VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY

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1 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY DIOGO A. GOMES, ADAM OBERMAN Abstract. The main objective of this paper is to prove new necessary conditions to the existence of KAM tori. These converse KAM conditions, can be used to detect gaps between KAM tori and Aubry-Mather sets. By exploring the connections between viscosity solutions of Hamilton-Jacobi equations, KAM and Aubry-Mather theories we develop a set of explicit a-priori estimates for smooth viscosity solutions that are valid in any space dimension, and can be checked numerically. We apply our results to detect non-integrable regions in several examples: a forced pendulum, two coupled penduli, and the double pendulum.. Introduction Let H : R n R n R n, H(p, x), be the Hamiltonian of a mechanical system. A classical procedure to determine solutions to the Hamiltonian dynamics () ṗ = D x H(p, x), ẋ = D p H(p, x), is the Hamilton-Jacobi method [AKN97], [Gol8]. This method has two main steps. The first one consists in computing a pair of functions (u(x, P ), H(P )), u : R n R n R, and H : R n R, which solves the Hamilton-Jacobi equation: () H(P + D x u, x) = H(P ); the second step is the construction of a change of coordinates X(p, x) and P (p, x), defined implicitly through the equations (3) p = P + D x u, X = x + D P u. In these new coordinates, the Hamiltonian dynamics becomes: (4) Ṗ =, Ẋ = D P H(P).

2 DIOGO A. GOMES, ADAM OBERMAN Thus, for each P the graph (5) p = P + D x u(x, P ), is invariant under the flow generated by (). Furthermore, this flow is conjugated to a translation as Ẋ is constant. If the Hamiltonian H(p, x) is Z n periodic in x, and u is a Z n periodic function, that is, H(p, x+k) = H(p, x), and u(x + k) = u(x), for all p, x R n, and k Z n, the graph (5) can be interpreted as an invariant torus. In this periodic case, it is natural to interpret H as a function H : T n R n R, where T n = R n /Z n is the n dimensional torus. A particularly important case is the one in which the Hamiltonian H(p, x) : T n R n R is smooth and strictly convex in p, that is, DppH > γ >. Throughout this paper we will work mostly in this setting. It is well known that the Hamilton-Jacobi method may fail in practice. Indeed, () may not admit smooth solutions, or (3) may not define a smooth change of coordinates. In particular, the Hamilton- Jacobi method may be valid for certain initial conditions of () but not everywhere. The main question we would like to address is whether, given initial conditions (p, x) for (), this integrability procedure can be carried out. More precisely, whether one can find a vector P, and pair u, H solving () such that p = P + D x u(x). If this is possible, complex behavior such as Arnold diffusion, or chaotic dynamics can be ruled out. The KAM theorem [AKN97] asserts that it is possible to use the classical Hamilton-Jacobi method for most initial conditions, provided the Hamiltonian H has the special structure H(p, x) = H (p) + ɛh (p, x), in which H satisfies non-degeneracy conditions, and ɛ is sufficiently small. Using viscosity solutions (for the general theory of viscosity solution we suggest the references [BCD97], [FS93], [Eva98]), one can prove that a weak form of integrability still holds, see, for instance, [Fat97a], [Fat97b], [Fat98a], [Fat98b], [CIPP98], [E99], [EG], [EG], [Gom], or [Gom3]. Indeed, there always exists a number H(P ) and a periodic viscosity solution u(x, P ) of () [LPV88]. Also, one can construct an invariant set contained in the graph (x, D p H(P + D x u, x)).

