Some Applications of Moser’s Twist Theorem

CHAPTER 5

Some Applications of Moser’s Twist Theorem Mark Levi 1 Department of Mathematics, Pennsylvania State University, United States E-mail: [email protected]

Contents 1. Background: the action-angle variables, the generating functions . . . 1.1. Examples of action-angle variables . . . . . . . . . . . . . . . 1.2. Action-angle variables for particles in Rn . . . . . . . . . . . . 1.3. A generating function construction of the action-angle variables 1.4. A mechanical interpretation of generating functions . . . . . . 1.5. The twist condition . . . . . . . . . . . . . . . . . . . . . . . . 2. Basic statements of KAM theory . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Moser’s twist theorem . . . . . . . . . . . . . . . . . . . . . . 2.2. The Kolmogorov–Arnold theorem . . . . . . . . . . . . . . . . 3. A variational approach to Moser’s twist theorem . . . . . . . . . . . . . . 3.1. The generating function of an area-preserving map . . . . . . . 3.2. Reduction to a difference equation . . . . . . . . . . . . . . . . 4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Arnold diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Partially supported by NSF grant DMS-9704554. HANDBOOK OF DYNAMICAL SYSTEMS, VOL. 3 Edited by H.W. Broer, B. Hasselblatt and F. Takens c 2010 Published by Elsevier B.V.

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1. Background: the action-angle variables, the generating functions In addition to a quick review of general definitions, this section contains the discussion of specific examples of action-angle variables. In this article we discuss Hamiltonian systems 

x˙ = Hy , y˙ = −Hx

(1)

defined by the Hamiltonian H : R2n → R. One can write (1) more compactly: (2)

z˙ = J ∇ H, where  J=

 0 I , −I 0

z=

  x , y

where I is the n × n identity matrix, ∇ H is the gradient treated as a column vector and where x, y are column vectors. A Hamiltonian system (1) is said to be completely integrable if it possesses n independent integrals in involution – that is, if there exist n functions Ik : R2n → R satisfying 1. {H, Ik } ≡ hJ ∇ H, ∇ Ik i = 0 (that is, dtd Ik = 0, where the derivative is taken along the flow of (1)). 2. {Ik , I j } = 0, and 3. the gradients ∇ I1 , . . . , ∇ In are linearly independent for all z in a neighbourhood of a level surface Ik = ck . The Liouville–Arnold theorem ([4]) Assume that (1) is completely integrable and that, in addition to the conditions 1-3 above, all the level sets Ik = ck are compact for any choice of constants ck , k = 1, . . . , n from an open set in Rn . Then the level set {z ∈ R2n : Ik (z) = ck , k = 1, . . . , n} is an n-torus, and there exists a symplectic transformation ϕ = z 7→ (I, θ) ∈ Rn × Tn in which the Hamiltonian becomes θ-independent: H ◦ ϕ −1 (I, θ) = K (I ), and our Hamiltonian system therefore takes the form  I˙ = 0 (3) θ˙ = ω(I ), where ω(I ) = K 0 (I ), where K 0 denotes the gradient. 1.1. Examples of action-angle variables In this section we give several specific examples of the action-angle variables.

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M. Levi y z0 z I

t

S x

x

Fig. 1. Action-angle variables for the convex Hamiltonian in R2 .

1.1.1. Convex Hamiltonians with n = 1 degree of freedom. consider the Hamiltonian system in R2  x˙ = Hy (x, y) y˙ = −Hx (x, y)

As our first example, let us

where we assume that the level curves of H are convex curves enclosing the origin, Figure 1. Now the action-angle map ϕ : z = (x, y) 7→ (I, θ ) is constructed as follows. Given any point z 6= 0 in the plane we define the action I = I (z) as the area enclosed by that level curve of H which passes through z: I I = y dx, Cz

where the level curve C z = {w : H (w) = H (z)} is oriented clockwise. The angle variable θ = θ (z) is defined as the proportion of the period it takes for the solution to reach z from the positive y-axis: θ = t (z)/T (z);

(4)

here T (z) is the period of the orbit passing through z and t (z) is the time of travel from the positive y-axis to z. Since this time is defined modulo T (z), the variable θ is defined modulo 1. The variable θ has a purely geometrical meaning without a reference to time. Namely, consider an infinitesimally thin ring between the level curve of H passing through z and a nearby level curve of H . Consider also the segment of this ring lying between the y-axis and the vertical line through z. The angle θ is then simply the ratio of the area of this segment to the area of the ring. This geometrical interpretation of θ is equivalent to the definition (4) due to the fact that the Hamiltonian flow in the plane is incompressible, and thus, loosely speaking, the area swept by a moving short segment (which sweeps out the above-mentioned ring) increases in direct proportion to the time. An equivalent construction of the action-angle variables via a generating function is given in Section 1.3.

Some applications of Moser’s twist theorem

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x=y

I

x

Fig. 2. The action variable for the case of periodic potential.

1.1.2. Periodic Hamiltonians with one degree of freedom. Let us now consider Hamiltonians satisfying the periodicity condition H (x, y) = H (x + 1, y). The prime example is the pendulum, or, more generally, a particle in a periodic potential; the corresponding Hamiltonian is given by the kinetic plus potential energy, the latter being periodic: H (x, y) =

1 2 y + V (x), 2

V (x + 1) = V (x),

and we limit our attention to this example. Let us consider the ‘fast’ trajectories, corresponding to the ‘tumbling’ motions in the particular example of the pendulum. These trajectories correspond to the energy levels H > max V chosen so that the particle has enough energy to climb the highest hill of the potential. Formally, from √ the conservation of energy y 2 /2 + V (x) = E (here y = x), ˙ we conclude that x˙ = y = ± 2(E − V ) 6= 0 for all time, and the corresponding trajectory wraps around the phase cylinder T × R = {(x mod 1, y)}, as shown in Figure 2. The action variable I corresponding to the point (x, y) is the area under the trajectory on the phase cylinder: Z 1 I = y dx, 0

where the integration is taken over the energy curve passing through the point (x, y), Figure 2. The angle variable θ is the time normalized by the period of one full revolution: Rx Z dx/y 1 x θ= dx/y = R01 , T 0 dx/y 0

where, again, the integration is taken over the same energy curve. Alternatively, the angle variable can be defined via infinitesimal areas as described in the preceding example. 1.2. Action-angle variables for particles in Rn Consider a particle in Rn moving under the influence of a potential force: x¨ = −∇V (x),

x ∈ Rn .

