Exponentially small estimates for KAM theorem near an elliptic equilibrium point
Abstract
We give a precise statement of KAM theorem for a Hamiltonian system in a neighborhood of an elliptic equilibrium point. If the frequencies of the elliptic point satisfy a Diophantine condition, with exponent $\tau$, and a nondegeneracy condition is fulfilled, we show that in a neighborhood of radius $r$ the measure of the complement of the KAM tori is exponentially small in $(1/r)^{1/(\tau+1)}$. This result is obtained by putting the system in Birkhoff normal form up to an appropriate order, and the key point relies on giving accurate estimates for its terms.
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EXPONENTIALLY SMALL ESTIMATES FOR KAM THEOREM NEAR AN ELLIPTIC EQUILIBRIUM POINT AMADEU DELSHAMS Departament de MatematicaAplicada I, Universitat Politecnica de Catalunya Diagonal 647, E-08028 Barcelona AND PERE GUTI ERREZ Departament de Matematica Aplicada II, Universitat Politecnica de Catalunya Pau Gargal lo 5, E-08028 Barcelona Abstract. We give a precise statement of KAM theorem for a Hamiltonian system in a neighb orho o d of an elliptic equilibrium p oint. If the frequencies of the elliptic p oint satisfy a Diophantine condition, with exp onent , and a nondegeneracy condition is fullled, we show that in a neighb orho o d of radius r the measure of the complement of the KAM tori is exp onentially small in (1 =r ) 1 = ( +1) . This result is obtained by putting the system in Birkho normal form up to an appropriate order, and the key p oint relies on giving accurate estimates for its terms. 1. Intro duction We consider a Hamiltonian, with n degrees of freedom, having the origin as an elliptic equilibrium p oint. In suitable canonical co ordinates, this system can b e put in the form H ( q p )= X s 2 H s ( q p ) (1) where H s is a homogeneous p olynomial of degree s in ( q p )forevery s 2, and H 2 ( q p )= 1 2 n X j =1 j q 2 j + p 2 j : (2) We are concerned with the existence of n -dimensional invariant tori in a neighborhood of the elliptic p oint. We rst see that the system (1{2) is nearly-integrable by putting it in Birkho normal form up to an appropriate degree K 4, provided the frequency vector =( 1 ::: n ) is nonresonantuptoorder K . A quantitativeversion of Birkho theorem, implicitly contained in 4], allows us to obtain estimates for the normal form. Like in 9], we consider action{angle variables in a neighb orho o d of radius r . Assuming a suitable nondegeneracy condition, we apply the known KAM theorem and show that most tra jectories in a neighb orho o d of radius r lie in invariant tori: we get for the relative measure of their complement an estimate of the typ e O r ( K ; 3) = 2 . We assume that satises a Diophantine condition: with given >n ; 1and > 0, j k j j k j 8 k 2 Z n nf 0 g (3)
403 where wewrite j k j = P n j =1 j k j j .Wesay to b e -Diophantine. Our main contribution is to show that in this case the estimates for the Birkho normal form allowustocho ose the degree K as a function of r , giving rise to an exp onentially small estimate of the typ e exp ( ; 1 r 1 = ( +1) ) for the measure of the complementoftheinvariantset.We remark that the estimates given in 4] do not allow to obtain the exp onent1 = ( + 1), but a worse one. Nevertheless, we shall see that an improvement of that estimates leads to the announced exp onent. 2. Estimates for the Birkho normal form Given K 4, assume that the frequency vector is nonresonantuptoorder K : k 6 =0 for k 2 Z n 0 < j k j K . The well-known Birkho theorem 1, 7] states that, in some neighb orho o d of the origin, there exists a canonical transformation ( K ) , near to the identity map, such that H ( K ) = H ( K ) is in Birkho normal form up to degree K : H ( K ) ( q p )= I + Z ( K ) ( I )+ R ( K ) ( q p )= h ( K ) ( I )+ R ( K ) ( q p ) (4) with Z ( K ) ( I )= X 4 s K s even Z s ( I ) R ( K ) ( q p )= X s K +1 R ( K ) s ( q p ) (5) where every Z s ( I ) (uniquely determined) is a homogeneous p olynomial of degree s= 2in the action variables I j = 1 2 q 2 j + p 2 j j =1 :::n and every R ( K ) s ( q p ) is a homogeneous p olynomial of degree s in ( q p ). Since h ( K ) ( I )is integrable, and in a neighb orho o d of radius r wehave R ( K ) = O r K +1 , it turns out that H ( K ) is a nearly-integrable Hamiltonian near the origin. However, to apply KAM theorem to H ( K ) we need quantitative estimates for its terms. As in 4], weintro duce the linear change to complex canonical co ordinates x j = 1 p 2 ( q j ; ip j ) y j = ; i p 2 ( q j + ip j ) j =1 :::n: Note that q , p are real if y = ix .We dene j ( x y ) j := max j =1 :::n q j x j j 2 + j y j j 2 .Given r> 0, the real and complex p olydisks of radius r centered at the origin will b e denoted B r and b B r , resp ectively. For a given homogeneous p olynomial f s ( x y )= X j l + m j = s f lm x l y m ,we dene the norm k f s k := X j l + m j = s j f lm j (we use the notation x l = x l 1 1 x l n n , y m = y m 1 1 y m n n ).
