V.I. Arnold's "pointwise" KAM Theorem

TL;DR

This paper reviews Arnold's 1963 KAM theorem proof, optimizing the scheme to derive sharp asymptotic conditions with explicit constants as perturbation strength approaches zero.

Contribution

It provides an optimized version of Arnold's scheme with explicit constants, leading to sharper asymptotic conditions in KAM theory.

Findings

Explicit constants are computed for the theorem.

Optimized scheme yields sharper asymptotic conditions.

Results improve understanding of small perturbations in Hamiltonian systems.

Abstract

We review V.I. Arnold's 1963 celebrated paper \cite{ARV63} {\sl Proof of A.N. Kolmogorov's theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian}, and prove that, optimizing Arnold's scheme, one can get "sharp" asymptotic quantitative conditions (as $\varepsilon\to 0$, $\varepsilon$ being the strength of the perturbation). All constants involved are explicitly computed.

Equations

$$H(y,x)=K(y)+\epsilon P(y,x),$$

$$|\omega \cdot k|:=\sum _{j=1}^d|{\omega }_j{k}_j|\ge \frac{\alpha }{|k{|}^{\tau }},\forall k\in {\mathbb{Z}}^d\backslash \{0\};$$

$$H(y,x)=\frac{1}{2}{y}^2+\epsilon (\cos x-1),$$

$$y=\pm \sqrt{2\epsilon (1-\cos x)},$$

$$|y_0|>2\sqrt{\epsilon }.$$

$$\frac{\epsilon }{{\alpha }^2}<\frac{1}{4}.$$

$$y_{\epsilon }(x):=\sqrt{y_0^2+2\epsilon (1-\cos x)}=y_0+v_{\epsilon }(x),$$

$$v_{\epsilon }(x):=\frac{2\epsilon (1-\cos x)}{y_0+\sqrt{y_0^2+2\epsilon (1-\cos x)}}.$$

$$\mathrm{osc}(y_{\epsilon })=\mathrm{osc}(v_{\epsilon })\ge v_{\epsilon }(\pi )-v_{\epsilon }(0)=\frac{4\epsilon }{y_0+\sqrt{y_0^2+4\epsilon }}=\frac{\epsilon }{y_0}\frac{4}{1+\sqrt{1+4\epsilon /y_0^2}},$$

$$\mathrm{osc}(v_{\epsilon })\ge \frac{4}{1+\sqrt{2}}\cdot \frac{\epsilon }{\alpha }$$

$$\frac{\epsilon }{{\alpha }^2}<c,$$

$$\mathrm{osc}(v_{\ast })\le C\cdot \frac{\epsilon }{\alpha },$$

$$H_j:=K_j+{\epsilon }^{2^j}{P}_j$$

$${\partial }_y{K}_j(y_j)=\omega :={\partial }_{y}K(y_0),\det {\partial }_y^2{K}_j(y_j)\ne 0,$$

$$\underset{j\to +\infty }{\lim }{\varphi }_0\circ \cdots {\varphi }_{j-1}(y_j,{\mathbb{T}}^n).$$

$$\det\begin{pmatrix}\partial^{2}_{y}K&\partial_{y}K\\ \partial_{y}K&0\end{pmatrix}\Big{|}_{y=y_{0}}\neq 0\,.$$

$${\Delta }_{\alpha }^{\tau }:=\left\{\omega \in {}^d:|\omega \cdot k|\ge \frac{\alpha }{|k{|}_1^{\tau }},\forall 0\ne k\in {\mathbb{Z}}^d\right\},$$

$${\mathbb{T}}_s^d$$

$$B_r(y_0)$$

$$D_r(y_0)$$

$$\mathbb{J}:=\left(\begin{array}{cc}0& -{\text{mathbbm}1}_d\\ {\text{mathbbm}1}_d& 0\end{array}\right).$$

$$\Vert \cdot {\Vert }_{r,s,y_0}:=\underset{D_{r,s}(y_0)}{\sup }|\cdot |.$$

$$\Vert \cdot {\Vert }_{r,y_0}:=\underset{D_r(y_0)}{\sup }|\cdot |,\Vert \cdot {\Vert }_s:=\underset{{\mathbb{T}}_s^d}{\sup }|\cdot |.$$

$$\varpi :=dy\wedge dx=d{y}_1\wedge d{x}_1+\cdots +d{y}_d\wedge d{x}_d,$$

$$\Vert \mathcal{L}\Vert :=\underset{x\in V_1\setminus \{0\}}{\sup }\frac{\Vert \mathcal{L}x{\Vert }_2}{\Vert x{\Vert }_1},\text{mbox}sothat\Vert \mathcal{L}x{\Vert }_2\le \Vert \mathcal{L}\Vert \Vert x{\Vert }_1\text{mbox}foranyx\in V_1.$$

$$D_{\omega }f:=\omega \cdot f_x=\sum _{j=1}^d{\omega }_j{f}_{x_j}.$$

$$f=\sum _{k\in {\mathbb{Z}}^d}{f}_k{e}^{ik\cdot x},f_k:=\frac{1}{(2\pi {)}^d}{\int }_{{\mathbb{T}}^d}f(x)e^{-ik\cdot x}dx,$$

$$⟨f⟩:=f_0=\frac{1}{(2\pi {)}^d}{\int }_{{\mathbb{T}}^d}f(x)dx,({\mathbf{p}}_{N}f)(x):=\sum _{|k{|}_1\le N}{f}_k{e}^{ik\cdot x},N>0.$$

$$\left\{\begin{array}[]{l}{\omega}\coloneqq{\partial}_{y}K(y_{0})\in{\Delta}^{\tau}_{\alpha}\,,\\ \\ \det({\partial}^{2}_{y}K(y_{0}))\not=0\;.\end{array}\right.$$