(5)

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v (x)

x

Fig. 3. An invariant 2-torus in R2 × T2 is an invariant vectorfield on the configuration torus.

Let us assume V to be periodic of period 1 in each of its variables, and let us treat each coordinate of x as an angle variable on the torus Tn = Rn (mod 1). For our purposes it suffices to consider the cases of n = 2 and n = 3 only since these two cases illustrate all the phenomena discussed in this article. In the simple special ‘unperturbed’ case of V = const. we have a free particle, and the associated Hamiltonian system (1) with y = x˙ and H = y 2 /2 is completely integrable. The action variables are the components of the velocity I = y = (y1 , . . . , yn ), and the angle variables are the coordinates of x. The geometrical interpretation of the invariant tori Without the loss of generality, let us consider the case n = 2. The invariant tori live in the phase space T2 × R2 . It is easier to visualize these tori as follows. Each invariant torus is given by {(x, y) : x ∈ T2 , y = (v1 , v2 ) = const.}. In other words, an invariant torus in T2 ×R2 corresponds precisely to a constant vectorfield on the torus T2 . The same interpretation survives a perturbation of the potential V ; an invariant torus (if it exists) in the perturbed case is again a vectorfield v(x) on the configuration torus T2 which is invariant under the dynamics of the system. The invariance means that if a solution x(t) of (5) satisfies x(t) ˙ = v(x(t)) for one value of t, then it satisfies the same relation for all t. In particular, an invariant torus with an irrational rotation number is manifested as a dense trajectory on T2 without self-intersections (see Figure 3).

1.3. A generating function construction of the action-angle variables We describe now an alternative construction of the action-angle variables, well suited to the non-autonomous case  x˙ = Hy (x, y, t) (6) y˙ = −Hx (x, y, t) referred to as the case of 1.5 degrees of freedom. As a side remark, for the higher degrees of freedom the action-angle variables are only defined in the exceptional case of complete

Some applications of Moser’s twist theorem

231

integrability (such as the preceding example), and the particular interpretation of these variables depends on the system. Let S(x, I, t) be the area (Figure 1) bounded by the level curve H enclosing the area I , the y-axis and the line x = const. More precisely, we consider the level curve of H enclosing the area I , and let C x,I,t be the arc of this curve lying above the segment [0, x]; we define Z S(x, I, t) = y dx. (7) C x,I,t

We can now define the action-angle variables (θ, I ) implicitly via Sx (x, I, t) = y,

S I (x, I, t) = θ.

(8)

This map is symplectic: indeed, using (8) gives dx ∧ dy = dx ∧ (Sx x dx + Sx I dI ) = Sx I dx ∧ dI, dθ ∧ dI = (S I x dx + S I I dI ) ∧ dI = S I x dx ∧ dI so that dx ∧ dy = dθ ∧ dI . The action-angle variables satisfy the Hamiltonian system with the Hamiltonian K (θ, I, t) = H ◦ ϕ −1 (θ, I, t) + St (x, I, t), where ϕ := (x, y; t) 7→ (θ, I ; t) is the map defined by (8), and where x = x(θ, I, t) is defined implicitly by (8). As a side remark, in the autonomous case, we have K 0 (I ) = 1/T , where T is the period of the trajectory enclosing the area I . This can be seen directly as follows. Let us consider two nearby orbits, i.e. two level curves γ : H = c and γ+ : H = c + ε. The width of the thin annulus between γ and γ+ , measured along the perpendicular to γ , is ε/|∇ H | (we drop the higher R order terms in ε throughout this paragraph), and the area of the annulus is thus dI = ε ds/|∇ H |, where ds is the length along γ . But the speed of the solution is |∇ H |, and thus ds = |∇ H |dτ , where τ is the time measured along the R solution (with the time t appearing in the right-hand side of (6) frozen). Thus dI = ε dτ = εT , and thus K 0 (I ) = dH/dI = 1/T , as claimed.

1.4. A mechanical interpretation of generating functions The generating function has an illuminating mechanical interpretation. Consider a seemingly very specific example shown in Figure 4. Two bars can slide without friction along the horizontal line. The bars are attached to each other and to the walls by springs, as shown in the figure. The potential P(x, X ) of the mechanical system is a function of the positions x, X of the sliding bars. Let us hold the bars in fixed positions x, X as shown; this requires that we

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M. Levi

y

Y

x

X

Fig. 4. The bars attached to springs can slide on the line. The forces required to keep the bars in respective positions x, X are denoted by y and −Y correspondingly. The potential energy P(x, X ) of the system is then a generating function for a symplectic map (x, y) 7→ (X, Y ). This mapping is defined by (9).

apply some force to prevent the bars from moving. The bars are trying to move with forces f = −Px (x, X ) and F = −PX (x, X ), where the subscripts denote partial derivatives. The last two relations follow from the definition of potential energy. These relations define the map ψ : (x, f ) 7→ (X, F), provided Px X 6= 0. Note that ψ preserves area, up to a change in sign: indeed, imagine holding the bar at x with the left hand, and the bar at X with the right hand. Then move both hands in some periodic fashion, returning to the original position. During the motion, both points: (x, f ) and (X, F) describe closed paths c and C in the plane, with C = ψ(c). But the net work we would do in such a motion is zero: Z Z f dx + F dX = 0, c

C

where the first term is the work done by the left hand and the second term is the work done by the right hand. But this mechanical statement expresses the fact that the two areas are equal, up to a sign. To avoid this sign reversal, we change the sign of one of the forces, by setting Y = −F. For the sake of uniformity of notation, let y = f (no sign change here). The resulting mapping ϕ : (x, y) 7→ (X, Y ) is defined by y = −Px (x, X ),

Y = PX (x, X ).