3 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY 3 The support of the Aubry-Mather measures [Mat89a], [Mat89b], [Mat9], [Mn9], [Mn96], which are the natural generalizations of invariant tori, is contained in this graph. Unfortunately, not every point is in the support of a Mather measure. Heuristically, one can think of the phase space as containing several sets: the set of all KAM torus, the supports of all Mather measures, heteroclinic and homoclinic connections between different components of Mather sets, and then a remaining part of the phase space which may contain elliptic periodic orbits and corresponding elliptic islands, areas in which the motion is irregular, possibly chaotic, and regions in which Arnold diffusion may occur. It is therefore of interest to study the existence of gaps in the Mather sets, which would prevent the Hamilton-Jacobi integrability and allow for more complex dynamics. There have been several attempt to study this problem, for instance, [MP85], [MMS89], [Mac89], [Kna9], [Har99]. The approach in [MP85], as well as in [MMS89], [Mac89], [Kna9], uses the well known fact that the Mather set is a Lipschitz graph - thus by detecting orbits that do not lie on a Lipschitz graph one proves the existence of gaps in the Mather set. These methods seem to work extremely well for one-dimensional maps, but do not extend easily for multi-dimensional problems. The paper [Har99], studies the discrete case, and uses the fact that orbits on the Mather set, being global minimizers, are also local minimizers. Therefore, with second order tests for critical points, one can show that certain orbits lie outside the Mather set. The main advantage is that this method work for maps in any dimension. Other approaches to prove non-integrability include, among others, renormalization methods, anti-integrable limit. These methods are fundamentally different from the ones considered in this paper. Also, we should mention that there are several related papers which use variational methods to prove analytic counterexamples to KAM theory, such as [For94] [Bes], and [GV4]. Measures supported in the gaps of Aubry-Mather sets were constructed in [For96]. The methods we use in this paper also explore the minimizing character of the orbits in the Mather set. However, they are quite different from the previous ones. The main idea is to identify conditions in which () does not admit a smooth solution, and therefore proving the failure of the Hamilton-Jacobi integrability method. These conditions consist in certain inequalities which can be checked numerically in a

4 4 DIOGO A. GOMES, ADAM OBERMAN very efficient way. The main advantage, however, is that this method is valid in any dimension and, when compared with the one in [Har99], is of a global nature as, in fact, we try to check whether the orbits are global minimizers, not just local minimizers. The plan of the paper is as follows: in section we review the necessary background from Aubry-Mather theory and viscosity solutions. In section 3 we prove explicit estimates for viscosity solutions. In section 4 we describe the converse KAM criteria. Explicit examples are discussed in section 5. The numerical results are presented in section 6.. Aubry Mather measures and Viscosity Solutions Let L(x, v), the Lagrangian, be the Legendre transform of the Hamiltonian, H(p, x), defined by L(x, v) = sup v p H(p, x). p In Aubry-Mather theory [Mat89a], [Mat89b], [Mat9], [Mn9], [Mn96] instead of looking for invariant tori, one looks for probability measures µ, the Mather measures, on T n R n which minimize the action (6) L(x, v) + P vdµ, and satisfy a holonomy condition: vd x φdµ =, for all φ(x) C (T n ). The supports of these measures are the Mather sets, and are the natural generalizations of invariant tori. Recent results [E99], [Fat97a], [Fat97b], [Fat98a], [Fat98b], [CIPP98], [EG], [EG], and [Gom] show that viscosity solutions of () encode many properties of Mather sets. In particular, if µ is a Mather measure and u solves () then L(x, v) + P vdµ = H(P ). Furthermore, the support of the Mather measure is a subset of the graph (7) (x, v) = (x, D p H(P + D x u, x)),

5 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY 5 for any viscosity solution of (). Finally, the Mather set is invariant under the flow generated by the Euler-Lagrange equations (8) d dt D vl(x, ẋ) D x L(x, ẋ) =, which are equivalent, by the Legendre transform p = D v L(x, ẋ), to (). In the Mather set, the asymptotic behavior of the Hamiltonian dynamics is controlled by viscosity solutions. Indeed, let (x, p) be any point in T n R n. Consider the solution (x, p) of the Hamilton equations () with initial condition (x, p). If the point (x, p) is a generic point of an ergodic component of a Mather set then x(t ) T Q, as T, for some vector Q R n. Furthermore, Q P H(P ), where P denotes the subdifferential. Thus Q = D P H(P ) if H is differentiable at P. 3. Explicit estimates for viscosity solutions In this paper, we need explicit estimates for viscosity solutions of (). Therefore, in this section we reprove some known results, but taking care of tracking the constants with detail. We are given a point (x, v), and we would like to investigate whether there is a vector P, and a smooth solution u to () such that this point lies in an invariant graph given by (7). The main difficulty with this approach is that P and u, if exist, are unknown. Our objective in this section is to prove estimates for P and u that only depend on (x, v) and other known quantities. These estimates are: bounds for second derivatives and Lipschitz constant of u, bounds for P in terms of the initial energy, and error estimates for the numerical computation of H(P ). In this section, and in most of the remaining part of the paper, we assume that the Lagrangian has the form (9) L(x, v) = g ij(x)v i v j + h i (x)v i V (x), in which g ij (x) is a positive definite metric, h i represents the magnetic field and V (x) is the potential energy.