$$T:={\partial }_y^{2}K(y_0{)}^{-1},\mathsf{P}:=\Vert P{\Vert }_{r,s,y_0},\mathsf{K}:=\Vert {\partial }_y^{2}K{\Vert }_{r,y_0},\mathsf{T}:=\Vert T\Vert ,\theta :=\mathsf{TK},$$

$$\boxed{ϵ:=\mathsf{KP}\frac{\epsilon }{{\alpha }^2}}.$$

Full text

1. 1 Introduction

2. 2 Notation and quantitative statement of Arnold’s Theorem

3. 3 Proof

4. 3.1 Arnold’s scheme: the basic step

5. 3.2 Arnold’s scheme: Iteration

6. 3.2.1 First step

7. 3.2.2 Subsequent steps, iteration and convergence

8. 3.3 Conclusion

9. Appendix

10. A Constants

11. B Kolmogorov’s non–degeneracy

12. C Reminders

13. C.1 Classical estimates (Cauchy, Fourier)

14. C.2 Implicit function theorem

1 Introduction

• a.

“One of the most remarkable of A.N. Kolmogorov’s mathematical achievements is his work on classical mechanics of 1954”: this is the beginning of V.I. Arnold’s celebrated paper Proof of A.N. Kolmogorov’s theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian [3], published in 1963, on the occasion of A.N. Kolmogorov’s 60th birthday. Few lines after, Arnold adds: “Its deficiency has been that complete proofs have never been published.

Even though one could argue whether Kolmogorov’s proof in [13] is “complete” or not (see, e.g., [7]), Arnold’s paper is certainly a milestone of modern dynamical systems, which not only contains a complete and detailed proof of Kolmogorov’s Theorem, but, also, introduces new original, technical ideas, of enormous impact in finite and infinite dimensional systems (for reviews, see, e.g., [4] or [11]).

• b.

Kolmogorov’s 1954 theorem in classical mechanics [13] (see, also, [7]), deals, as is well known, with the persistence, for small ${\varepsilon}$, of Lagrangian invariant tori of analytic integrable systems governed by a nearly integrable Hamiltonian

[TABLE]

where $(y,x)\in{}^{d}\times{\mathbb{T}}^{d}$ are standard symplectic action–angle variables. In short, the theorem says that:

for small ${\varepsilon}$, non–degenerate Diophantine unperturbed Lagrangian tori persist

Let us recall that “Diophantine” means that the unperturbed torus $\mathcal{T}_{{\omega},0}\coloneqq\{y_{0}\}\times{\mathbb{T}}^{d}$, which is invariant for the flow $\phi_{K}^{t}$ governed by the integrable Hamiltonian $K$, is such that the frequency ${\omega}\coloneqq K_{y}(y_{0})$ is Diophantine, i.e., it satisfies, for some ${\alpha},{\tau}>0$,

[TABLE]

“non–degenerate” means that the Hessian of $K$ at $y_{0}$ is invertible; finally, “persists” means that $\mathcal{T}_{{\omega},0}$ deforms, for positive small enough ${\varepsilon}$, into a a Lagrangian111A Lagrangian manifold is a submanifold of dimension $d$ on which the restriction of the two form $\sum_{j=1}^{d}dy_{j}\wedge dx_{j}$ vanishes. torus $\mathcal{T}_{{\omega},{\varepsilon}}$ invariant for $\phi_{H}^{t}$.

The scheme on which Arnold’s proof of Kolmogorov’s theorem is based, while sharing two basic ideas of Kolmogorov’s approach – namely, the use of a quadratic symplectic iterative method and the idea of keeping fixed the Diophantine frequency of the motion – is quite different from Kolmogorov’s scheme in the following respects.

First, for a fixed frequency, Arnold constructs an embedded, Lagrangian invariant torus obtained as a limit of symplectic transformations on action domains shrinking to a single point; in contrast, Kolmogorov conjugates the given Hamiltonian to a complete normal form admitting a Lagrangian invariant torus with the prescribed frequency.

A key difference between these two approaches is that, Arnold, at each step of the iteration, needs to control only a finite number of small divisors222To work with a finite number of divisors, Arnold introduces a Fourier cut–off (depending, in view of analyticity, logarithmically on the size of the perturbation), an idea which has been widely followed also in infinite dimensional Hamiltonian perturbation theory., which however depend on actions (this being the reason for the shrinking to one point of the action domains), while in the denominators appearing in Kolmogorov’s scheme there enters only the prefixed Diophantine frequency, allowing one to control at once all small divisors, and also to work with smaller and smaller domains, which contain a fixed open set, allowing one, in the end, to get a genuine symplectic transformation.

A clever quantitative revisitation of Kolmogorov’s scheme ([18]) shows that such a scheme leads to optimal asymptotic estimates (as ${\varepsilon}\to 0$). We shall show below that this is true also for Arnold’s original “pointwise” scheme.

• c.

Kolmogorov’s and Arnold’s schemes are “‘pointwise” in the sense that they deal with the continuation of a single prefixed unperturbed Lagrangian torus with Diophantine frequency. This is in contrast with versions of the KAM theorem333Striclty speaking, there does not exists a KAM Theorem (“KAM” standing for the initials of A.N. Kolmogorv, V.I. Arnold and J.K. Moser), however, normally, it refers to (variations of) Kolmogorov’s theorem. Here, we follow this tradition. dealing with the persistence of sets of simultaneously persistent invariant tori, see [3], [16], [15], [9]. We point out that, actually, Arnold’s original formulation of the KAM theorem in [3] belongs to this second kind of theorem as it states the existence of a set of simultaneously invariant tori, however, the proof is pointwise in nature and its scheme is exactly the scheme we follow closely here. Typically, especially when one is concerned with lower dimensional invariant tori, it is not possible to construct a single torus with some pre–assigned property, but, rather, one obtains “Cantor” families of persistent tori (compare, e.g., [11]).

• d.