(9)

We showed by a mechanical argument that this mapping preserves area: Z Z Z y dx = Y dX = y dx. c

ϕ(c)

c

The interpretation extends trivially to higher dimensions. In most applications the generating function P arises not as the energy of a system of springs, but rather as the action of a Lagrangian (the difference of the kinetric and potential energies) of a moving mechanical system: P(x, X ) = min x(·)

Z

t1

L(x(t), x(t))dt, ˙

x(t0 ) = x,

x(t1 ) = X.

t0

As a side remark, any such action can be realized as the potential energy of a certain elastic spring in a potential field. In that case the momenta defined by p(t0 ) = −Px = L x˙ |t=t0 , p(t1 ) = PX = L x˙ |t=t1 acquire the meaning of forces at the ends of this spring!

Some applications of Moser’s twist theorem

233

(t )

(t + t ) z (t ) z (t + t )

Fig. 5. Towards the proof of the twist condition (10).

1.5. The twist condition Twist is a crucial property required for application of KAM theory. In this section we give a simple sufficient condition for twist for Hamiltonian systems with one degree of freedom. We consider the two cases: (1) the integral curves are closed in the plane, as in, say, a cubic potential, and (2) the integral curves encircle the cylinder, as in the case of the pendulum. Convexity of the potential and the twist Let us first consider an important class of ‘Newtonian’ Hamiltonians H (x, y) = y 2 /2 + V (x) with V either sub- or super-quadratic, in the sense made precise shortly. If V is a quadratic function, the system is linear and there is no twist. It it thus natural to expect that the non-quadratic growth of the potential should lead to twist; the following theorem makes this intuition precise. Recall that the action I associated with a point (x, y) is the area enclosed by the level curve through (x, y), while the angle θ = t/T is the time, measured in the units of the period that it takes for the solution to travel from the positive y-axis to (x, y). The system with the Hamiltonian H , written in the action-angle form (3) has ω = T1 , and the twist condition ω0 (I ) 6= 0 amounts to T 0 (I ) 6= 0, implying the monotone dependence of the period on the area, or on the amplitude of the periodic solution. Here is a convenient sufficient condition for the twist. T HEOREM 1. If the potential V satisfies V0 < V 00 , x

(10)

for all x 6= 0 in some interval [−a, a], then the period T of solutions of the Hamiltonian system  x˙ = y (11) y˙ = −V 0 (x) is a monotonic decreasing function of the amplitude, for those solutions which satisfy −a < x(t) < a for all t. If the opposite of the inequality (10) holds, then the period is a monotonic increasing function of the amplitude (see Figure 5).

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For the pendulum we have V 0 = sin x, and the opposite of (10) holds: sin x/x > cos x, or | tan x| > |x| (for x 6= 0). We conclude that the period of the pendulum is a monotonic increasing function of the amplitude, for the class of oscillatory solutions. In the cases of potentials V (x) = x 4 , or V (x) = x 2 + x 4 , or more generally, Pnsuperquadratic 2k V (x) = k=0 ak x , ak ≥ 0, our criterion (10) shows that the period is a decreasing function of the energy, while for the subquadratic potential V (x) = x α with 1 < α < 2 the period increases with the energy. P ROOF OF T HEOREM 1. Let us write the linearization of (11) along a solution z = col(x, y): ( ξ˙ = η (12) η˙ = −V 00 (x)ξ. The linearized solution ζ (t) = col(ξ, η) describes the infinitesimal difference between two nearby solutions of (11). Consider now any solution z(t) of (11) along with the associated solution ζ (t) of the linearized system, with ζ (0) k z(0). If the vector ζ (t) turns clockwise faster than z(t) at t = 0, then the solutions on larger curves travel with higher angular velocity and thus the period of a solution z(t) is a decreasing function of the amplitude, or of the area enclosed by the orbit in the phase plane. The condition on the angular velocities of z and ζ can be written as   d η y η d y > , whenever = . dt x dt ξ x ξ This condition reduces to the desired condition on V without reference to solutions, as follows. Carrying out the differentiations, we obtain x y˙ − x˙ y ηξ ˙ − ξ˙ η > 2 x ξ2

when

y η = . x ξ

Using (11) and (12) to eliminate the derivatives, and using the collinearity: obtain (10).

y x

=

η ξ,

we 

Twist in planar Hamiltonian systems The same idea leads to a twist condition for Hamiltonian vector fields in the plane (not necessarily of the kinetic+potential form): (

x˙ = Hy (x, y) y˙ = −Hx (x, y),

(13)

  0 1 or z˙ = J ∇ H (z), where z = col (x, y), J = −1 0 and ∇ ≡ grad. Assume that the level curves of H are closed, convex and enclose the origin, which is the rest point: ∇ H (0, 0) = 0. Then all solutions of (13) are periodic. Denote by T (E) the period of the solution on the level curve H (z) = E.

Some applications of Moser’s twist theorem

235

T HEOREM 2. In the assumptions of the last paragraph, if hH 00 (z)z, zi < h∇ H (z), zi, i.e. Hx x x 2 + 2Hx y x y + Hyy y 2 < Hx x + Hy y,

(14)

then T (E) is a monotonic increasing function of E. If the opposite of (14) holds, then T (E) is a monotonic decreasing function. The proof is exactly the same as for the preceding theorem. Hamiltonians on a cylinder Consider now the class of Newton-type Hamiltonians y 2 /2 + V (x) with periodic potentials V (x + 1) = V (x). T HEOREM 3. The time advance map ϕ t : (x, y) 7→ (X (t; x, y), Y (t; x, y)) for the system (11) with a periodic potential V is a monotonic twist map in the sense that ∂ X (t; x, y) > 0 ∂y

(15)

if t > 0, for all (x, y) satisfying y 2 /2 + V (x) > supx∈R V.

(16)

P ROOF. Let Z (t) = (X (t; x, y), Y (t; x, y)) be a solution of (11) with initial conditions (x, y) satisfying (16). This means that the solution has enough energy to clear every hill of the potential and thus will travel always to the left or always to the right in the phase plane. Assume, without the loss of generality, that ∂t∂ X (t; x, y) > 0. This means that the velocity Z˙ = ( X˙ , Y˙ ) lies in the right-half plane for all t. Note also that Z˙ is a solution of the linearized system (12), and we thus established that the linearized system possesses a solution that lies in the right half-plane for all time. To finish the proof, note that the y-derivative ζ = (ξ, η) =

∂ Z (t; x, y) ∂y

(17)

(we suppress the initial conditions (x, y) from now on) satisfies the linearized system. Consider the time-dependent sector S(t) defined by the vectors Z˙ (t) and (0, 1). Since the vector field (12) crosses the positive η-axis into the first quadrant, any solution of (12) starting in S(0) at t = 0 lies in S(t) at time t > 0. Thus ζ (t) lies in S(t). But both rays forming S(t) lie in the right-half plane, and thus ζ (t) lies in the right half-plane, which proves (15). 

