives in mathematics (Birkhauser, Basel,
10. Y. STERNFELD,Dimension, superposition of functions on separation of points, in compact metric11. V. M . TIKHOMIROV,Widths of sets in function spaces and the theory o f best approxim ations,12. A. G . VITUSHKIN,On the 13th problem of Hilbert, Dokl. Akad. Nauk SSSR 95 (1954) 701-704.13. A. G. VITUSHKIN,heory of the transmission and processing of information (Pergamon Press, New14. A. G . VITUSHKINnd G.M. HENKIN,Linear superpositions of functions, Uspekhi Mat. Nauk 22
1984), pp. 509-550.spaces, Israel Math . J . 49 (1985) 13-53.Uspekhi M at . Nauk. 15 no . 3 (93) (1960) 81-120.
York, 1961; Russian edition: 1959).(1967) 77-124.
KAM-THEORYH. K . MOFFATTIn 1953 and 1954 Kolmogorov wrote two papers [1953c 1954al on the generaltheory of dynamical systems, with importan t applications to H am iltonian mechanics.Both papers were precisely four pages in length, the limit permitted by Dokludy Aku d.Nauk SSSR at that time. Their influence on the subsequent development of thesubject has however been out of all proportion to their length; indeed the secondpaper, whose title may be translated On the preservation of quasi-periodic orbitsunder a small change of Ham iltonian conta ins the essence of what has subsequentlycome to be known as KAM-theory (after Kolmogorov, Arnold and Moser).
Kolmogorov stated the first critical theorems in this field and outlined the essentialingredients in their pro of; it was left to V. I. Arnold [l] nd J. Moser  to com pletethe proofs and to extend somewhat the circumstances to which Kolmogorovstheorems apply. KAM-theory lies at the heart of recent new understanding of thephenom enon of chaos in Ham iltonian systems (see, for example, Percival  in theproceedings of the Royal Society Discussion Meeting on Dynamical Chaos held inFebruary 1987), and Kolmogorovs con tribution in 1954 may be seen, with the benefitof hindsight, as providing the most im portan t break through in this subject since thefundamental difficulties were first recognized by PoincarC  in 1892. I say w ith thebenefit of hindsight because it was not until the development of the high-speedcomputers of the 1970s and 1980s tha t the full significance of KA M-theory could beproperly appreciated. There were in fact rather few citations of Kolm ogorovs paperson this subject up to ab ou t 1970; and since then the papers of Arnold and Moser,being more accessible to English-speaking readers, are those that are most widelyknown. There can be no doub t however that Kolmogorov was the ultimate source ofinspiration for these new developments.It is worth noting that Kolmogorov presented an account of this work, referringto both of the papers mentioned above, in a lecture at the International C ongress ofMathem aticians held in Am sterdam 2-9 September 1954; this lecture, Generaltheory of dynamical systems and classical mechanics appeared (in Russian) inVol. 1 of the Proceedings of the Congress, published in 1957. Kolmogorovs famouspaper On the preservation of quasi-periodic orbits. ..was received by Dokludy on31 August 1954 (and it appeared before the year was out!). One may surmise that itwas partly the stimulus of preparing a n important invited lecture to the Inte rnationa lCongress that promoted the breakthrough for which Kolmogorov had alreadyprepared the ground in his earlier (1953) paper.
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72 ANDREI NIKOLAEVICH KOLMOGOROVWhat, then, was the nature of this breakthrough? Kolmogorov considers anautonomous Hamiltonian system with Hamiltonian H(q,,p,, 0) where a runs from 1to 5 (the number of degrees of freedom), and 0 is a perturbation parameter; heassumes fo r simplicity hat H i s an analytic function of its arguments, although it isclear that he recognises that this assumption is stronger than absolutely necessary.After some preliminary statement of notation? he states his Theorem 1 (referred tolater by Arnold as Kolmogorovs theorem - ee, for example, V. I. Arnold Mathe-matical methods of classical mechanics, Springer-Verlag, 1978), which in view of itsgreat historic interest we state here in full (in translation from the Russian original) :
where m and I , are constants such that for suitable constants c > 0 and1H( q , r ,O)= m + k ~ e + j @ES(q)papB+ O(b13) .a aB
q > 0 the inequality
is satisfied for all integer vectors n. Furthe rmore, let the determinant formed
Then there exist analytic functions &(Q,R, ) and Ga(Q,P, ) defined for allsufficiently small 8 and for all points (Q, P) n some neighbourhood V of theset T,, such that the associated contact transformation4,= Q ,+eF,(Q,P, 1, ~a = Pa+eG,(Q, P,
of V into V E G reduces H o the form
( M ( 0 )does not depend on Q and P).Recognizing that the import of this theorem may be lost on the inexpert reader,Kolmogorov immediately provides a vital word of explication which again is worthquoting in full:
It is easy to understand the importance of Theorem 1 for mechanics. Itshows that the s-parameter family of quasi-periodic motions
q, = t l a t + qp , p, = 0existing at 8 = 0 cannot disappear under conditions (2) and (3) as a resultof a small change in the Hamiltonian H: here is merely a shift of the s-
t Vector notation is used, for examp le,p = @A with scalar product fp , q) = Caqaqand M*= ( p , p ) .Thespace G is the product of an s-dimensional torus T and a domain S of W. It IS assumed tha t p = 0 iscontained in S; is the set of points in G for which p = 0. The Theorem 1 as quoted above contains anobvious misprint which is faithfully transcribed from the original Russian version.
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OBITUARY 73dimensional torus & which is covered by the trajectories of these motionsinto the torus P = O , which remains covered with trajectories of quasi-periodic motions with the same set of frequencies I , , . ,&.
