# Kolmogorov-Arnold-Moser Theory

(Redirected from KAM theory)
Post-publication activity

Curator: Luigi Chierchia

Kolmogorov-Arnold-Moser (KAM) theory deals with persistence, under perturbation, of quasi-periodic motions in Hamiltonian dynamical systems.

An important example is given by the dynamics of nearly-integrable Hamiltonian systems. In general, the phase space of a completely integrable Hamiltonian system of $$n$$ degrees of freedom is foliated by invariant $$n$$-dimensional tori (possibly of different topology). KAM theory shows that, under suitable regularity and non-degeneracy assumptions, most (in measure theoretic sense) of such tori persist (slightly deformed) under small Hamiltonian perturbations. The union of persistent $$n$$-dimensional tori (Kolmogorov set) tend to fill the whole phase space as the strength of the perturbation is decreased.

The major technical problem arising in this context is due to the appearance of resonances and of small divisors in the associated formal perturbation series.

## Classical KAM theory

The main objects studied in KAM theory are $$d$$-dimensional embedded tori $$\mathcal{T}^d$$ invariant for a Hamiltonian flow $$\phi^t_H: \mathcal{M}^{2n}\to\mathcal{M}^{2n}\ ,$$ where $$t\in\mathbb{R}$$ denotes the time variable and $$H=H(p,q)$$ is a (smooth enough or analytic) Hamiltonian function depending on $$2n$$ symplectic (or canonical) variables $$p=(p_1,...,p_n)$$ and $$q=(q_1,...,q_n)$$ defined on the phase space $$\mathcal{M}^{2n}\ .$$ This means that if $$(p_0,q_0)\in\mathcal{T}^d\ ,$$ then $$\phi^t_H(p_0,q_0)\in\mathcal{T}^d$$ for any $$t\in\mathbb{R}\ ,$$ $$\phi^t_H(p_0,q_0)=(p(t),q(t))$$ denoting the solution of the (standard) Hamilton equations $\tag{1} \left\{\begin{array}{l}\dot p = - \partial_q H(p,q)\\ \dot q = \partial_p H(p,q)\end{array}\right.\quad{\rm with\ initial\ data }\quad \left\{\begin{array}{l} p(0)=p_0\\ q(0)=q_0\end{array}\right. .$

Here, the dot represents time derivative, while $$\partial_z$$ denotes the gradient with respect to the $$z$$ variables.

A $$d$$-dimensional (embedded and smooth or analytic) invariant torus for $$\phi_H^t\ ,$$ with $$2\le d\le n\ ,$$ is called a KAM torus if:

• the flow $$\phi^t_H$$ on $$\mathcal{T}^d$$ is conjugated to a linear translation $$\theta \to \theta + \omega t\ ,$$ where $$\theta=(\theta_1,...,\theta_d)$$ belongs to the standard $$d$$-dimensional torus $$\mathbb{T}^d=\mathbb{R}^d/(2\pi \mathbb{Z})^d\ ;$$ the vector $$\omega=(\omega_1,...,\omega_d)\in\mathbb{R}^d$$ is called the frequency vector;
• the frequency vector $$\omega$$ is rationally independent and "badly" approximated by rationals, typically in a Diophantine sense:

$\tag{2} \exists\ \gamma, \tau>0\ {\rm such\ that} \quad |\omega\cdot k|:= |\sum_{j=1}^d \omega_j k_j|\ge \frac{\gamma}{\|k\|^\tau}\ ,\ \forall\ k\in\mathbb{Z}^d\backslash\{0\}\ .$

From measure theory, it follows that the set of Diophantine vectors in $$\mathbb{R}^d$$ is of full Lebesgue measure.

Note that the case $$d=1$$ corresponds to periodic trajectories of period $$2\pi/\omega$$ (this case is normally excluded in classical KAM theory since does not involve small divisors). On the other hand, the case $$d=n$$ corresponding to maximal KAM tori is particularly relevant.