2. Basic statements of KAM theory In this section we state the two results: Moser’s twist theorem and the Kolmogorov–Arnold theorem. These and related results form what is now referred to as the Kolmogorov–Arnold –Moser (KAM) theory. In the following section we describe a variational approach to

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Moser’s twist theorem, referring to the paper of Salamon and Zehnder [48] for a similar variational proof of the Kolmogorov–Arnold theorem. 2.1. Moser’s twist theorem Consider a perturbation of the twist mapping: ϕ : (θ, I ) 7→ (θ + α(I ), I ) + r (θ, I )

(18)

where the perturbation r is small and periodic in θ of period 1. It is assumed that all the functions are defined for all I ∈ (0, 1) and for all θ ∈ R, and, moreover, that the twist condition d α(I ) > 0 dI holds for all I ∈ (0, 1). The mapping ϕ is a lift of an annulus map. In the unperturbed case where r = 0 the annulus 0 < I < 1 is foliated by invariant circles I = const. It is easy to produce an arbitrarily small area-preserving perturbation that will destroy all the invariant circles – for example, a radial perturbation r = (0, ε). We forbid such ‘nonexact’ perturbations by insisting that ϕ preserves area under any circle I = f (θ ), where f is periodic of period 1. In the statement below, | · |s denotes the C s -norm of a function, i.e. the supremum of the sum of absolute values of all derivatives up to order s. T HEOREM 4 (Moser’s Twist Theorem). Given any ε > 0, τ > 0 and s ≥ 1, let ω satisfy the Diophantine conditions ω − p ≥ ε 1 , (19) q q 2+τ for all integers p, q with q > 0, and α(0) + ε < ω < α(1) − ε. Let α(I0 ) = ω. Then the mapping (18) has an invariant curve with the rotation number ω: γ : u 7→ (u, I0 ) + ρ(u), where ρ is a periodic function of period one with |ρ|s < ε, provided the following hypotheses are satisfied. 1. Every non-contractible closed curve on the annulus intersects its image under ϕ. 2. The twist is bounded from below and from above: c0−1 ≤ α 0 (I ) ≤ c0 . 3. The perturbation r is smaller than a certain constant δ0 depending on ε, s, c0 : |r |0 < δ0 , with a bound on the higher derivatives up to order 4: |α|4 + |r |4 < c0 .

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Some applications of Moser’s twist theorem

Moreover, the mapping ϕ restricted to the invariant curve γ is conjugate to a rotation: ϕ(γ (u)) = γ (u + ω). As stated, this theorem is actually a strengthening of Moser’s original theorem [40]; see Herman [19] for further references and details. 2.2. The Kolmogorov–Arnold theorem The Hamiltonian of a completely integrable Hamiltonian system written in the action-angle variables depends on the action alone. We will now consider perturbations of a completely integrable Hamiltonian system with the Hamiltonian H (I, ϕ) = h(I ) + R(I, ϕ),

(20)

where the remainder R is going to be small. We assume h and R to be real analytic functions of their arguments. To make the concept of smallness precise, we introduce a supremum norm as follows. Given ρ > 0, we define def

kRkρ = sup |R(I, ϕ)|, where the supremum is taken over the complex neighbourhood |Im I | ≤ ρ, |Im ϕ| ≤ ρ. We also need a vector version of the Diophantine condition. Given positive constants C and σ , we will say that the vector ω ∈ Rn is of (C, σ )-type if it satisfies |hω, ki| ≥

C |k|σ

(21)

for all nonzero vectors k ∈ Zn . T HEOREM 5 (Kolmogorov–Arnold). Assume that the Hamiltonian (20) is a real analytic function on the set |Im I | ≤ ρ, |Im ϕ| ≤ ρ for some ρ > 0. Assume that for some I0 the frequency vector h 0 (I0 ) = ω satisfies Diophantine conditions (21) for some positive C, σ . Finally, assume that the Hessian h 00 satisfies |det h 00 (I )| ≥ µ > 0 in a neighbourhood of I0 . Then, if kRkρ is small enough, the Hamiltonian system corresponding to the Hamiltonian (20) possesses an invariant n-torus Tω given by I = I0 + f (ϕ) where f is analytic and small, and any solution with initial conditions on Tω is dense on Tω . In fact, the flow on Tω is analytically conjugate to the rigid translation θ 7→ θ + ωt on the n-torus. 3. A variational approach to Moser’s twist theorem In the variational approach to KAM theory one reduces finding an invariant torus in Rn ×Tn to solving the Euler–Lagrange equation of Percival’s variational functional. A physical

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motivation for this approach is suggested by the Frenkel–Kontorova model. For the case of mappings, covered by Moser’s twist theorem, the existence of KAM tori is reduced to solving a simple-looking second order difference equation. In this section we outline this reduction for area-preserving maps. We do not discuss here the solution of the difference equation itself; the details can be found in [25] for maps and in [48] for flows.

3.1. The generating function of an area-preserving map Consider an area-preserving twist mapping ϕ : R2 → R2 of the covering plane of the cylinder R(mod 1) × R. Assume that ϕ : (x1 , y1 ) 7→ (x2 , y2 ) has a generating function h: h 1 (x1 , x2 ) = −y1 , h 2 (x1 , x2 ) = y2 .

(22)

Here h 1 , h 2 are the derivatives with respect to the first or second argument of h. Similarly we will denote the second derivatives by h 11 , h 12 , h 22 . A large class of area-preserving mappings is given by (22), according to the following theorem. This class includes all area preserving maps that are covered by Moser’s twist theorem. T HEOREM 6. Any smooth twist cylinder map ϕ satisfying the monotonic twist condition ∂ x2 /∂ y1 > 0 possesses a generatingRfunction h with R h 12 < 0 such that the map is given by (22) implicitly. The map is exact: ϕ(γ ) ydx = γ ydx, where γ is an arbitrary smooth noncontractible circle on the cylinder, if and only if h(x1 + 1, x2 + 1) = h(x1 , x2 ).