6 6 DIOGO A. GOMES, ADAM OBERMAN First, we recall the well known fact that the energy E(x, v) = g ij(x)v i v j + V (x) is conserved by the Euler-Lagrange equations. Note that the energy does not depend on the magnetic field h. The proof of this fact can be found in any book on classical mechanics, for instance [Gol8]. The Hamiltonian corresponding to L is given by H = gij (x)(p j h j )(p i h i ) + V (x), in which g ij is the inverse of g ij, that is g ik g kj = δ i j. Furthermore, along solutions of () H(p, x) = E(x, ẋ) for p j = g ij ẋ i h j. Our choice of Lagrangians is quite general, as many important examples have the form (9). Of course, estimates similar to the ones in this section are true in a much more general setting, but we prefer to lose some generality and have them as explicit as possible. To write our estimates as explicitly as possible, we assume the following bounds for the metric g ij, the magnetic field h i and potential V : () as a matrix, c g ij (x) c, furthermore D x g ij (x) c 3, and D xxg ij (x) c 4 ; () h i (x) c 5, D x h i (x) c 6, and D xxh i (x) c 7 ; (3) V (x) c 8, D x V c 9, D xxv c. With these bounds, we can estimate the velocity ẋ at any time by the energy: Proposition. Proof. Since V, ẋ (E)/. c / c ẋ g ijẋ i ẋ j E. The energy only depends on the initial conditions, thus the previous result implies that one can obtain bounds on the velocity for all time that depend only on the initial conditions.

7 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY 7 Viscosity solutions of () have an interpretation in terms of control theory: a function u is a viscosity solution of () if and only if it satisfies the following fixed point identity: () u(x) = inf x( ) t L(x, ẋ) + P ẋ + H(P )dt + u(x(t)), in which the infimum is taken over Lipschitz trajectories x( ), with initial condition x() = x. If a point (x, v) is in the integrable region, that is,there exists P and u solving () such that v = D p H(P + D x u, x), then H(P ) = E, so H can be determined directly from the initial condition. However, both u and P are unknown. First we prove an estimate for P : Proposition. Let Then C = sup L(x, ω)dx. ω = T n P H(P ) + C. Proof. Observe that from the control theory representation formula H(P ) lim T T T L(x, ẋ) + P ẋ, for any trajectory x( ). Let ω be an arbitrary vector such that ω =, and ω k = for k Z n implies k =, that is, the flow ẋ = ω is ergodic on torus. Let x(t) = ωt. Then H(P ) lim T T This yields: T L(x, ẋ) + P ẋ = P ω + L(x, ω)dx, T n P H(P ) + sup L(x, ω)dx. ω = T n The next objective is to study the regularity of u. Proposition 3. Let u be a viscosity solution of (). Then u is semiconcave, that is, u(x + y) u(x) + u(x y) C y.

8 8 DIOGO A. GOMES, ADAM OBERMAN For y k, the constant C is estimated by: [ c ( c ( ) 4 H 4 + 4c 3) c + k c ( ) / H + c 7 + k + 4c 6k + c c c c 3 ] /. Remark. The main point of this lemma is that the constant C can be estimated explicitly in terms of bounds for the Lagrangian and the energy of an optimal trajectory. Remark. The constant in the lemma is bounded uniformly in y for bounded values of y, for large values of y one can use the fact that u is periodic to get a better estimate. Proof. Fix x R n, and choose any y R n. We claim that u(x + y) + u(x y) u(x) + C y, for the constant C given in the statement. Let x( ), x() = x, be an optimal trajectory for u(x). Clearly, for any trajectory y( ) such that y() = y u(x + y) + u(x y) u(x) t [L(x(s) + y(s), ẋ(s) + ẏ(s)) + L(x(s) y(s), ẋ(s) ẏ(s)) L(x(s), ẋ(s))] ds. Define y(s) = y t s. Observe that t L(x ± y, ẋ ± ẏ) L(x, ẋ ± ẏ) ± D x L(x, ẋ ± ẏ)y + C y. The constant C has three contributions: one comes from the bounds for the metric g ij g ij(x + y)(ẋ i + ẏ i )(ẋ j + ẏ j ) g ij(x)(ẋ i + ẏ i )(ẋ j + ẏ j ) D xg ij (x)(ẋ i + ẏ i )(ẋ j + ẏ j )y + 8 D xxg ij ( ẋ i + ẏ ) y ; the second one corresponds to the magnetic field, h i (x + y)(ẋ i + ẏ i ) h i (x)(ẋ i + ẏ i ) D x h i (x)y(ẋ i + ẏ i ) + D xxh i y ẋ i + ẏ i ; and the last one from the potential energy: V (x + y) V (x) D x V (x)y + D xxv y (t s) t.