The smallness condition, i.e., how small the perturbation has to be in order for the perturbed invariant torus to exist, depends on local analytic properties of $K$ (and on the analytic norm of $P$). In particular, the main quantitative “competition” is between ${\varepsilon}$ and the size of the small divisors appearing in the iterative scheme, the size of which may be measured by the “homogeneous Diophantine constant” ${\alpha}$ (compare Eq. (2)) of the prefixed frequency ${\omega}=K_{y}(y_{0})$.

The most important quantitative relations may be easily understood by looking at explicitly solvable examples, i.e., at integrable systems.

To illustrate this point, let us consider, for example, a simple pendulum with gravity ${\varepsilon}$,

[TABLE]

viewed as an ${\varepsilon}$–perturbation of the non–degenerate Hamiltonian $K(y)\coloneqq\frac{1}{2}y^{2}$, (here, $d=1$). The energy zero level $\{H=0\}$ corresponds to the separatrix, i.e.,

[TABLE]

which shows immediately that in the region $\mathcal{S}:=\{|y|\leq 2\sqrt{\varepsilon}\}$ there are no homotopically trivial invariant tori (curves) or, equivalently, no Lagrangian invariant curves, which are graphs over the angle variable (“primary tori”). In other words, the region of action space where unperturbed curves $\{y_{0}\}\times{\mathbb{T}}$ may be continued into invariant Lagrangian invariant curves, which stay out of the “singular region” $\mathcal{S}$ are such that:

[TABLE]

Now, the resonant relations $|K_{y}(y_{0})\cdot k|$ become, in this one–dimensional example, simply $|y_{0}||k|$ and the Diophantine condition is, therefore, equivalent to requiring that ${\alpha}=|y_{0}|$ (recall (2)), and the necessary condition (4) becomes:

[TABLE]

Another fact that can be easily extracted from this example concerns the oscillations of (primary) invariant tori444A primary Lagrangian torus is a graph over the angles $\{(y,x)|\ y=U(x)\,,x\in{\mathbb{T}}^{d}\}$ and its oscillation is given by $\sup_{x,x^{\prime}}|U(x)-U(x^{\prime})|$..

For $y_{0}>0$ the invariant (primary) curves are given by

[TABLE]

with

[TABLE]

Thus, one has that

[TABLE]

which, in view of (5), yields the relation

[TABLE]

Below, we shall prove that the enhanced Arnold’s scheme leads to a smallness condition of the type (compare (14) below)

[TABLE]

(for an ${\varepsilon}$ and ${\alpha}$ independent constant $c$), which is in agreement with (5).

Furthermore, we shall also show that Arnold’s scheme leads to a bound on the oscillations of persistent tori given as graphs $\{y=y_{0}+v_{*}(x),x\in{\mathbb{T}}^{d}\}$ of the form (compare (16) below)

[TABLE]

(for an ${\varepsilon}$ and ${\alpha}$ independent constant $C$), which, in view of (6), is seen to be optimal (as far as the dependence upon ${\varepsilon}$ and ${\alpha}$ is concerned), showing the “quantitative sharpness” of Arnold’s scheme, on which the proof presented below is based.

Condition (7) is also the fundamental quantitative relation needed to evaluate the measure of the Kolmogorov’s set, i.e., the union (in a prefixed bounded domain) of all primary tori. Indeed, (7) leads to a bound on the Lebesgue measure of the complement of the Kolmogorov’s set by a constant times $\sqrt{\varepsilon}$ (compare [16], [15]), which again, comparing with the simple pendulum (3) – that has a region (the area enclosed by the separatrix) of measure $16\sqrt{\varepsilon}$ free of primary tori – is seen to be asymptotically optimal. It has to be remarked, however, that obtaining such an estimate is quite delicate and far from trivial (for a more detailed discussion on this point, see [5], [14], [10]).

• e.

As is well known, Arnold’s scheme is an iterative Newton scheme yielding a sequence of “renormalised Hamiltonians”

[TABLE]

so that $H_{0}=H$ is the given nearly integrable Hamiltonian (1) and, for any $j$, $K_{j}$ is integrable (i.e., depends only on the action variable $y$), real–analytic in a $r_{j}$–ball around a point $y_{j}$ close to $y_{0}$ and satisfies:

[TABLE]

which means that at each step the frequency is kept fixed and that the integrable Hamiltonian $K_{j}$ is non–degenerate. The sequence of Hamiltonians $H_{j}$ are conjugated, i.e., $H_{j+1}=H_{j}\circ\phi_{j}$, with $\phi_{j}$ symplectic, closer and closer to the identity. The persistent torus $\mathcal{T}_{{\omega},{\varepsilon}}$ is then obtained as the limit

[TABLE]

The symplectic transformations $\phi_{j}$’s are obtained by solving the classical Hamilton–Jacobi equation so as to remove quadratically the order of the perturbation. In doing this one cannot take into account all small divisors (which are dense) and therefore Arnold introduces a Fourier cut–off ${\kappa}_{j}$, which allows him to deal with a finite number of small divisors. In view of the exponential decay of Fourier coefficients, ${\kappa}_{j}$ can be taken \sim\big{|}\log\big{(}e^{2^{j}\|P_{j}\|}\big{)}\big{|}, which introduces a logarithmic correction555For full details, see § 3.1 below, and in particular “Step 1: Construction of Arnold’s transformation”., that does not affect the convergence of the scheme. All this is well known.

The problem is to equip the scheme with “optimal” quantitative estimates, which may lead, at the end, to the above sharp asymptotic bounds. This involves careful choices of various parameters entering the scheme (see § 3.2) and, in particular, it is crucial to treat the first step in a different way with respect to the remaining steps: this technical, but important, aspect is explained in Remark 4 below.

• f.