3.2. Reduction to a difference equation The problem of finding an invariant curve of a map ϕ given by (22) reduces to a difference equation, as follows. We seek an invariant curve in the parametric form w(θ ) = (u(θ ), v(θ)), where u(θ + 1) = u(θ) + 1 is monotonic in θ, and v(θ + 1) = v(θ ). We seek w satisfying ϕ(w(θ )) = w(θ + ω),

(23)

with a prescribed rotation number ω. T HEOREM 7. The curve w(θ) = (u(θ), v(θ)) satisfies the invariance condition (23) under the map ϕ given by (22) if and only if the horizontal coordinate u(θ ) satisfies the second order difference equation E(u(θ )) ≡ h 1 (u(θ), u(θ + ω)) + h 2 (u(θ − ω), u(θ )) = 0. The vertical coordinate v is then given by v(θ) = −h 1 (u(θ ), u(θ + ω)).

(24)

Some applications of Moser’s twist theorem

P ROOF. Since ϕ is given by (22), the condition (23) is equivalent to the system  h 1 (u, u + ) = −v h 2 (u, u + ) = v + ,

239

(25)

where u = u(θ ), u + = u(θ + ω), etc. Replacing θ by θ − ω in the second equation and adding it to the first, we obtain (24).  R EMARK 8. The Equation (24) is the Euler–Lagrange variational equation for the R1 variational problem δ 0 h(u, u + )dθ = 0 (Percival’s variational principle). R EMARK 9. The mean value of u θ E(u) is zero: Z 1 u θ E(u)dθ = 0

(26)

0

as follows from the invariance of the Lagrangian u(θ ) 7→ u(θ + c), or from the identity u θ E(u) =

R1 0

h(u, u + )dθ under θ-translations

∂ h(u, u + ) − ∇(u θ h 2 (u − , u)) ∂θ

where ∇ f = f (θ + ω) − f (θ). Integration gives the claim. E XAMPLE . Consider the standard map x2 = x1 + y1 + V 0 (x1 ) y2 = y1 + V 0 (x1 ); with a periodic function V of period 1. The generating function is h(x1 , x2 ) =

1 (x1 − x2 )2 + V (x1 ), 2

and the Euler–Lagrange difference equation (24) takes a particularly simple form u(θ + ω) − 2u(θ) + u(θ − ω) = V 0 (u(θ)). Here is a quick sketch outlining how a statement on the difference equation (24) leads to a proof of Moser’s twist theorem. One first proves that near an approximate solution u 0 (θ ) of (24) (approximate in the sense that E(u(θ )) is sufficiently small in a certain norm) there exists an exact solution u. This statement requires some assumptions on E, one of which is the Diophantine nature of ω. Now this theorem implies Moser’s twist theorem as follows. Consider a generating function for Moser’s twist map, fix a Diophantine ω and consider the corresponding Euler–Lagrange operator E. The obvious fact that any unperturbed circle is ‘approximately invariant’ under the twist map implies that the function u 0 corresponding to this circle is an ‘approximate solution’ of E(u) = 0. There exists therefore an exact solution near u 0 , and this exact solution gives rise to an invariant curve, thus proving Moser’s twist theorem. All the details of this can be found in [25] (or [48] in the Hamiltonian case).

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4. Applications This section lists some applications of KAM theory. Our goal here is to avoid generality and to rather focus on maximally simple representative examples. More general statements, as well as more detailed history, can be found in the references mentioned in the text. E XAMPLE 1. Consider the particle in a periodic potential V in Rn : x¨ = −ε∇V (x),

x ∈ Rn ,

(27)

where V is periodic of period one in each of its variables. To be specific, let us fix the energy x˙ 2 /2 + V (x) = 1. We ask the same questions as in the previous example. (1) For ε = 0 every action remains constant: x˙ = const. Is it true that for ε 6= 0 every action x˙ remains close to its initial value for all time? (2) For ε = 0 x(t) is a quasiperiodic function of t. What can be said for ε 6= 0? The best known answer to (1) is this: For n = 2, the actions stay close to their initial values for all solutions, for ε sufficiently small. For n ≥ 3, for most (in the sense of Lebesgue measure) initial conditions the actions remain nearly constant and the corresponding solutions are quasiperiodic, for ε sufficiently small. However, there exist potentials V such that some actions do not remain close to their original values, no matter how small ε is. This phenomenon of action drift is referred to as Arnold diffusion, and we mention a little more on it at the end of this chapter. E XAMPLE 2. The system x¨ + (2 + cos t) sin x = 0 can be interpreted as a pendulum whose pivot undergoes periodic oscillations in the vertical direction. It doesn’t seem unreasonable to conjecture that resonance will cause some solutions to gain speed without bound. Do such unbounded solutions exist? KAM theory gives a negative answer, as described later. This system is not close to completely integrable, and it may seems surprising that the KAM theory applies. It turns out, however, that this system is close to a completely integrable one in the range of large |x|. ˙ E XAMPLE 3. Consider the system x¨ + x 3 = ε cos t. For small ε this system is obviously close to a completely integrable one and KAM theory applies directly, with the resulting claim that there the Poincar´e map ϕε : (x, x) ˙ t=0 7→ (x, x) ˙ t=2π has ‘many’ invariant circles. Any solution starting between two such circles stays between them, thus remaining bounded for all time. In addition, any solution starting on an invariant circle is quasiperiodic with two frequencies. But what if ε is not small, say, ε = 1, or ε = 10? Surprisingly, even in that case all solutions stay bounded for all time. This result, due to Morris [39], follows from the fact that the Poincar´e map is close to an integrable one near infinity.

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E XAMPLE 4 (A Large Perturbation). The perturbation in the preceding example was of bounded magnitude. Let us consider a much more violently perturbed oscillator: x¨ + (2 + cos t)x 3 = 0. This perturbation is vastly stronger than the one in the previous example. Nevertheless, even here the solutions are all bounded for all time, as has been shown in [21,11,23]. It turns out that even in this case the system is close to a completely integrable one near infinity, after an appropriate (somewhat delicate) reduction. A more general theorem for potentials above examples and much more.