9 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY 9 Note that in the previous estimates we used y y, to estimate the second order terms. Therefore, we have the following estimate for C : C ( c ( ) 4 H 8 + c 3) + k + c ( ) / 7 H + k + c 6k + c c t c t t 3. Note that the term c 3 comes from the integration in time. Also we have Thus Therefore L(x, ẋ + ẏ, s) + L(x, ẋ ẏ, s) L(x, ẋ, s) + c ẏ. L(x + y, ẋ + ẏ, s) + L(x y, ẋ ẏ, s) L(x, ẋ, s) + C y + c y t. u(x + y) u(x) + u(x y) C y t + c y t. By choosing t = c C we obtain u(x + y) u(x) + u(x y) C y with C = c C. Proposition 4. Let u be a solution of (). Then u is Lipschitz with Lipschitz constant estimated by [ c ( c ( 4 H 4 + 4c 3) + 9n ) c c +c 7 ( H c + 9n c ) / ] / + 4c 69n + c c n, c 3 in which c n is a constant that depends only on the dimension n. Proof. It is well known that a periodic semiconcave function φ is Lipschitz. To prove this fact, observe that Dφ(x) = Dφ(y) + D φ(sx + ( s)y)(x y)ds.

10 DIOGO A. GOMES, ADAM OBERMAN Since φ is periodic, there exists y such that x y 3 n, Dφ(y) =, and c n Dφ(x) Dφ(x) (x y) C x y, where c n is a constant that depends only on the dimension n. Therefore Dφ(x) c n C. Furthermore, in the estimate for the semiconcavity constant C we may take k = 3 n. Proposition 5. Assume that u is a smooth solution of the Hamilton- Jacobi equation (). Then u is semiconvex, that is u(x + y) u(x) + u(x y) C y, and the constant C, for y k, can be estimated by [ c ( c ( ) 4 H 4 + 4c 3) c + k c ( ) / H + c 7 + k + 4c 6k + c c c c 3 ] /. Remark. The main point of assuming that a solution u is smooth is that, then, the method of characteristics is valid, and so for every point x there is a global characteristic. Proof. Fix x R n and choose any y R n. We claim that u(x + y) + u(x y) u(x) C y. Let x(t) be an optimal trajectory, x = x(t), and x = x(). Then Therefore u(x ) = t u(x y) u(x) + u(x y) t [L(x(s), ẋ(s)) + P ẋ] ds + u(x). [ L(x(s) + y(s), ẋ(s) + ẏ(s)) L(x(s) + y(s), ẋ(s) + ẏ(s))+ +L(x(s), ẋ(s))] ds, if we choose y(s) = y s. From this inequality and proceeding exactly t as in proposition 3 we obtain the estimate. Given a value P, there are efficient numerical methods to compute H(P ), and control the error [GO4] (for an alternative scheme, consult [Qia3]). The algorithm in [GO4] is based on the representation