V.I. Arnold pointed out that his proof extended with little changes to the iso–energetically non–degenerate case, i.e., when the energy is prescribed and the unperturbed Hamiltonian satisfies the condition666The matrix in (10) is a $(d+1)\times(d+1)$ matrix, where the upper right corner $\partial_{y}K$ has to be interpreted as a column vector, while the lower left corner is a raw vector and the zero is a scalar. The condition expresses the fact the map $(y,{\lambda})\mapsto({\lambda}\partial_{y}K,K)$ is locally invertible.

[TABLE]

Indeed, it would not be difficult to adapt our improved Arnold’s scheme also to the iso–energetically non–degenerate case, proving the sharpness of the asymptotic smallness conditions also in this case.

• g.

Finally, we mention that the quantitative estimates provided in this paper could be used to improve the (exponentially long) stability time of “nearly–invariant tori”, introduced in [12].

2 Notation and quantitative statement of Arnold’s Theorem

• $\bullet$

For $d\in{\mathbb{N}}\coloneqq\{1,2,3,...\}$ and $x,y\in{\mathbb{C}}^{d}$, we let $x\cdot y\coloneqq x_{1}\bar{y}_{1}+\cdots+x_{d}\bar{y}_{d}$ be the standard inner product; $|x|_{1}\coloneqq\displaystyle\sum_{j=1}^{d}|x_{j}|$ be the $1$–norm, and $|x|\coloneqq\displaystyle\max_{1\leq j\leq n}|x_{j}|$ be the sup–norm.

• $\bullet$

${{\mathbb{T}}^{d}}\coloneqq{{}^{d}}/2{\pi}{{\mathbb{Z}}^{d}}$ is the standard $d$–dimensional (flat) torus.

• $\bullet$

$\pi_{1}\colon{{\mathbb{C}}^{d}}\times{{\mathbb{C}}^{d}}\ni(y,x)\longmapsto y$ and $\pi_{2}\colon{{\mathbb{C}}^{d}}\times{{\mathbb{C}}^{d}}\ni(y,x)\longmapsto x$ are the projections on the first and second component respectively.

• $\bullet$

For ${\alpha}>0$, ${\tau}\geq d-1\geq 1$,

[TABLE]

is the set of $({\alpha},{\tau})$–Diophantine numbers in d.

• $\bullet$

For $r,s>0$, $y_{0}\in{\mathbb{C}}^{d}$, we denote:

[TABLE]

• $\bullet$

If ${\mathbbm{1}}_{d}\coloneqq{\,\rm diag\,}(1)$ is the unit $(d\times d)$ matrix, we denote the standard symplectic matrix by

[TABLE]

• $\bullet$

For $y_{0}\in{}^{d}$, $\mathcal{A}_{r,s}(y_{0})$ denotes the Banach space of real–analytic functions with bounded holomorphic extensions to $D_{r,s}(y_{0})$, with norm

[TABLE]

We also denote:

[TABLE]

• $\bullet$

We equip ${{\mathbb{C}}^{d}}\times{{\mathbb{C}}^{d}}$ with the canonical symplectic form

[TABLE]

and denote by $\phi_{H}^{t}$ the associated Hamiltonian flow governed by the Hamiltonian $H(y,x)$, $y,x\in{\mathbb{C}}^{d}$, i.e., $z(t)\coloneqq\phi_{H}^{t}(y,x)$ is the solution of the Cauchy problem $\dot{z}=\mathbb{J}\nabla H(z)$, $z(0)=(y,x)$.

• $\bullet$

Given a linear operator $\mathcal{L}$ from the normed space $(V_{1},\|\cdot\|_{1})$ into the normed space $(V_{2},\|\cdot\|_{2})$, its “operator–norm” is given by

[TABLE]

• $\bullet$

Given ${\omega}\in{{}^{d}}$, the directional derivative of a $C^{1}$ function $f$ with respect to ${\omega}$ is given by

[TABLE]

• $\bullet$

If $f$ is a (smooth or analytic) function on ${\mathbb{T}}^{d}$, its Fourier expansion is given by

[TABLE]

(where, as usual, $\,e\coloneqq\exp(1)$ denotes the Neper number and $i$ the imaginary unit). We also set:

[TABLE]

${\bf p}_{N}$ being the Fourier projection onto the Fourier modes with $|k|_{1}\leq N$; notice that $\langle\cdot\rangle={\bf p}_{0}(\cdot)$.

We are ready to formulate a quantitative version of Arnold’s Theorem777To avoid to introduce too many symbols, we use capital straight style for positive constants ($\mathsf{P},\mathsf{K},\mathsf{T},\mathsf{C},...$), while, usually, capital normal style is used for functions or matrices ($K,P,H,T,...$). :

Theorem A Let $d\geq 2$; ${\tau}\geq d-1$; ${\alpha},r,{\varepsilon}>0$; $0<s_{*}<s\leq 1$; $y_{0}\in{{}^{d}}$; $K,P\in\mathcal{A}_{r,s}(y_{0})$; $H:=K+{\varepsilon}P$. Assume that

[TABLE]

Define:

[TABLE]

and denote by ${\epsilon}$ the rescaled smallness parameter:

[TABLE]

There exist constants $1<\mathsf{C}<\mathsf{C}_{*}$ depending only on $d$ and $\tau$, such that, if $a\coloneqq 6{\tau}+3d+8$ and

[TABLE]

then, there exists a real–analytic embedding

[TABLE]

where $\phi_{\rm e}$ is the trivial embedding

[TABLE]

such that the $d$–torus

[TABLE]

is a Lagrangian torus satisfying

[TABLE]

Furthermore,

[TABLE]

Remarks and addenda

• (i)

$\theta$ is a measure of the local “torsion” and is a number greater than or equal to one:

[TABLE]

• (ii)

Notice that the estimate on $v_{*}$ in (16) implies that the maximal action oscillation of the torus $\mathcal{T}_{{\omega},{\varepsilon}}$ is bounded by a constant times ${\alpha}{\epsilon}$, which in view of (13), is $\sim{\varepsilon}/{\alpha}$ as advertised in (8).