The following theorem [23] includes all of the

T HEOREM 10. Assume that the function V (x, t) = V (x, t + 1) tends to ∞ as |x| → ∞ and that it satisfies conditions (28)–(30) below. Then the equation x¨ + Vx (x, t) = 0 is near-integrable for large energies; more precisely, for any 0 < ω < 1 satisfying the Diophantine conditions ω − p ≥ K |q|−5/2 for all 0 6= q, p ∈ Z q (where K > 0 is independent of p, q), the Poincar´e map P : (x, x) ˙ t=0 7→ (x, x) ˙ t=1 possesses countably many invariant circles with rotation number ω. These circles cluster at infinity in the (x, x)-plane, ˙ and their relative measure near infinity approaches full measure. Each of these circles is a section of an invariant torus in the extended phase space (x, x, ˙ t mod 1). The flow restricted to each torus is quasiperiodic with basic frequencies 1 and k + ω, where k is an integer. All integers k ≥ k0 (ω) are represented by an invariant torus. In particular, 1. All solutions are bounded for all time: supR (|x| + |x|) ˙ < ∞. 2. Most solutions with large amplitudes are quasiperiodic, i.e. most (in the sense of Lebesgue measure) initial conditions with large |x(0)| + |x(0)| ˙ give rise to quasiperiodic solutions: x(t) = f (t, ωt) where f is a function periodic in both variables. 3. Any sufficiently large number is a rotation number for some solution, i.e. there exists ρ0 such that for any ρ > ρ0 there exists a solution xρ (t) with that rotation number.1 Conditions on the potential V (x, t) For some positive constants c, c1 , a and 0 ≤ µ < 1 1 100 ( 2 − a) and for all x and t we have |∂xk ∂tτ V | ≤ c|x|−k V 1+µ ,

k + τ ≤ 6,

1 1 − + c1 ≤ ∂x W ≤ a < , 2 2

where

(28) W = V /Vx ,

1 This statement is a consequence of the Aubry–Mather theory, [6,32].

(29)

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and |∂xk ∂tτ U |,

|∂xk ∂tτ W | ≤ c|x|1−k ,

k + τ ≤ 5,

where U = Vt /Vx .

(30)

R EMARK . Even a very special case of µ = 0 includes, together with the polynomial, potentials with exponential growth e.g. V (x, t) = (2 + cos t) cosh x. If µ > 0 then potentials with spatial oscillations are allowed as well; an example is V = p(t)(x + √ cos x)2n , where n is sufficiently large. More details on this as well as the proof can be found in [23]. Littlewood’s counterexample In the early 1960s Littlewood produced an example of a periodically forced oscillator with superquadratic potential: x¨ + V 0 (x) = p(t),

p(t + 1) = p(t),

lim V 0 (x)/x → ∞

x→∞

possessing unbounded solutions. A geometrical consequence of the superquadratic growth of V is a twist property of the period map of the plane, in the sense that the angle by which an initial vector (x, y) rotates during one period tends to infinity as |(x, y)| → ∞. It follows that the period map does not satisfy the assumptions of Moser’s twist theorem. Littlewood’s example has a discontinuous p. Zharnitsky [51] extended Littlewood’s counterexample to the considerably more delicate case of continuous forcing p. The (nonfatal) errors in Littlewood’s original paper [27] were corrected by Y. Long [30]. A much simplified analysis of Littlewood’s counterexample with an optimal estimate on a possible V which makes such an example possible, is given in [24]. E XAMPLE 5 (A Quasiperiodic Perturbation). Consider the system in Example 4 with quasiperiodic time dependence, replacing 2 + cos t with f (t) = F(ω1 t, . . . , ωn t), where F(ξ1 , . . . , ξn ) > 0 is a function periodic of period one in each of its n variables, and where ωk are rationally independent frequencies satisfying, moreover, an appropriate Diophantine condition: x¨ + f (t)x 3 = 0. √ For example, one can take f (t) = 3 + cos t + cos 2t. Even in this case all solutions are bounded for all time, and most solutions with large initial conditions are quasiperiodic. A much more general class of oscillators with quasiperiodic forcing has been analyzed in [26]. E XAMPLE 6 (Fermi–Ulam’s Ping-pong). A particle bounces between two parallel walls, moving along a line perpendicular to the walls. The walls oscillate periodically, both with the same period. Between the collisions the particle moves with constant velocity. The collisions are perfectly elastic, which means that, at the moment of collision, the particle changes only the sign of its velocity when viewed from the reference frame of the wall: v2 − w = −(v1 − w), where w is the speed of the wall and v1 , v2 are the speeds before and after collision, all measured in the frame attached to the origin. This gives v2 = −v1 + 2w:

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Fig. 6. An invariant circle for a billiard map corresponds to a caustic.

the ball gains twice the speed of the wall in collision. It is natural to ask whether there are initial conditions for which the particle can have unbounded speed, i.e. whether it is possible for a particle to gain an unbounded amount of energy from the moving walls. If the motion of the walls is sufficiently smooth, then the answer, according to KAM theory, is negative: for any initial condition the velocity is bounded for all time. The proof can be found in [16,47,21]. A similar result for quasiperiodically moving walls has been proven in [52]. E XAMPLE 7. Instead of a particle bouncing between two walls, consider a ball bouncing up and down on a periodically oscillating racket. If the oscillations of the racket are smooth, then any sufficiently small amplitude bounce will remain of small amplitude, bounded from both above and below, for all time. Most small amplitude oscillations are quasiperiodic [52]. E XAMPLE 8. Consider a billiard with a smooth boundary close to an ellipse, in the C k+1 norm. The corresponding billiard map is then C k -close to the integrable billiard map of the ellipse. Integrability of the latter map manifests itself in the fact that all segments of a trajectory are tangent to the same confocal ellipse (as shown in Figure 6) or hyperbola. If k ≥ 4 + ε, Moser’s invariant curve theorem, as strengthened by Herman [19], applies. The resulting invariant curves of the billiard mapping correspond to quasiperiodic billiard orbits. The infinite family of segments of such an orbit has an envelope, called the caustic, Figure 6. The caustics cluster at the boundary, and their density, in the sense of measure, approaches one near the boundary. Moreover, one has stability in the following sense: if one chord of an orbit does not meet a caustic, then the entire orbit is forever trapped in the ring outside the caustic. The first application of Moser’s twist theorem to billiards is due to Lazutkin [22]. Zharnitsky gave a simplified proof; we refer to [53] for further details and references. This result is intimately related to the previous one of the ball bouncing on a periodically oscillating racket, as follows. Consider a billiard particle travelling close to the boundary. To better keep track of this particle, let us consider its shadow on the boundary: the foot of the perpendicular from the particle to the boundary. Let us put ourselves into the reference frame of that foot, and imagine the inward normal direction as the direction vertically up.