11 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY formula for H(P ) () H(P ) = inf φ Cper sup H(P + D x φ, x) x due to [CIPP98] (see also, for a more general setting, [LS] and [Gom5]). Let T h be a set of piecewise linear finite elements, with the diameter of each element bounded by h. Let H h (P ) be the numerical approximation computed by: The main error estimate is: H h (P ) = inf esssuph(d x φ, x). φ T h Proposition 6. For any convex Hamiltonian H(p, x) for which () has a viscosity solution H H h (P ). Furthermore, if () has a smooth viscosity solution then H h (P ) H(P ) + Ch. The constant C depends only on bounds for the Hamiltonian, the energy level, but not on P, and can be estimated by () n C D x H(p, x), + Dxxu D p H(p, x),, in which f(p, x), = sup f(p, x), x T n, p <R and R is a bound for P + D x u. Proof. The first claim, that is, H = inf sup H(P + D x ψ, x) inf esssuph(p + D x φ, x), ψ C (T n ) x φ T h x can be proved in the following way: to each φ T h we associate a function ψ = φ η ɛ C (T n ). Then the convexity of H implies sup x x H(P + D x ψ, x) esssuph(p + D x φ, x) + O(ɛ), x since H(P + D x (φ η ɛ )(x), x) H(P + D x φ(y), y)η ɛ (x y)dy + O(ɛ).

12 DIOGO A. GOMES, ADAM OBERMAN Since ɛ is arbitrary, we get the desired inequality. Suppose u is smooth viscosity solution. Construct a function φ u T h by interpolating linearly the values of u at the nodal points. At a node x we have D xj φ(x i ) = u(x + he j) u(x) h = D xj u(x + she j )ds In each triangle T i, the oscillation of the derivative of u can be estimated by which implies Thus at a node x we have D x u(x) D x u(y) D xxu x y D xj φ(x i ) D xj u(x) D xxu h. H(D x φ u (x), x) H(P ) Dxxu h D p H(p, x), n. We also have, for all points y in the triangle, Then H(D x φ u (x), x) H(D x φ u (x), y) D x H(p, x), x y. esssup H(P + D x φ u, x) H(P ) + D x H(p, x), h + Dxxu h D p H(p, x), n. 4. Detection of Non-Integrability In this section we put together the previous estimates and discuss several possible ways to detect gaps on Mather sets and thus nonintegrability by using properties of viscosity solutions. A main advantage of our methods is that they are completely rigorous, work in any space dimension, and can be implemented numerically. Our objective is to decide whether an arbitrary point (x, v) in T n R n lies outside any invariant tori given by a smooth solution of ().

13 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY 3 Theorem. Suppose that (x, v) belongs to the graph of a viscosity solution u of () for some P. Let x( ) be the solution to the Euler- Lagrange equations with initial conditions (x, v). Let S P (t) = t L(x, ẋ) + P ẋ + H(P )ds. Then, there are constants C, C such that (3) sup S P C, t (4) sup t S P (t) x(t) x() C. Furthermore, if u is C then there exists a constant C such that (5) S P (t) (D v L(x(), ẋ()) + P )(x(t) x() + k) sup sup C t k Z n x(t) x() + k. The constants C = u, C = Du and C = D u can be estimated from known properties of H and the initial point (x, v). Finally, if the solution is C 3 and there are uniform third derivative bounds C 3 for the solutions to () then (6) sup t sup k Z n S P (t) DvL(x(),ẋ()) DvL(x(t),ẋ(t)) (x(t) x() + k) C x(t) x() + k 3 3. Proof. If (x, v) belongs to the graph of a viscosity solution u of (), and x( ) is the corresponding solution of the Euler-Lagrange flow, then for all t u(x) = t L(x, ẋ) + P ẋ + H(P )ds + u(x(t)). First, since u is bounded, there exists a constant C such that sup u(x) u(x(t)) C, t therefore we have (3). An improved version of this estimate follows from the fact that u is periodic and Lipschitz, and so if C is the Lipschitz constant of u then sup t u(x) u(x(t)) x(t) x() C in which x y = inf k Z n x y+k is the periodic distance. Therefore we have (4).