• (iii)

All numerical constants are explicitly “computed” during the proof. A complete list of them, including the definitions of $\mathsf{C}_{*}$ and $\mathsf{C}$, is given in Appendix A.

• (iv)

The torus $\mathcal{T}_{{\omega},{\varepsilon}}$ is Kolmogorov non–degenerate. More precisely, $H$ can be put in Kolmogorov’s normal form with non–degenerate quadratic part: there exists a symplectic transformation $\phi$ close to $\phi_{\rm e}$, for which

[TABLE]

for details, see Appendix B.

• (v)

The value of ${\epsilon}_{*}$ in (14) is not optimal. In Remark 5 a better (still not optimal) value is given.

• (vi)

The dependence of the invariant torus $\mathcal{T}_{{\omega},{\varepsilon}}$ on ${\varepsilon}$ is analytic. More generally, if $H=H(y,x;z)$ is real–analytic also in $z\in V$, $V$ being some open set in ${\mathbb{C}}^{m}$, and all the above norms are uniform in $z\in V$, then the invariant torus $\mathcal{T}_{{\omega},z}$ is real analytic in $V$. This is an obvious corollary of Weierstrass’s theorem on uniform limits of holomorphic functions, in view of the uniformity of the limits in the proof.

3 Proof

3.1 Arnold’s scheme: the basic step

The next Lemma describes Arnold’s basic KAM step, on which Arnold’s scheme is based. Its quantitative formulation involves a few constants, which are defined as follows:

[TABLE]

Lemma 1

Let888$K$ and $P$ stand, here, for generic real analytic Hamiltonians which, later on, will respectively play the roles of $K_{j}$ and $P_{j}$, and $\mathsf{y},\,r$, the roles of $y_{j},\,r_{j}$ in the iterative step. $r>0,\,0<2{\sigma}<s\leq 1$, $\mathsf{y}\in{}^{d}$, $K,P\in\mathcal{A}_{r,s}(\mathsf{y})$ and consider the Hamiltonian parametrised by ${\varepsilon}>0$

[TABLE]

Assume that

[TABLE]

and let $\mathsf{K}$, $\mathsf{T}$ and $\mathsf{P}$ be positive numbers such that

[TABLE]

*where $T\coloneqq K_{yy}(\mathsf{y})^{-1}$.

Now, let ${\lambda},\check{r},\bar{r}$ be positive number such that:*

[TABLE]

where

[TABLE]

Finally, define

[TABLE]

Then, if

[TABLE]

there exist $\mathsf{y}^{\prime}\in{{}^{d}}$ and a symplectic change of coordinates

[TABLE]

*such that *

[TABLE]

where

[TABLE]

Moreover, letting

[TABLE]

the following estimates hold:

[TABLE]

where

[TABLE]

Observe that

[TABLE]

so that (20) implies

[TABLE]

which, in particular, implies that ${\lambda}>1$ and ${\kappa}>4$.

Proof

**Step 1: Construction of Arnold’s transformation **

We seek a near–identity symplectic transformation

[TABLE]

with $D_{r_{1},s_{1}}(\mathsf{y}^{\prime})\subset D_{r,s}(\mathsf{y})$, generated by a generating function999Following the classical approach of Arnold, we use generating functions to construct symplectic transformations. Of course one could also use the equivalent method of time–one Hamiltonian flows (or Lie series). of the form $y^{\prime}\cdot x+{\varepsilon}g(y^{\prime},x)$, so that

[TABLE]

such that

[TABLE]

By Taylor’s formula, we get101010Recall (§2) that ${\left\langle\cdot\right\rangle}$ stands for the average over ${{\mathbb{T}}^{d}}$ and that ${\bf p}_{N}$ is the Fourier projection onto modes with $|k|_{1}\leq N$.

[TABLE]

with ${\kappa}>0$, which will be chosen large enough so that $P^{(3)}=O({\varepsilon})$ and

[TABLE]

By the non–degeneracy condition $\det K_{yy}(\mathsf{y})\neq 0$, for ${\varepsilon}$ small enough (to be made precised below), $\det{\partial}_{y^{\prime}}^{2}K^{\prime}(\mathsf{y})\neq 0$ and, therefore, by the standard Inverse Function Theorem (see, e.g., Lemma A.2), there exists a unique $\mathsf{y}^{\prime}\in D_{r}(\mathsf{y})$ such that the second part of (26) holds. In view of (27), in order to get the first part of (26), we need to find $g$ such that $K_{y}(y^{\prime})\cdot g_{x}+{\bf p}_{{\kappa}}P(y^{\prime},\cdot)-\widetilde{K}(y^{\prime})$ vanishes; such a $g$ is indeed given by

[TABLE]

provided that

[TABLE]

But, in fact, since $K_{y}(\mathsf{y})$ is rationally independent, then, given any ${\kappa}>0$, there exists $\bar{r}\leq r$ such that

[TABLE]

The last step is to invert the function $x\mapsto x+{\varepsilon}g_{y^{\prime}}(y^{\prime},x)$ in order to define $P^{\prime}$. By the Inverse Function Theorem, for ${\varepsilon}$ small enough, the map $x\mapsto x+{\varepsilon}g_{y^{\prime}}(y^{\prime},x)$ admits a real–analytic inverse of the form

[TABLE]

so that the Arnold’s symplectic transformation is given by

[TABLE]

Hence, (26) holds with

[TABLE]

**Step 2: Quantitative estimates

**First of all, notice that from the definitions of $\bar{r}$ and $\check{r}$ it follows that

[TABLE]

We begin by extending the “Diophantine condition w.r.t. $K_{y}$” uniformly to $D_{\bar{r}}(\mathsf{y})$ up to the order ${\kappa}$. Indeed, by the Mean Value Inequality and $K_{y}(\mathsf{y})={\omega}\in{\Delta}^{\tau}_{\alpha}$, we get, for any $0<|n|_{1}\leq{\kappa}$ and any $y^{\prime}\in D_{\bar{r}}(\mathsf{y})$,