404 Prop osition 1 Let H ( x y )= P s 2 H s bea real Hamiltonian with H 2 = I , and assume that k H s k c s ; 2 d for s 3 . Given K 4 , assume that j k j K 8 k 2 Z n 0 < j k j K (6) with 0 < K 1 . Then, there exists a real canonical transformation ( K ) ,near to the identity map, such that H ( K ) = H ( K ) is in the Birkho normal form (4{5) up to degree K . With some constants c 1 , c 2 , one has: a) kZ s k c 2 c s ; 2 1 ( s ; 2)! 3 s ; 1 for 4 s K and s even. b) R ( K ) s c 2 c s ; 2 1 ( K ; 3)!( K ; 2) s ; K +1 3 K ; 1 s ; K +1 K for s K +1 . c) The transformation ( K ) is analytic on b B r K where we dene r K := K c 1 K : These estimates rely on the results obtained in 4], although a direct application would giveworse estimates, with s ; 3 K instead of 3 s ; 1 in the denominators. The improvement comes from the fact that, in the construction of the normal form, the only small divisors which app ear up to the obtainmentof Z s corresp ond to the orders 3 :::s ; 1. This is crucial in order to get the rightexponent in the estimates given in the last section. 3. Applying KAM theorem We rst recall a usual statement of KAM theorem. Let us consider a nearly-integrable Hamiltonian written in action{angle variables H ( I )= h ( I )+ f ( I ) with 2 T n and I 2G R n .To show that most of the tra jectories of H lie in n -dimensional invariant tori, one usually imp oses one of the following nondegeneracy conditions on the frequency map ! = r h : det @! @I ( I ) 6 =0 or det @! @I ( I ) ! ( I ) ! ( I ) > 0 ! 6 =0 for every I 2G .We call these conditions Kolmogorov nondegeneracy and isoenergetic nondegeneracy, resp ectively. We denote V ( G ) a complex neighb orho o d of radius r around G . Theorem 2 (KAM theorem) Consider the Hamiltonian H = h ( I )+ f ( I ) , analytic for 2 T n and I 2V ( G ) , with f of size " . Assume that ! = r h is Kolmogorov or isoenergetical ly nondegenerate on G .Let > 0 given. For some constants C 1 , C 2 , C 3 ,if " C 1 2 C 2 then there exists I T n G l led with n -dimensional invariant tori of H , satisfying mes ( T n G ) nI ] C 3 (diam G ) n ; 1 : (7)
405 For more detailed statements and pro ofs, see 8, 9, 2, 3, 5]. Given , it turns out that the invariant tori of H come from invariant tori of the unp erturb ed system h with frequencies satisfying a Diophantine condition of the typ e (3), with a xed . Cho osing p " , the measure of the complement in (7) b ecomes O ( p " ). Now, our aim is to apply KAM theorem to the Hamiltonian H ( K ) = h ( K ) + R ( K ) intro duced in (4{5). We put this Hamiltonian in action{angle variables byintro ducing the known canonical change q j = q 2 I j cos j p j = q 2 I j sin j j =1 :::n: However, KAM theorem cannot b e applied in a direct way b ecause the change to action{ angle variables is not analytic at the hyp erplanes I j = 0. Like in 9], this fact forces us to remove a neighborhood of these hyp erplanes. Thus, to obtain invariant tori in the neighborhood B r ,we consider for the action variables the domain G r := ( I 2 R n : I 2 j I j 1 r 2 2 ) with > 0(we use the notation I a to mean that I j a for j =1 :::n ). With a suitable choice of , this reduction of the domain do es not aect essentially the measure estimates given in the next prop osition. To apply KAM theorem to H ( K ) ,we also have to require that the frequency map ! ( K ) = r h ( K ) is Kolmogorov or iso energetically nondegenerate. In fact we only assume the nondegeneracy at the origin itself, since this suces to ensure it in a small neighborhood. The condition we imp ose involves the vector and the matrix A := @ 2 Z 4 @I 2 : Nevertheless, wepoint out that higher order conditions are also p ossible. Prop osition 3 In the same situation of proposition 1, assume also det A 6 =0 or det A > 0 ! 