14 4 DIOGO A. GOMES, ADAM OBERMAN Since p = P + D x u, and p = D v L(x, ẋ), we have D x u(x) = D v L(x, ẋ) P. If the solution u is C, the previous estimate can be improved since we have a priori estimates for second derivatives sup t u(x) + D x u(x)(x(t) x + k) u(x(t) + k) sup C k Z n x(t) x() + k, and u(x(t) + k) = u(x(t)), this yields (5), and if we assume a bound C 3 for the third derivative of the viscosity solution we have (6). To prove that a point (x, v) does not lie in any invariant tori, we will proceed by contradiction. That is, we assume that (x, v) is in an invariant torus, the previous equalities must hold. Unfortunately all these inequalities involve the values P and H(P ), both of them unknown. However, the value H(P ) can be well approximated by the minimax representation formula (), provided that to P corresponds and invariant tori. The error of the approximation depends on the energy, which can be estimated by the initial condition (x, v), but is independent of P. The solution of (8) can be computed with arbitrary precision using a suitable numerical solver. Therefore, given a number P, we can test the inequalities (3), (4), or (5). Corollary. If a point (x, v) belongs to the graph of a smooth viscosity solution then (7) inf P (8) inf P and (9) inf P sup t sup t sup S P (t) C, t S P (t) x(t) x() C, S P (t) D v L(x(), ẋ())(x(t) x() + k) sup C k Z n x(t) x() + k. Remark. In practice, it may be enough to choose a single value for t, for instance the terminal time T. There is a heuristic explanation why the terminal time (or any large value of t) may be enough. Let (x, v) be a generic point in an ergodic component of the Mather set, and x the corresponding trajectory.

15 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY 5 Then, there exists a vector Q for which as T, and T T x(t ) T Q, L(x, ẋ)dt L(Q). Recall that L(Q) is the infimum of the average action over all trajectories with rotation number Q. The function L(Q) = P Q H(P ) is the Legendre transform of H(P ). If the solution x has rotation number Q but () lim inf T T T L(x, ẋ)dt > L(Q) + ɛ, for some ɛ >, the initial condition is not a generic point in an ergodic component of the Mather set. Therefore, proving the existence of gaps of the Mather sets. If () holds, then, since L(Q) P Q H(P ) =, T L(x, ẋ) + P ẋ + H(P )ds > T ɛ, as T. Therefore (7), (8) and (9) cannot be satisfied. 5. Explicit Examples In this section we consider two examples: the one dimensional pendulum, and linear Hamiltonians, which can be studied explicitly. 5.. The one dimensional pendulum. The Hamiltonian corresponding to a one-dimensional pendulum with mass and length normalized to be, is H(p, x) = p cos πx. For this Hamiltonian one can determine the solution of (). Indeed, for each P R, and for a.e. x R, the solution u(p, x) satisfies (P + D x u) = H(P ) + cos πx. This implies H(P ), and so D x u = P ± (H(P ) + cos πx), for a.e. x R.

16 6 DIOGO A. GOMES, ADAM OBERMAN Thus u = x P + s(y) (H(P ) + cos πy)dy where s(y) =. Because H is convex in p and u is a viscosity solution, it is semiconcave. So, the only possible discontinuities in the derivative of u are the ones that satisfy D x u(x ) D x u(x + ) >. Therefore s can change sign from to at any point but jumps from to can happen only when (H(P ) + cos πx) =. If we require - periodicity there are two cases, first if H(P ) > the solution is C since (H(P ) + cos πy) is never zero. These solutions correspond to invariant tori. In this case, P and H(P ) satisfy the equation P = ± (H(P ) + cos πy)dy. It is easy to check that this equation has a solution H(P ) whenever that is P ( + cos πy)dy, P 4 π.734. When this inequality fails, H(P ) = and s(x) can have a discontinuity. Indeed, s(x) jumps from to when x = + k, with k Z, and there is a point x defined by the equation s(y) ( + cos πy)dy = P, such that s(x) jumps from to at x + k, k Z. Therefore, for initial conditions (x, p) such that the energy E <, there is no corresponding vector P and solution u(x) so that H(P + D x u, x) = E, thus these energy levels should be non-integrable. In fact, we can detect this behavior using our methods. We have S P (T ) = = T T L(x, ẋ) + P ẋ + H(P ) = (P p(t))ẋ(t) H(p(t), x(t)) + H(P ).