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p

K

Tp

Fig. 7. The outer billiard.

With this choice, we are now in an accelerating frame, and the particle will no longer seem to be travelling with constant acceleration. Instead, it will seem to us as if the particle were under the influence of centrifugal gravity which forces the particle to bounce up and down repeatedly. In short, we are dealing with a particle bouncing on a plate under the influence of periodically varying gravity. E XAMPLE 9. The outer billiard map, also referred to as the Neumann billiard map, is defined as follows. We are given a closed curve K , Figure 7. From a point p outside K we draw the tangent line l to K in the counterclockwise direction, and let T p 6= p be the point on l equidistant with p from the point of tangency. Are there any points p whose iterates T k p escape to infinity? If the boundary K is sufficiently smooth and its curvature is never zero, then the answer is negative, [17,21]. The proof relies on the non-obvious observation that near infinity T is a near-integrable map, provided K is a ‘nice’ curve. A brief background on adiabatic invariants Before describing in the next paragraph a result due to Arnold, we give a brief sketch on adiabatic invariants. To focus on a simple case, consider a slowly changing Hamiltonian system 

x˙ = Hy (x, y, εt) y˙ = −Hx (x, y, εt).

(31)

Since the derivative along solutions is small: dtd H = ε∂τ H (x, y, τ )τ =εt = O(ε), we expect H to change by O(1) during time 1/ε, along a typical solution. It turns out, however, that despite the ‘large’ change of H , the area enclosed by the energy curve H (x, y, τ ) = const. associated with the initial and the final points of the solution (x, y)(t) changes little (by O(ε)). Rather than giving a rigorous statement and proof [3], we mention that two geometrical properties are responsible for this phenomenon: (1) the incompressibility of the Hamiltonian flow, and (2) the ergodicity of the motion of solutions of a frozen system along their energy curves. A beautiful and much earlier explanation of adiabatic invariance is due to Einstein, at least for the special case of a pendulum. This explanation is based on the observation that as we (say) shorten the length of the pendulum, we do not only raise the pendulum, but also work against the extra centrifugal tension of the string. Using the conservation of energy then leads to the conclusion of adiabatic invariance of the area, i.e. of the action. More details on this can be found in [43].

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E XAMPLE 10 (Adiabatic Invariants for all Time). Another application of KAM theory is the following. Consider a slowly changing time-periodic Hamiltonian system (31), where H (x, y, τ ) = H (x, y, τ + 1). Assume, for simplicity, that H (0, 0, τ ) = 0 for all τ and that the level curves H (x, y, τ ) = c are closed for all τ and for all c > 0. Assume also that the frozen system has twist, i.e. that ∂ T (H, τ )/∂ H 6= 0 for all H > 0 and all τ , where T (H, τ ) is the period of the solution with energy H . Finally, assume that H is sufficiently smooth – say, real analytic. Under these assumptions Arnold’s theorem states that, for ε sufficiently small, the system possesses an adiabatic invariant valid for all time (and not just for time of order 1/ε). For more details we refer to [1,5] and references therein. 5. Arnold diffusion In conclusion, we mention briefly the phenomenon of ‘Arnold diffusion’: the ‘drift’ of action for near-integrable systems. This drift can happen only to the orbits that do not lie on KAM tori, and for small perturbations these drifting orbits form a small portion of the phase space. In fact, full proof of existence of such orbits in ‘typical’ systems is still not available, although great progress has been made and Mather announced a solution. There is by now a large literature on the subject; we mention [2,7–10,12,13,15,16,20,29, 31,33–38,44,45,49,50]. References [1] V. Arnold, On the behavior of an adiabatic invariant under a slow periodic change of the Hamiltonian, Soviet Math. Dokl. 3 (1962), 136–139. [2] V. Arnold, Instabilities in dynamical systems with several degrees of freedom, Sov. Math. Dokl. 5 (1964), 581–585. [3] V. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd edn, SpringerVerlag, New York, Berlin, Heidelberg (1988). [4] V. Arnold, Mathematical Methods of Classical Mechanics, 2nd edn, Graduate Texts in Mathematics, Vol. 60 (1989). [5] V.I. Arnold, V.V. Kozlov and A.I. Neishtadt, Mathematical aspects of classical and celestial mechanics, Dynamical Systems III, 3rd edn, Encyclopaedia of Mathematical Sciences, Vol. 3, Springer-Verlag, Berlin (2006); Translated from the Russian original by E. Khukhro. [6] S. Aubry and P.Y. LeDaeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground states, Physica 8D (1983), 381–422. [7] P. Bernard and G. Contreras, A generic property of families of Lagrangian systems, Ann. of Math. (2) 167 (3) (2008), 1099–1108. [8] M. Berti and Ph. Bolle, A functional analysis approach to Arnold diffusion, Ann. Inst. H. Poincare 19 (4) (2002), 395–450. [9] U. Bessi, An approach to Arnold’s diffusion through the calculus of vartiations, Nonlinear Anal. 26 (6) (1996), 1115–1135. [10] U. Bessi, L. Chierchia and E. Valdinoci, Upper bounds on Arnold diffusion times via Mather theory, J. Math. Pures Appl. 80 (1) (2001), 105–129. [11] L. Bin, Boundedness for solutions of nonlinear Hill’s equations with Periodic Forcing Terms via Moser’s twist Theorem, J. Differential Equations 79 (1989), 304–315. [12] J. Bourgain and V. Kaloshin, On diffusion in high-dimensional Hamiltonian systems, J. Funct. Anal. 229 (1) (2005), 1–61. [13] C.-Q. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom. 67 (3) (2004), 457–517.