6 =0 : (8) Let r K denedasinpart (c) of proposition 1. For some constants c 3 , c 4 ,if 0 <r c 3 r K (9) then there exists T ( K ) r B r l led with invariant tori of H ( K ) , satisfying mes h B r nT ( K ) r i c 4 7 r r K ! ( K ; 3) = 2 mes B r : (10) This result is obtained applying KAM theorem on the domain V ( G r ), with " r K +1 and r ( K +1) = 2 . In fact, this is a more elab orated version of a result given in 9], where a measure estimate like (10) is obtained, also with the exp onent( K ; 3) = 2.
406 4. The Diophantine case Finally,we assume that the frequency vector satises the Diophantine condition (3) with given and .We then take K = K in (6) and hence condition (9) is fullled if we cho ose K ( =r ) 1 = ( +1) , leading to an exp onentially small estimate. Theorem 4 Let H ( x y )= P s 2 H s bea real Hamiltonian with H 2 = I , and assume k H s k c s ; 2 d for s 3 . Assume that is -Diophantine, with >n ; 1 and > 0 . Assume also one of the nondegeneracy conditions (8) .For some constants c 5 , c 6 , c 7 ,if 0 <r c 5 then there exists T r B r l led with invariant tori of H , satisfying mes B r n T r ] c 6 exp ( ; c 7 r 1 = ( +1) ) mes B r : A related result has b een announced in 6] where, for a xed KAM torus of a nearlyintegrable Hamiltonian, it is shown that in a neighb orho o d of radius r there exist many invariant tori, and the measure of their complement is exp onentially small in 1 =r . Acknowledgements This work has b een partially supp orted by the EC grant ERBCHRXCT940460. Research by Amadeu Delshams is also partially supp orted by the spanish grant DGICYT PB94{ 0215 and the catalan grant CIRIT GRQ93{1135, and researchbyPere Gutierrez is also partially supp orted by the U.P.C. grant PR9409. References 1. G. D. Birkho (1927). \Dynamical systems" . Am. Math. Soc. Col loq. Publ. 9 . American Mathematical So ciety, New York. 2. H. W. Bro er and G. B. Huitema (1991). \A pro of of the iso energetic KAM{theorem from the `ordinary' one". J. Di. Eq. 90 , 52{60 . 3. A. Delshams and P. Gutierrez (1995). \Eective stability and KAM theory". Submitted to J. Di. Eq. 4. A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simo (1989). \Eective stability for a Hamiltonian system near an elliptic equilibri um p oint, with an application to the restricted three b o dy problem". J. Di. Eq. 77 , 167{198 . 5. P.Gutierrez (1995). \Estabilitat efectiva i tors invariants de sistemes hamiltonian s quasi-integrabl es" . Do ctoral thesis, Universitat de Barcelona. 6. A. Morbidell i and A. Giorgill i (1994). \Sup erexp onenti al stability of KAM tori". To app ear in J. Stat. Phys. 7. J. Moser (1968). \Lectures on Hamiltonian systems". Memoirs Am. Math. Soc. 81 , 1{60 . 8. A. I. Neishtadt (1982). \Estimates in the Kolmogorov theorem on conservation of conditional l y p erio dic motions". J. Appl. Math. Mech. 45 , 766{772 . 9. J. Poschel (1982). \Integrability of Hamiltonian systems on Cantor sets". Comm. Pure Appl. Math. 35 , 653{696 .