17 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY 7 For E < the trajectories are periodic, thus Tn P ẋ =, for any multiple T n of the period. We have ẋ = p(t), therefore Tn p(t)ẋ(t) = Tn p(t). Since H(P ) H(p, x) E >, the integral is unbounded. Tn H(P ) H(p(t), x(t))dt 5.. Linear Hamiltonians. It is well known that there may not exist smooth solutions to the linear Hamilton-Jacobi equation ω (P + D x u) + V (x) = H(P ), with ω R n. The Hamiltonian H(p, x) = ω p + V (x) is convex in p but not strictly convex. However as we will show, our methods still detect non-integrability in some cases. The identity ω (P + D x u) + V (x) = H(P ), is a necessary condition for the existence of solutions of the Hamilton- Jacobi equation. Thus H(P ) = ω P + H(), with H() = V (x), which we can assume to be zero. The Lagrangian corresponding to the Hamiltonian is { V (x) if v = ω L(v, x) = + otherwise. The equation of the dynamics are ẋ = ω ṗ = π sin(πx ). Therefore, the action S P (T ) is given by S P (T ) = T V (x(t))dt,

18 8 DIOGO A. GOMES, ADAM OBERMAN with x(t) = x() + ωt. We consider three examples. The first one: H(p, s) = (, ) p + cos(πx ), that is, ω = (, ). In this case, one can construct a smooth solution to the Hamilton-Jacobi equation. Therefore S P (T ) = T cos(πx () + πt)dt, is bounded uniformly in T for any value of x (), which can be checked directly. The second case is ω = (, ), and the equation reads (, ) Du + cos(πx ) =. This is a resonant example since (, ) is rationally dependent. However one can still find a smooth solution to the Hamilton-Jacobi equations by using Fourier series. The action S P (T ) = T cos(πx () + πt)dt, is again bounded uniformly in T for any value of x (). Resonant linear Hamiltonians as the previous one may fail to have a viscosity solution. An example is The variational formula yields (, ) Du + sin(πx ) = H. H() = inf φ sup H(Dφ, x) =. x which is a contradiction. And, in fact, the action is S P (T ) = T which is unbounded in T for x () sin(πx ())dt, 6. Computational Examples In this section we consider several Hamiltonian systems and try to study their integrability numerically. The first two examples which fit directly our framework, are two coupled penduli, and the double pendulum, which is known to have chaotic behavior. In the last example,

19 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY 9 a forced pendulum, our estimates do not apply directly, but can easily be modified so that we can also study its non-integrable regions. To compute numerically H we have used the numerical implementation of the minimax formula () in [GO4], using recursive mesh refinement for speed up. To implement the ODE s we used MATLAB s solver, and used energy conservation to verify accuracy. We only consider the first method to detect non-integrability, and plot for T = the value inf P S P (T ). 6.. Coupled penduli. The Hamiltonian for two coupled penduli is given by H(p x, p y, x, y) = p x + p y + cos πx + cos πy + ɛ cos π(x y), and the corresponding Lagrangian is L(v x, v y, x, y) = v x + vy cos πx cos πy ɛ cos π(x y). The equations of the dynamics are ṗ x = π sin πx πɛ sin π(x y) ṗ y = π sin πy + πɛ sin π(x y) ẋ = p x ẏ = p y. To plot the non integrable regions we choose an initial point (x, y) = (, ) and then vary the values of p x and p y. To compute H we have used a grid. The case with no coupling, ɛ =, figure is simply a two dimensional version of the pendulum example. When coupling is positive, ɛ =. in figure, there is numeric evidence of resonances between the center equilibrium in one pendulum and periodic orbits in the other. Error estimates. For ɛ., the second derivative of the potential D V This yields a bound for D xxu 8.58, which implies the L bound for u u.5. Furthermore, we have D x H.65, and for the initial conditions we consider ( p x, p y 3) we have D p H 4.4.

20 DIOGO A. GOMES, ADAM OBERMAN residual p x p y Figure. Residual for coupled pendulum (no coupling, ɛ =.). residual p x.5.5 p y Figure. Residual for coupled pendulum (ɛ =.). With a grid of nodes h =. Thus the error term in the computation of H is bounded by 3.3. This means that residuals over 37.3 indicate that the system is non integrable. Note that these bounds are really coarse, as the error in computing H is in fact quite small than our estimates [GO4]. 6.. Double pendulum. The Lagrangian for the double pendulum is L(v x, v y, x, y) = v x + v y + v x v y cos π(x y) + cos πx + cos πy.

