INTEGRABILITY : A DIFFICULT ANALYTICAL PROBLEM

Generically hamiltonian systems are nonintegrable . However there are few tools in order to prove that a given system is noninttegrable . Por two degrees of freedom the usual methods rely upon the appearance of trans versal homoclinic or heteroclinic orbits . The transversal character is shown through evaluation of integrals along orbits . Such computation requi res the knowledgement of a one parameter family of periodic orbits and an explicit solution for the unperturbed (integrable) case . Due to the dependence of the form exp(-C/£ k ) of the angle measuring transversality with respect to the perturbation parameter, none of the approximations of pertur bation theory is enough to establish nonintegrability . §1 . The meaning of integrability and nonintegrability . Let H(q,p) be a ha miltonian of n degrees of freedom . Por the sake of simplicity we take R2n as phase space . A first integral F of the associated hamiltonian system is a smooth function such that the Poisson bracket (F,H) is identically zero . Let Fi, j = 1-k smooth functions . We say that they are in involution if (Fi ,F j ) = 0 Vi,j . From now on we take F1 =H. A hamiltonian system is said integrable if there are n functionally independent smooth global functions in involution . We refer to [1] and (3] for basic definitions and results . We recall that what Hamilton-Jacobi theory intends is to convert a given system in an integrable one . Under the preceding conditions if on the level set L ={P J .=C j , j = 1= n} where C 1 , . . ., Cn are real values, the forms DFj , j = 1= n aje independent and L is compact, then it is diffeomorphic to the n-dimensional torus Tn . (Taking out the compactness condition we get some Rk x Tn-k ; eventual dependence of DF J ., j = 1+ n , along subsets can produce that L be a union

miltonian of n degrees of freedom .Por the sake of simplicity we take R2n as phase space .A first integral F of the associated hamiltonian system is a smooth function such that the Poisson bracket (F,H) is identically zero .
Let Fi, j = 1-k smooth functions .We say that they are in involution if (F i ,F j ) = 0 Vi,j .From now on we take F 1 =H .A hamiltonian system is said integrable if there are n functionally independent smooth global functions in involution .We refer to [1] and (3] for basic definitions and results .
We recall that what Hamilton-Jacobi theory intends is to convert a given system in an integrable one .
Under the preceding conditions if on the level set L ={P J .=C j , j = 1= n} where C 1 , . .., Cn are real values, the forms DF j , j = 1= n aje independent and L is compact, then it is diffeomorphic to the n-dimensional torus T n .(Taking out the compactness condition we get some Rk x Tn-k ; eventual dependence of DF J ., j = 1+ n , along subsets can produce that L be a union of tori along such subsets) .Then there exist action-angle variables (I,~o) in D n x T n , a neighbourhood of T n such that H flow is given by I (t) = I (0) , %o (t) =%0 (0) + wt cy vector .Therefore we get a linear the torus if w is incommensurable or dependent) or in a lower dimensional if the Z -module generated by the w's is 1-dimensional) .The solution can be obtained through quadratures .The statements above constitute the Liouville-Arnold theorem .A standing problem is how many first integrals has, generica-lly, a hamiltonian system .Some numerical experiments [12] and the destruc-tion of symmetries by genericity seem to favourish the fact that only H re mains as first integral .
Let us look for the behaviour of the solutions in the nonintegrable case .We first consider the easiest nontrivial case : n =2 .The levels H = e are 3-dimensional hypersurfaces H .We restrict our study to some fixed H . Let 1: be a 2-dimensional manifold in H transversal to the flow at a e e point P .Let us suppose that the orbit through P cuts again L transversally (this is the case if such orbit is periodic) .Then we .candefine (locally) a map T : E-',) given by : find the next cut ( Suppose that T has a hyperbolic fixed point P : Spec (DT(P)) = {a, 1/Ín},1ñ1> 1 .Then Hartman's theorem [14] assures that the behaviour near P is essentially the one given by the linear part .Even in this case the linearizing change of variables is analytic [30] .We have invariant stable and unstable mani-folds (W s (P), Wu (P)) that can be globalized .In a similar way, some piece of analytic invariant curve y near P can be extended by iteration of T and T 1 .
However if G is a first integral near P (i .e .G(T(P)) = G(P) in a neighbourhood U of P) this function can not be extended in general, for instance, if WS (P) comes near P again .This happens if Wu (P)(1 Ws (P) # 0 .A point belon-ging to both manifolds is called homoclinic .If P,Q are fixed hyperbolic and S s Wu (P) 0 W s (Q) the point S is called heteroclinic and similar problems can occur .