[TABLE]

so that, by Fourier estimates (Lemma A.1–(ii)), we have

[TABLE]

[TABLE]

where

[TABLE]

Analogously,

[TABLE]

and, by Cauchy’s estimate (Lemma A.1–(i)) we get

[TABLE]

where

[TABLE]

Also,

[TABLE]

[TABLE]

Next, we prove the existence and uniqueness of $\mathsf{y}^{\prime}$ in (26). Let $U_{{\varepsilon}}\coloneqq\{\eta\in{\mathbb{C}}:|\eta|<2{\varepsilon}\,\}$ and consider the map:

[TABLE]

Then

• •

$F(\mathsf{y},0)=0,\quad F_{y}(\mathsf{y},0)^{-1}=K_{yy}(\mathsf{y})^{-1}=T$.

• •

For any $(y,\eta)\in D_{\check{r}}(\mathsf{y})\times U_{{\varepsilon}}$,

[TABLE]

• •

Recalling ${\sigma}\leq\frac{1}{{2}}$, we have

[TABLE]

Therefore, we can apply the Inverse Function Theorem (Lemma A.2). Hence, there exists a function $g\colon U_{{\varepsilon}}\to D_{\check{r}}(\mathsf{y})$ such that its graph coincides with $F^{-1}(\{0\})$. In particular, $\mathsf{y}^{\prime}\coloneqq g({\varepsilon})$ is the unique $y\in D_{\check{r}}(\mathsf{y})$ satisfying $0=F(y,{\varepsilon})={\partial}_{y}K^{\prime}(y)-{\omega}$, i.e., the second part of (26). Moreover,

[TABLE]

so that

[TABLE]

Next, we prove that ${\partial}^{2}_{y}K^{\prime}(\mathsf{y}^{\prime})$ is invertible. Indeed, by Taylor’ formula, we have

[TABLE]

and, by Cauchy’s estimate,

[TABLE]

Hence ${\partial}_{y^{\prime}}^{2}K^{\prime}(\mathsf{y}^{\prime})$ is invertible with

[TABLE]

and

[TABLE]

Next, we prove estimate on $P_{+}$. We have,

[TABLE]

so that, for any $(y^{\prime},x)\in D_{\bar{r},\bar{s}}(\mathsf{y})$,

[TABLE]

and thus

[TABLE]

[TABLE]

and by Fourier estimates (Lemma A.1–(ii)), we have,

[TABLE]

Hence,

[TABLE]

Finally, we prove that, given $y^{\prime}\in D_{\bar{r}}(\mathsf{y})$, the function $\psi_{\varepsilon}(x)=x+{\varepsilon}g_{y^{\prime}}(y^{\prime},x)$ has an analytic inverse111111Observe that $\psi_{\varepsilon}(id+{\varepsilon}u)=id$ is equivalent to $u=-g_{y^{\prime}}(y^{\prime},id+{\varepsilon}u)$, i.e., $u$ is a fixed–point of the map $u\mapsto-g_{y^{\prime}}(y^{\prime},id+{\varepsilon}u)$.. Consider the Banach’s space

[TABLE]

For any $u\in\mathcal{B}$ and any $x^{\prime}\in{\mathbb{T}}^{d}_{s^{\prime}}$, we have ${\rm\,Im\,}(x^{\prime}+{\varepsilon}u(x^{\prime}))\leq s^{\prime}+{\varepsilon}\,\|u\|_{s^{\prime}}\leq s^{\prime}+{\varepsilon}\,\overline{\mathsf{L}}\stackrel{{\scriptstyle{\rm(\ref{cond1Bisv2})}}}{{\leq}}s^{\prime}+{\sigma}/6=s^{\prime\prime}.$ Hence, the functional $f\colon\mathcal{B}\ni u\mapsto-g_{y^{\prime}}(y^{\prime},{\rm id}+{\varepsilon}u)$ is well–defined and smooth. Moreover, for any $u\in\mathcal{B},$

[TABLE]

Thus, $f\colon\mathcal{B}\to\mathcal{B}$. Furthermore, for any $u_{1},u_{2}\in\mathcal{B}$,

[TABLE]

Hence, $f$ is a contraction. Therefore, by the Banach–Caccioppoli fixed–point Theorem, $f$ has a unique fixed–point $\widetilde{{\varphi}}_{\varepsilon}\in\mathcal{B}$; $\widetilde{{\varphi}}_{\varepsilon}$ is obtained as the uniform limit $\displaystyle\lim_{n}f^{n}(0)$ (as $0\in\mathcal{B}$). Thus, as $f^{0}=f$ is real–analytic on $D_{\bar{r}}(\mathsf{y})\times{\mathbb{T}}^{d}_{s^{\prime}}$, by Weierstrass’s Theorem on the uniform convergence of analytic functions, $\widetilde{{\varphi}}_{\varepsilon}$ is real–analytic on $D_{\bar{r},s^{\prime}}(\mathsf{y})$. The rest of the claims on $\phi^{\prime}$ and $P^{\prime}$ are then obvious.

3.2 Arnold’s scheme: Iteration

Let $d$, $\tau$, $H$, $K$, $P$, $T$, ${\varepsilon}$, ${\alpha}$, $r$, $s$, $s_{*}$, ${\mathsf{P}}$, $\mathsf{K}$, $\mathsf{T}$, $\theta$, ${\epsilon}$ be as in Theorem A. Set $K_{0}\coloneqq K\;,\ P_{0}\coloneqq P\;,\ H_{0}\coloneqq H$. Then, starting from $H_{0}$, we shall iterate infinitely many times Lemma 1.

The very first step being quite different from all the others, it shall be done separately.