21 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY residual p x p y Figure 3. Residual for the double pendulum, initial conditions x = y = and the corresponding Hamiltonian: H(p x, p y, x, y) = p x p x p y cos(π(x y)) + p y cos (π(x y)) cos πx cos πy. The equations of the dynamics are ṗ x =π (p x + p y) cos θ + p x p y (5 + cos θ) ( + cos (θ)) sin θ+ + 4π sin πx ṗ y =π (p x + p y) cos θ p x p y (5 + cos θ) ( + cos θ) sin θ+ + π sin πy ẋ = p x p y cos θ cos θ ẏ = p y p x cos θ, cos θ in which θ = π(x y). Error estimates. In this case, the bounds we obtain for the error of the effective Hamiltonian (more than ), are too large for the numerical results to actually prove the non-integrability of the double pendulum. However, as the numerical experiments in [GO4] suggest, the actual errors should be quite small and therefore is likely that most points shown in figure 3 belong to the non-integrable region.

22 DIOGO A. GOMES, ADAM OBERMAN 6.3. Time-periodic Hamiltonians. Another example is a periodic time-dependent, one space dimension Hamilton-Jacobi equation: u t + H(D x u, x, t) = H. There exists a unique value H for which this problem admits spacetime periodic viscosity solutions, see for instance [EG]. Moreover this solution is Lipschitz. Note also that P = (P t, P x ) but H(P ) is linear in P t so we may as well consider just the problem inf φ sup φ t + H(P x + D x φ, x, t) = H(P x ). (x,t) Although this problem is not exactly in the form discussed previously in the paper, the estimates can be adapted. The estimates for the numerical computation of H carry through, as they only depend on the convexity, of the Hamiltonian H(p, q, x, t) = q + H(p, x, t). The only problem consists in estimating the semiconcavity constant. For that we use the variational formula and obtain T [ ] u(x + y, ) u(x, ) + u(x y, ) T + t D V (x, t) y T By optimizing over T we get Dxxu(x, D V (x, t) ). 3 In this example we set up a forced pendulum corresponding to the time-dependent Hamiltonian H(p, x) = p + ( + ɛ sin πt) cos πx. The equations of motion are ṗ = π( + ɛ cos πt) sin πx ẋ = p. The corresponding Lagrangian is L(x, v, t) = v cos πx ɛ cos πx cos πt. The figures 4-6 show, from no forcing (ɛ = ) to large forcing (ɛ =.4)

23 VISCOSITY SOLUTIONS METHODS FOR CONVERSE KAM THEORY 3 residual x p x.5 Figure 4. Residual for forced pendulum (no forcing, ɛ = ). residual x p x.5 Figure 5. Residual for forced pendulum (ɛ =.). residual x p x Figure 6. Residual for forced pendulum (ɛ =.4).

24 4 DIOGO A. GOMES, ADAM OBERMAN the evolution of the heteroclinic region of the pendulum and show its break-up. Error estimates. We have the following bounds (for ɛ.4) D xxu 8.58, D x H 8.79, Dp H(p, x) 5. (this was estimated from the numerical computations). Thus, with a grid we have that the error for H is bounded by.98, the bound for u is.5 which means that residuals over.3 correspond to non-integrable regions. With a slightly larger grid, we believe that it would be possible to further improve this bound and classify even more points as non-integrable. 7. Conclusions In this paper we have developed a set of necessary conditions for the existence of KAM tori. These conditions can be easily implemented numerically and the error terms coming from the discretization can be estimated explicitly. Therefore the numerics provide a rigorous proof of non-existence of KAM torus. Both in explicit examples, as well as in more complex cases, we are able to detect behavior such as resonances and heteroclinic break-up. The main numerical problems, are due to the fact that the estimates for the error in the numerical computation for H overestimate the error, in fact, as the simulations in [GO4] show, the errors are, in fact, quite small. To sum up, we believe that these methods are an effective way to study in practice the integrability of Hamiltonian systems. References [AKN97] V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt. Mathematical aspects of classical and celestial mechanics. Springer-Verlag, Berlin, 997. Translated from the 985 Russian original by A. Iacob, Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Dynamical systems. III, Encyclopaedia Math. Sci., 3, Springer, Berlin, 993; MR 95d:5843a]. [BCD97] M. Bardi and I. Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser Boston Inc., Boston, MA, 997. With appendices by Maurizio Falcone and Pierpaolo Soravia. [Bes] Ugo Bessi. An analytic counterexample to the KAM theorem. Ergodic Theory Dynam. Systems, ():37 333,.

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