The intersections of globalized invariant curves can produce to the folding of Wu is rela-a submanifold of R2 .Wintner integrals G (obtained locally and pieare not able to isolate the "in-and the boundary of this set can have of a point standing on G = g "fill" a cantorian sets .The lack of integrability due ted to the fact that Wu is a manifold but not [33] stated this fact saying that the cewise continuated) are nonisolating : they ner" and "outer" parts of a set G < g positive measure .Therefore, iterates strip .
In a different approach, using perturbation theory, the pro--blems of small divisors, overlapping of resonances, etc .are typical of nonintegrable systems .For two degrees of freedom the shift of Bernoulli can be included as a subsystem of the hamiltonian system [24] .Then, in particular, an infinity of periodic orbits (P .0.) of arbitrary high period exist as well as oscillatory and quasirandom motions .§2 .Some analytical results .We beinn with a few historical comments .Newton formulated the n-body problem and found the 10 classical integrals .Some 200 years were elapsed in a fruitless search of additional integrals .Later in the XIX century, Bruns proved that no more independent first integrals can be found being algebraic functions of q and p, and Painlevé stated even the nonexistence of additional first integrals algebraic in p .Poincaré [26] in turn showed the lack of integrals analytical in q, p, M for the restricted three body problem with mass parameter tA besides the Jacobi integral .
If we restrict ourselves to analytic hamiltonians near an equi librium point then the existente of integrals is related to the problem of normal forms started by Birkhoff (see [23] ) .Let H = H 2 +H 3 +H 4 + . . .an analy tical function near an equilibrium point that we take as the origin .Hk is 2 the homogeneous part o .fdegree k .Suppose H 2 = 1 o( J (q ?+ p?) and define n 2 J = jk e Z j(k,d) = 0} as the Z-module of the resonantes (here (,) is the inner product) .Theorem (Gustavson [13]) : There is a formal change of variables (q,p)-(j,h) such that the new hamiltonian r is of the form (Gustavson As examples of integrable systems we can display all the problems found in elementary textbooks in mechanics .Nonintegrability is dis-played by systems with n = 2 possessing transversal homoclinic or heteroclinic orbits [2, 8, 15,18,19,24,32] .However for n> 2 there are examples with transversal homoclinic orbits that are integrable [10] . Nonintegrability is related to the divergente of the transfor mation to normal form .In fact, for n = 2, Rüssmann [28] proved that if d2 /ad ¢ Q and G = G 2 + G 3 + . ., is a first integral with G2 = 2 1 f j (q~+ p2 ) , d 1 2 g # 0 then we have convergente when going to the B .N .F .r 1 P21 For the relation between divergente and destruction of inva-riant curves see [27] .§3 .Detecting nonintegrability .Faced to a definite problem, how to decide about integrability?Here genericity is useless .A hopeless approach is trying to get enough first integrals .However this is not be recommended ex cept if there is a strong evidente (numerically, see later) of such exis-tence .That was the way Hénon followed to show that the Toda lattice with equal masses is integrable [16] .
If weproceed numerically the Poincaré map is a useful device .
If n = 2 and the iterates of a point are scattered along a line we have an evidente of nonintegrability .However, if the system is very near an integrable one it could be difficult to decide whether or not the points are on a curve .A much finer criterion is to look for transversal homoclinic points [18,19,24,31,32] or for a chain [4] of transversal heteroclinic points .We return later to that topic .If n> 2 to visualize the Poincaré map we need some "stroboscopic" device [22] or different cuts of E [11] .
A dimension-independent method consists in the computation of the Lyapunov numbers .Let Ot be a (hamiltonian) (]ow and DOt the differen-tial with respect to initial conditions(DOt is the solution of the firstor-2n 2n der variational equations ; in coordinates D Ot = A (t) e 'f (R , R ) , A (O) = I) .
The maximal Lyapunov number 1 1 (P) is the maximum rate of growth of the length of a tangent vector at P under Dot , i .e .1i (P) = lim In JIA(t)'1 2 /t w_e re we recall that IIA(t)11 2 = (4(A(t) AT(t))~z(S= spectral radius) .The rema¡ñ ing Lyapunov numbers 1 ., j =2-2n are defined in the following way : let J J be the maximum rate of growth of the j-dimensional measure of a j-dimensional subspace of the tangent space at P under the action of Do. .Thenl j = = .2./.2 .See [5] for the effective computation of all the Lyapunov num--J j-1 bers (taking care of scaling, orthogonalization, etc .) .he important fact for detecting nonintegrability relies in the Theorem : Integrability => all the Lyapunov numbers are zero .