This style is not untypical of Kolmogorov : a theorem stated in full formality andat some length, followed by an informal and illuminating indication of its realmeaning. This is further followed, not by a proof of the theorem (although it is hardto believe that Kolmogorov would have stated it as a theorem if he had not been100% confident of its provability) but by a summary of the procedure by which thecontact transformation ( q , p ) + (Q , P)may be constructed, in the course of whichdiscussion, the need for and meaning of the conditions (3) and (4 ) of the theorem ismade abundantly clear. For a system with two degrees of freedom, the condition (3)takes the formCn,A,+n A >-nlqwhich, with n, and n, integers, means that the frequency ratio A J I , must besufficiently irrational, a condition which appears also in the earlier (1953) paper. Ifresonances occur through vanishing of n , I , + n, I , (for any n,, n,) then Kolmogorovsprocedure fails, just as Poincarts attempt to analyse non-integrable systems byperturbation analysis had failed some 60 years before. But Kolmogorovs recognitionof the need for a condition of the type (*) was the crucial flash of insight that enabledvital progress to be made.
The torus P = 0 that survives the perturbation in the above theorem is of coursewhat later came to be known as the KAM-torus; and Kolmogorov argued further inhis (1954) paper that for small values of 6 he displaked tori obtained in accordancewith Theorem 1 fill the greater part of the region G, a statement which he thenrefined to a Theorem 2 (not a conjecture!) relating to the Lebesgue measure of the setof quasi-periodic orbits that survive the perturbation. And all this in four pages ! Thedegree of crystallization of thought in these four pages is truly remarkable and canrarely have been surpassed.References
1. V. I. ARNOLD,Small denominators 11, Proof of a theorem of A . N. Kolmogorov on the preservationof conditionally-periodic motion s under a small perturbation of the Hamiltonian ,Russian Mach.Surveys 18 (1963) no. 5, 9-36 (translated from Uspekhi Mat. Nauk 18 (1963) no. 5, 13-40).2. J . MOSER,On invariant curves o f area-preserving mappings of an annulus, Nachr. Akad. Wiss.Gottingen l(1962) 1-20.3. I . C. PERCIVAL,Chaos in hamiltonian system s, Proc. Roy. Soc. London A413 (1987) 131-143.4. H . P O I N C ~ ,es mdhodes nouvelles de la m6chanique cdleste (Gauthier-Villars, Pan s, 1892).
ENTROPY IN ERGODIC THEORY 2 THE INITIAL YEARSWILLIAMARRY
The years 1954-59 were especially fruitful even for one renowned for so manysingular contributions to mathematical research. There was profusion as always (infact 52 separate items are listed in Kolmogorovs bibliography, including books andother expository work); but more important, these were the years in which he:(i) proposed and solved his famous perturbation theorem for Hamiltonian
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74 ANDREI NIKOLAEVICH KOLMOGOROVsystems (the beginning of KA M -Kolmogorov, Arnold, Moser - heory [1954a, b]
(ii) extended Shannons work to con tinuous state processes (along w ithDobrushin, Gelfand, Pinsker, Yaglom and others) (see for example );(iii) introduced &-entropyand other related ideas into the problem of measuringthe massiveness of function spaces [1959b];(iv) made the first major inroads to Hilberts 13th problem concerning therepresentability of continuo us functions of several variables by the superposition offunctions of fewer variables (the reduction from three to two variables was made bythe 3rd year student Arnold  after which Kolmogorov reduced the variables fromtwo to one together with addition) [1957b, 1959bl;(v) introduced entrop y into ergodic theory and solved the principle outstandingproblem of that time [1958f, 1959al.
All but (i) above are concerned with some variation on the theme of entropy.It is a specifically ergodic theoretic point of view that ho lds together the problemsof stochastic processes and those of dynamical systems. But it was a specificallyKolmogorov point of view that enabled him to make a smooth transition betweenthese areas and Hilberts 13th problem in the sam e period. The prob lems of carryingforward Shannons ideas on entropy from finite state processes to continuous stateprocesses are natural enough and indeed had been probed by Shannon and hiscow orkers . But the successful application of these ideas to formally deterministicsystems represented a significant leap forw ard.Or perhaps one should speak of Kolmogorovs lead forward. For althoughKolmogorovs two papers [1958f, 1959al prepared the basic groundw ork for futuredevelopments, it was Sinai who shaped these ideas into a more serviceable form.Perhaps today when we witness a surfeit of papers and a number of significantresults relating to chaos and strange attractors the idea that deterministicdynamical systems can exhibit randomness or stochasticity is no longer surprising,but there is no dou bt tha t present day discoveries in these areas are the florescence ofthe seed sown by Kolmogorov.
It would be too much to claim that Kolmogorov foresaw the main features ofdevelopment in dynamical systems over the subsequent 30 years, but his papers onentropy in its various guises, combined with his emerging ideas on complexity, dosuggest a totally new point of view from which to understand dynamics. Today wemight say that, Laplace notwithstanding, a human scale of observation provides evendeterministic systems with the main features of stochastic processes and therebyimportant invariants. The invariants are computed from observations of the systemthrough a finitely partitioned lens, so to speak.Ergodic theory is concerned with the behaviour of discrete or continuous timedynamical systems for which there is a measure (volume) which remains invariantthrough the passage of time. In the discrete case such a dynamical system amountsto a single measure preserving transformation T :X +X together with its iteratesT = F - l o T. The measure m (which we assume to be a probability measure, thatis, m ( X ) = 1) enjoys the invariance property
J f l x ) dm = Jf lTx) dmwhenever f s integrable.