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[14] R. de la Llave, A tutorial on KAM theory, Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), A. Katok, R. de la Llave, Ya. Pesin and H. Weiss, eds, Proc. Sympos. Pure Math., Vol. 69, Amer. Math. Soc., Providence, RI (2001), 175–292. [15] A. Delshams, R. de la Llave and T. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model, Mem. Amer. Math. Soc. 179 (844) (2006), viii+141 pp. ´ [16] R. Douady, Stabilit´e ou instabilit´e des points fixes elliptiques, Ann. Sci. de l’E.N.S. 4e s´erie 21 (1) (1988), 1–46. [17] R. Douady, Ph. D. Thesis, Ecole Polytechnique (1988). [18] J. F´ejoz, D´emonstration du th´eor`eme d’Arnol’d” sur la stabilit´e du syst´eme plan´etaire (d‘apr´es Michael Herman), Michael Herman Memorial Issue, Ergodic Theory Dynam. Systems 24 (5) (2004), 1521–1582. Updated version at http://www.institut.math.jussieu.fr/˜fejoz/. [19] M.R. Herman, Sur les courbes invariantes par les diff´eomorphismes de l’anneau II, Asterisque 144 (1986). [20] V. Kaloshin and M. Levi, An example of Arnold diffusion for near-integrable Hamiltonians, Bull. Amer. Math. Soc. (N.S.) 45 (3) (2008), 409–427. [21] S. Laederich and M. Levi, Invariant curves and time-periodic potentials, Ergodic Theory Dynam. Systems 11 (1991), 365–378. [22] V.F. Lazutkin, Existence of caustics for the billiard problem in a convex domain, Math. USSR-Izvestiya 7 (1973), 185–214. [23] M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Commun. Math. Phys. 143 (1991), 43–83. [24] M. Levi, On Littlewood’s counterexample of unbounded motions in superquadratic potentials, Dynamics Reported, 1 (New Series) (1992), 113–124. [25] M. Levi and J. Moser, A Lagrangian proof of the invariant curve theorem for twist mappings, Proc. Symp. Pure Math. 69 (2001), 733–746. [26] M. Levi and E. Zehnder, Boundedness of solutions for quasiperiodic potentials, SIAM J. Math. Anal. 26 (5) (1995), 1233–1256. [27] J.E. Littlewood, Unbounded solutions of an equation y¨ + g(y) = p(t) with p(t) periodic and g(y)/y → ∞ as y → ±∞, J. London Math. Soc. 41 (1966), 497–507. [28] P. Lochak, Canonical perturbation theory: an approach based on joint approximations, Uspekhi Mat. Nauk 47 (6(288)) (1992), 59–140; Translation in Russian Math. Surveys 47 (6) (1992), 57–133. [29] P. Lochak and J.-P. Marco, Diffusion times and stability exponents for nearly integrable analytic systems, Cent. Eur. J. Math. 3 (3) (2005), 342–397. [30] Y. Long, An unbounded solution of a superlinear Duffing’s equation, Acta Math. Sinica (N.S.) 7 (4) (1991), 360–369. [31] J.-P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publ. Math. Inst. Hautes Etudes Sci. (96) (2002), 199–275. [32] J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology 21 (1985), 457–476. [33] J. Mather, Modulus of continuity of Peierls’ barrier, Periodic Solutions Hamiltonian Systems and Related Topics, P. Rabinowitz, ed., NATO ASI Series C, Vol. 209 (1987), 177–202. [34] J. Mather, Action minimizing invariant measures for positive definite Lagrangians, Math. Z. 207 (1991), 169–207. [35] J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier 43 (1993), 1349–1386. [36] J. Mather, Arnold diffusion. I: Announcement of results, J. Math. Sci. 124 (5) (2004), 5275–5289. [37] J. Mather, Arnold diffusion, II, preprint, 2006, 160 pp. [38] J. Mather, Total disconnectedness of the quotient Aubry set in low dimensions, Comm. Pure Appl. Math. 56 (8), 1178–1183. [39] G.R. Morris, A case of boundedness in Littlewood’s problem on oscillatory differential equations, Bull. Austr. Math. Soc. 14 (1976), 71–93. [40] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachrichten der Akademie der Wissenschaften, G¨ottingen, Math.-Phys, Klasse IIa, 1962, pp. 1-20. [41] J. Moser, A stability theorem for minimal foliations of the torus, Ergodic Theory Dynam. Systems 8 (1988), 151–188.

Some applications of Moser’s twist theorem

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[42] J. Moser and E. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, Vol. 12 (2005). [43] K. Nakamura, Quantum Chaos, Cambridge University Press (1993). [44] N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspekhi Math. Nauk 32 (1) (1977), 5–66. [45] L. Niederman, Prevalence of exponential stability among nearly integrable Hamiltonian systems, Ergodic Theory Dynam. Systems 27 (2007), 905–928. [46] J. P¨oschel, Integrability of Hamiltonian Systems on Cantor Sets, Comm. Pure Appl. Math. 35 (1982), 653–696. [47] L.D. Pustyl’nikov, Stable and oscillating motions in non-autonomous dynamical systems, Trans. Moscow Math. Soc. 14 (1978), 1–101. [48] D. Salamon and E. Zehnder, KAM theory in configuraition space, Comm. Math. Helv. 64 (1989), 84–132. [49] D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity 17 (5) (2004), 1803–1841. [50] J. Xia, Arnold diffusion: a variational construction, Proc of the ICM, Vol. II (Berlin, 1998) (1998), 867–877. [51] V. Zharnitsky, Breakdown of stability of motion in superquadratic potentials, Comm. Math. Phys. 189 (1) (1997), 165–204. [52] V. Zharnitsky, Instability in Fermi-Ulam “ping-pong” problem, Nonlinearity 11 (6) (1998), 1481–1487. [53] V. Zharnitsky, Invariant tori in Hamiltonian systems with impacts, Comm. Math. Phys. 211 (2) (2000), 289–302. [54] V. Zharnitsky, Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem, Nonlinearity 13 (4) (2000), 1123–1136.