We see that integrable systems have a "parabolic" character in the same sense that a fixed parabolic point of a diffeomorphism .
Following a result of Pesin [25] the entropy of the flow is given by h = l Y-1 .(P) .However a direct computation of h phase space 1 .>o i can be harder than that of th l e Lyapunov numbers .
For n = 2 nonintegrability .followsif the Poincaré map has the smale horseshoe embedded as a subsystem [24] .At some level of.energy h for a hamiltonian it is possible to show the existence of transversal homo clinic and chains of heteroclinic orbits and, therefore, of such embedding by simple topological considerations .See f .i .[8) for.the Hénon-Heiles (HH) problem and for the potential 2 (ql + q2) -2 q1 x,q2 .However for those and other systems nonintegrability is detected numerically for smaller values of h, far away of the value for which the zero velocity curve becomes open .
The HH problem is obtained through perturbation of a harmonic oscillator .With a suitable scaling we have H = H 2 + E H 3 on the levelH =1, 2 ) is the harmonic oscillator and H 3 a homogeneous third order term (or, in general, an analytic function beginning with terms of third order at least) .We realize that is a resonant hamiltonian, being here j(k,-k)j k e Z} .For E>0 all the orbits are periodic .In fact H=1 is S 3 and it is easily obtained than the space of orbits is S 2 .For a n-di mensional harmonic oscillator we have S 2n-1 and P n-1 ( :), respectively .Perturbation methods or the Gustavson N .F .allow to establish the existence of a finite number (except for degenerate cases) of families of simple P .O .[17, 321 .We have a map near the identity in S 2 that can be seen as the approximate time one flow of an integrable hamiltonian system .The rest points are associated to families of P .O . of the original hamiltonian .For the HH pro-blem there are 8 such families .The stability status of some of the orbits can be a delicate question because they appear as parabolic up to high order term, ¡ ..e ., the eigenvalues of the associated Poincaré map being of the form 1 + O(E 4 ), we need several terms to detect the hyperbolicity .The effect of this "slow" hyperbolicity is seen through the following result .
Theorem : Let p be the angle at one homoclinic point between W u (P) and W s (P) where P is a hyperbolic point of a planar diffeomorphism T depending on a pa rameter E .If the eigenvalues of DT (P) are A = 1 ± 0(¿ k ) then p -A r £ exp(-B/¿ k ) for EJO, where A,B,r are constante .Equivalently we can put fi(E)= A Er exp(-lnB(E) ) .A similar behaviour is found for suitable heteroclinic points .
As a consequence of the theorem S(E)« ¿rn for all natural m when E+0 and using a theory of perturbations with respect to 1 we can not find p analytically .
Examples of analytical computation of transversality of homo-clinic (heteroclinic) points are found in [2], related to the problem of diffusion, [z4] (for the Sitnikov problem) and [18,19] for the restricted problems of there bodies, planar and collinear, and general collinear problem .
In the first and third referentes the computation is obtained through the use of a second perturbation parameter .
Proof of the theorem : we suppose that one of the branches WP' 1 of the unstable manifold of P coincides with one of the branches W P,2 of the stable manifold, or that we have coincidente WP' 1 _°wQ' 2 for the heteroclinic case .
As we obtain the coincidente taking only a .finite number of terms of the B .N .F .or of the G .N .F .we must compute the variation of the manifold due to the suppressed terms .We get an expresion of the form I --Es 'R tos t " f(t) where f (t) is of the type exp (-1ln a 't~) for t -±w, and ln a =' E k .Scaling With this theorem in mind we return to the HH problem .In the nu merical survey where the problem is introduced [15] it is reported that for small (E = h) it seems to be a foliation by inv riant curves .For h = 0 .11some curves dissappear and at h =0 .16no invariant curve perhaps, at a very small scale) .A numerical computation produces the values of o<= tg P/2 as a function of E given related results for the 2 (q2+ q2) -1 £ 1 q 2 q2 (E=h) for which the lack of integrability was nondetected in in table 2 .

2
We see that in fact we have nonintegrability for is hard (or even impossible) to detect it for small energy due ly small angle p .One can ask for the importance of very small all h but it to the extreme angles .For instance, for HH and h = 0 .01 a rough extrapolation gives p = 0(10 -250 ), and this is nonsense for the physical and numerical points of view .We can proceed in the converse way .The width of the "random" zone is of the order of We define a 6-approximate first integral (for a diffeomorphism T : M--» as a function F such that IF (Tk (Q)) -F (Q) I< j , dk s Z, d Q E M .Then we say that T is J-1ntegrable, i .e ., we neglect zones of width 04) .We can ask for the maxi mum value of h in the HH problem for which the system is J -integrable .We set r X = A £ exp(-B/¿ 4 ) with values óf A,B,r obtaiñed analytically (as in or through a rough numerical estímate .For instance, using the second values A = 1 .74E4,B = 0 .0554,r = B are obtained .Then X= 10 -20 gives z> h = 0 .036,j .e., if zones of width 0(10 -20 ) are taken as curves, we that HH is integrable up to h =0 .036 .