Before starting, let us give some definitions121212Recall the definitions of ${\nu}$ and $\mathsf{C}_{4}$ given at the beginning of § 3.1..

[TABLE]

We also set, for $j\geq 0$:

[TABLE]

Observe that

[TABLE]

Note, also, that, since $\hat{\epsilon}_{0}$ is proportional to ${\varepsilon}$, $\mathsf{P}_{1}$ is independent of ${\varepsilon}$.

3.2.1 First step

Lemma 2

Assume

[TABLE]

Then, there exist $y_{1}\in D_{r_{0}}(y_{0})$ and a real–analytic symplectic transformation

[TABLE]

such that, for $H_{1}\coloneqq H_{0}\circ\phi_{0}$ , we have

[TABLE]

and

[TABLE]

**Proof **Since

[TABLE]

and

[TABLE]

we get

[TABLE]

Thus,

[TABLE]

Therefore, Lemma 1 implies Lemma 2.

3.2.2 Subsequent steps, iteration and convergence

For $j\geq 1$, define

[TABLE]

Thus, for any $j\geq 1$, one has

[TABLE]

i.e.,

[TABLE]

Once the first step is completed, all the following steps do not need any other condition. Actually, the first condition in (41) is no longer necessary and the second condition needs to be strengthen merely a little bit more. To be precise, the following holds.

Lemma 3

Assume ${\rm(\ref{HjBis0v2})}\div{\rm(\ref{estfin2Bis00011v2})}$ and

[TABLE]

Then, one can construct a sequence of symplectic transformations

[TABLE]

so that

[TABLE]

*converges uniformly.

More precisely, ${\varepsilon}^{2^{j-1}}P_{j-1}$, $\phi^{j-1}\coloneqq\phi_{1}\circ\phi_{2}\circ\cdots\circ\phi_{j-1}$, $K_{j-1}$, $y_{j-1}$ converge uniformly on $\{y_{*}\}\times\displaystyle{\mathbb{T}}^{d}_{s_{*}}$ to, respectively, [math], $\phi^{*}$, $K_{*}$, $y_{*}$ and $H_{1}\circ\phi^{*}=K_{*}$ with $\phi^{*}$ real–analytic for $x\in\displaystyle{\mathbb{T}}^{d}_{s_{*}}$ and $\det{\partial}^{2}_{y}K_{*}(y_{*})\neq 0$. Finally, the following estimates hold for any $i\geq 1$:*

[TABLE]

Remark 4

Notice that $\mathsf{P}_{1}$ is actually independent of ${\varepsilon}$ (and, in particular, of $\log{\epsilon}^{-1}$), while $\mathsf{P}_{j}$ for $j\geq 2$ does depend on $\log{\epsilon}^{-1}$ through ${\lambda}_{*}$. This is a crucial point, which allows, at the end, to get optimal bounds on the displacement of the persistent invariant torus from the unperturbed one.

**Proof **First of all, notice that, for any $i\geq 1$,

[TABLE]

where

[TABLE]

For a given $j\geq 2$, let $(\mathscr{P}^{j})$ be the following assertion:

there exist $j-1$ symplectic transformations131313Compare (21).

[TABLE]

and $j-1$ Hamiltonians $H_{i+1}=H_{i}\circ\phi_{i}=K_{i+1}+{\varepsilon}^{2^{i+1}}P_{i+1}$ real–analytic on $D_{r_{i+1},s_{i+1}}(y_{i+1})$ such that, for any $1\leq i\leq j-1$,

[TABLE]

and

[TABLE]

Assume $(\mathscr{P}^{j})$, for some $j\geq 2$ and let us check $(\mathscr{P}^{j+1})$. Fix $1\leq i\leq j-1$. Then,

[TABLE]

and, similarly,

[TABLE]

which prove the two first relations in (59) for $i=j$. Also

[TABLE]

so that

[TABLE]

Moreover,

[TABLE]

Thus, by last relation in (60), for any $1\leq i\leq j-1$,

[TABLE]

which proves the fourth relation in (59) for $i=j$. Furthermore, by exactly the same computation as above, one gets

[TABLE]

which proves the last relation in (59) for $i=j$. It remains only to check that the fifth relation in (59) holds as well for $i=j$ in order to apply Lemma 1 to $H_{i}$, $1\leq i\leq j$ and get (60) and, consequently, $(\mathscr{P}^{j+1})$. In fact, we have141414Notice that $\left(\log t\right)^{2s}\leq t^{1/2}\;,\quad\forall\;t\geq\,e,\quad\forall\;s\geq 1/4,$ so that ${\epsilon}_{0}(\log{\epsilon}_{0}^{-1})^{2{\nu}}\stackrel{{\scriptstyle{\rm(\ref{condBisv2Prt})}}}{{\leq}}\sqrt{{\epsilon}_{0}}\leq\,e^{-1/2}<1$, which in turn proves the r.h.s. inequality in (62).

[TABLE]

so that

[TABLE]

To finish the proof of the induction, i.e., to construct an infinite sequence of Arnold’s transformations satisfying (59) and (60) for all $i\geq 1$, one needs only to check $(\mathscr{P}^{2})$. Thanks to151515Observe that for $j=2$, $i=1$. ${\rm(\ref{HjBis0v2})}\div{\rm(\ref{estfin2Bis00011v2})}$, we just need to check the two last inequalities in ${\rm(\ref{bbbBisv2})}_{i=1}$. But, in fact, this is contained in the above computation. Then, we apply Lemma 1 to $H_{1}$ to get ${\rm(\ref{bes06v2})}_{i=1}$ and ${\rm(\ref{C.1Bisv2})}_{i=1}$, which achieves the proof of $(\mathscr{P}^{2})$.