Another analytical method to make apparent the nonintegrabili--h1 S1 h2 . . .a ty is to show the existence of solenoids .Let  (with the ~r topology, r ~>4) in óe such that for every He ¡L and every solenoid la there is a minimal set, un der the hamiltonian flow, homeomorphic to F-a .

E
The intuitive idea associated to the described solenoids is the existence of islands inside the islands .Near an elliptic fixed point we have a stable island (we think in the case n = 2 for simplicity) .Inside the invariant curves given by Moser twist theorem there are elliptic periodic points where the whole structure is repeated .However it can be very difficult to check the existence of such chains of islands, smaller and smaller, for a con crete system .

References [1]
[2) [3] [4] [5] [6] can be expressed as HM, the where w = (D I H) T is the frequenflow on T n , quasi periodic and dense on nonresonant (i .e., w1 wn are Z -intorus otherwise (in particular periodic Near the integrable systems the KAM theory [4] ensures the existence of slighty distorted invariant tori (the resonant ones) .They not fill completely the available phase space and if n>2 some slow escape across the tori is possible : the so called Arnold diffusion .do In the situation opposite to the integrable systems we found the ergodic ones .A hamiltonian systems is called ergodic if it is ergodic in (almost) all the levels of the energy .Letting aside (functions of) H the only first integrals are the constante .In the .integrable case the flow is confined to a n-dimensional manifold almost everywhere (if D I H is nondegenerate) and dense there .In the ergodic_ case the flow is dense in their energy level .The real world is neither integrable non ergodic .A mathematical statement is this direction is due to Markus and Meyer .Let é( be the set of `e r hamiltonians with the <C r topology .It is a Baire space .A propiety Q is called generic in Y if A ={HSX IH satisfies P} is a set of the second category .Then the theorem [20] asserts : Ha-miltonian systems are generically neither integrable non ergodic . ic k Normal Form) r = F c km 11 k Y m where ~,r = 1r + i 7r, ~k = ~1 1 " . . ." t n n and k-m E J .(Equivalently (H 2 ,[') = 0) .If the dimension of J is r we get n-r formal first integrals .The question of obtaining more first integrals is sometimes refered as the search of the third integral [9] because for problems of galactic dynamics we already know the energy and momentum integrals .If J=0 (c('s Z-independent) then k=m and therefore f=['(I) (Bir-2 2 khoff N .F .[6]), where I= (i1, . ..,In)T,I r = Ir + 7r' and the system is formally integrable .What about convergente?For n = 2 we have the following result (Siegel [29j) : Let J¿ be the set of analytic hamiltonians, H e M , H = k 2 c~t k 7 m .It is nota res , Y meZ triction (use scaling if necessary) to suppose Ir, km{1 < 1 .Define a very fine topology T in the following way : Given H and E= LEkm1 the ball of radius E centered in H is the set B E (H) _ {H* E Ée I ¡ c km -ckm 1 < Ekm d k m} .Theorem : with the topology 'C the set of hamiltonian systems in Y showing divergence when going to the B .N .F . is dense .A coarser topology cr' can be defined throug B F , N (H) = ¬ H*eal1 Ickm c* km 1< ¿ km for ¡k¡+ Iml < N} .Then we can produce finite changes to B .N .F .without convergente problems .The set of integrable systems is dense in W with respect to T' .
From = H I , I= 0, we get the variational equasolutions are P=0, Q=I, M=I, N=H II t, and AAT = N N Í) = O(t 2 ) from

t=
Ek t we get I = Es-k IR tos (Z E -k ) f (L) dr= Re fR Es-k exp (ir E k ) f (Z) dL = = Es-k Re (2TTi Res) where the summation is extended to the residues of f in the upper semiplane .Let us suppose that the goles are a .+ib ., b .> 0 k J 7 i k) residue c +id . .Then I = E s-Re 21Ti(c .+id.)exp((ia .-b.)E -kdominant term is of the form stated in the theorem .
{F2= c2{ is a curve, except for degenerate cases .The refore the iterates of P under T are on this curve .If H is perturbed and in tegrability is lost the points are scattered in a more or less narrow strip around the curve .It seems that they fill a region of positive measure accor_