Next, we prove that $\phi^{j}$ is convergent by proving that it is a Cauchy sequence. For any $j\geq 4$, we have, using again Cauchy’s estimate (and noting that $2^{i-1}\geq i,\,\forall\;i\geq 0$),

[TABLE]

Therefore, for any $n\geq 1,\,j\geq 0$,

[TABLE]

Hence, by (51), $\phi^{j}$ converges uniformly on $\{y_{*}\}\times{\mathbb{T}}^{d}_{s_{*}}$ to some $\phi^{*}$, which is then real–analytic map in $x\in{\mathbb{T}}^{d}_{s_{*}}$.

To estimate $|\mathsf{W}_{0}(\phi^{*}-{\rm id})|$ on $\{y_{*}\}\times{\mathbb{T}}^{d}_{s_{*}}$, observe that , for $i\geq 1$,

[TABLE]

and therefore

[TABLE]

Moreover, for any $i\geq 1$,

[TABLE]

which iterated yields

[TABLE]

Therefore, taking the limit over $i$ completes the proof of (56) and hence of Lemma 3.

3.3 Conclusion

We can now complete the proof of Theorem A. Let

[TABLE]

Observe that

[TABLE]

Then,

[TABLE]

and

[TABLE]

Hence, (14) implies the smallness conditions (41) and (51). Therefore, Lemma 2 and 3 hold. Now, set $\phi_{*}\coloneqq\phi_{0}\circ\phi^{*}$ and observe that, uniformly on $\{y_{*}\}\times{\mathbb{T}}^{d}_{s_{*}}$,

[TABLE]

Moreover, for any $i\geq 1$,

[TABLE]

and then passing to the limit, we get

[TABLE]

Thus, the triangle inequality gives

[TABLE]

which proves the bounds on $\|u_{*}\|$ and $\|v_{*}\|$ in (16). Let us now prove the bound on ${\partial}_{x}u_{*}$ in (16). Set

[TABLE]

Then, for any $j\geq 0$, we have

[TABLE]

so that

[TABLE]

which implies

[TABLE]

and then letting $j\rightarrow\infty$, we get the estimate on ${\partial}_{x}u_{*}$.

Remark 5

As it is easy to check, Theorem A holds under the milder condition ${\epsilon}\leq{\epsilon}_{\sharp}$ where

[TABLE]

Notice that ${\epsilon}_{*}<{\epsilon}_{\sharp}$.

Indeed, condition

[TABLE]

guaranties the convergence of Arnold’s scheme, while condition

[TABLE]

ensures that the Torus $\mathcal{T}_{{\omega},{\varepsilon}}$ is a Lagrangian graph (over the “angle” variables).

Acknowledgements L.C. has been supported by the ERC grant HamPDEs under FP7 n. 306414 and the PRIN national grant “Variational Methods in Analysis, Geometry and Physics”. The authors are indebted with an anonymous referee for valuable suggestions and corrections.

Appendix

Appendix A Constants

For convenience, we collect here the list of constants appearing in the proof of Theorem A.

Recall that ${\tau}\geq d-1\geq 1$ and notice that all $\mathsf{C}_{i}$’s are greater than $1$ and depend only upon $d$ and ${\tau}$.

[TABLE]

Appendix B Kolmogorov’s non–degeneracy

Let

[TABLE]

Since $\|{\partial}_{x}u_{*}\|_{s_{*}}\stackrel{{\scriptstyle{\rm(\ref{est})}}}{{\leq}}1/2$, then ${\rm id}+u_{*}$ is a diffeomorphism of ${{\mathbb{T}}^{d}}$. Letting

[TABLE]

we have

[TABLE]

In [17] it is proven that the map

[TABLE]

is symplectic. Then,

[TABLE]

with:

[TABLE]

which show that $\langle Q_{yy}(0,\cdot)\rangle$ is invertible.

Appendix C Reminders

C.1 Classical estimates (Cauchy, Fourier)

Lemma A.1

[6]* Let $p\in{\mathbb{N}},\,r,s>0,y_{0}\in{{\mathbb{C}}^{d}}$ and $f$ a real–analytic function $D_{r,s}(y_{0})$ with*

[TABLE]

*Then,

(i) For any multi–index $(l,k)\in{\mathbb{N}}^{d}\times{\mathbb{N}}^{d}$ with $|l|_{1}+|k|_{1}\leq p$ and for any $0<r^{\prime}<r,\,0<s^{\prime}<s$,161616As usual, ${\partial}_{y}^{l}\coloneqq\frac{{\partial}^{|l|_{1}}}{{\partial}y_{1}^{l_{1}}\cdots{\partial}y_{d}^{l_{d}}},\,\forall\,y\in{{}^{d}},\,l\in{{\mathbb{Z}}^{d}}$.*

[TABLE]

(ii)* For any $k\in{{\mathbb{Z}}^{d}}$ and any $y\in D_{r}(y_{0})$*

[TABLE]

C.2 Implicit function theorem

Lemma A.2

[8]* Let $r,s>0,\,n,m\in{\mathbb{N}},\,(y_{0},x_{0})\in{\mathbb{C}}^{n}\times{\mathbb{C}}^{m}$ and171717Here, $D^{n}_{r}(z_{0})$ denotes the ball in ${\mathbb{C}}^{n}$ centered at $z_{0}$ and with radius $r$.*

[TABLE]

be continuous with continuous Jacobian matrix $F_{y}$. Assume that $F_{y}(y_{0},x_{0})$ is invertible with inverse $T\coloneqq F_{y}(y_{0},x_{0})^{-1}$ such that

[TABLE]

Then, there exists a unique continuous function $g\colon D^{m}_{s}(x_{0})\to D^{n}_{r}(y_{0})$ such that the following are equivalent

• $(i)$

$(y,x)\in D^{n}_{r}(y_{0})\times D^{m}_{s}(x_{0})$* and $F(y,x)=0$;*

• $(ii)$

$x\in D^{m}_{s}(x_{0})$* and $y=g(x)$.*

Moreover, $g$ satisfies

[TABLE]