### Kolmogorov normal forms and Kolmogorov's Theorem

Let $$H$$ be a real-analytic Hamiltonian on $$\mathcal{M}^{2n}=U\times \mathbb{T}^n$$ (with $$U$$ an open region in $$\mathbb{R}^n$$) and assume that $$\mathcal{T}^n$$ is a maximal KAM torus for $$H$$ and that it is a (Lagrangian) graph over the angle variables. Then there exists a symplectic transformation $$\phi: (y,x)\to(p,q)$$ (i.e., a diffeomorphism preserving the canonical 2-form $$\displaystyle \sum_{i=1}^n dp_i\wedge dq_i$$) transforming $$H$$ in Kolmogorov normal form: $\tag{3} H\circ\phi(y,x)=K(y,x):=E+\omega\cdot y + Q(y,x)$

for some number $$E$$ (the energy level of the KAM torus), some Diophantine frequency vector $$\omega$$ and $$Q$$ a function vanishing together with its first $$y$$-derivatives at $$y=0\ .$$ In the "new" variables $$(y,x)\ ,$$ the $$n$$-torus $$\{0\}\times\mathbb{T}^n$$ is obviously a KAM torus for the transformed Hamiltonian $$H\circ\phi\ .$$

One says that the Kolmogorov normal form $$K$$ in (3) is non-degenerate if the Hessian matrix (with respect to $$y$$) of the average of $$Q$$ over $$\mathbb{T}^n$$ is regular.

Kolmogorov's Theorem (Kolmogorov, 1954) Let $$K$$ be a real-analytic non-degenerate Kolmogorov's normal form and let $$P$$ be a real-analytic function in a neighborhood of $$\{y=0\}\times\mathbb{T}^n\ .$$ Then, there exists $$\epsilon_0>0$$ and for any $$|\epsilon|<\epsilon_0$$ a real-analytic symplectic transformation $$\phi_\epsilon\ ,$$ close to the identity, such that, if $$H_\epsilon$$ denotes the perturbed Hamiltonian $$K+\epsilon P\ ,$$ then $$H_\epsilon\circ\phi_\epsilon$$ is a non-degenerate Kolmogorov's normal form with the same frequency vector of $$K\ .$$

Thus, in particular, $$\mathcal{T}_\epsilon^n:=\phi_\epsilon(\{0\}\times \mathbb{T}^n)$$ is a (real-analytic, non-degerate) KAM torus for $$H_\epsilon$$ and such a torus is $$\epsilon$$-close to $$\{0\}\times\mathbb{T}^n\ .$$

### Nearly-integrable Hamiltonian systems

A nearly-integrable Hamiltonian system is a Hamiltonian system governed by a Hamiltonian function of the form $$H_\epsilon(y,x)=K(y)+\epsilon P(y,x)$$ with $$y=(y_1,...,y_n)$$ (action variables) varying in a domain $$B\subset \mathbb{R}^n$$ and $$x=(x_1,...,x_n)$$ (angle variables) varying in the standard $$n$$-dimensional torus $$\mathbb{T}^n\ .$$ For $$\epsilon=0\ ,$$ equations (1) give $$\dot y=0$$ and $$\dot x=\partial_y K(y)\ ,$$ hence $$y=y_0=$$ constant and $$x = x_0 + \omega_0\, t$$ (mod $$2\pi$$), with $$\omega_0:= \partial_y K\big(y_0\big)\ .$$ Thus the torus $$\{y_0\}\times \mathbb{T}^n$$ is invariant for the flow $$\phi^t_{K}\ ,$$ and if $$\omega_0$$ is Diophantine and $$\partial_y^2 K(y_0)$$ is invertible, then such a torus is a non-degenerate KAM torus for $$H_0=K\ .$$ Since $$K(y)$$ can be expanded by Taylor's formula as $K=K(y_0)+\omega_0\cdot (y-y_0)+ \frac{1}{2} \partial_y^2 K(y_0) (y-y_0) \cdot(y-y_0)+ O(|y-y_0|^3|),$ it follows from Kolmogorov's Theorem that for $$\epsilon$$ small enough such tori persist, giving rise to non-degenerate KAM tori for $$H_\epsilon$$.

### Moser's differentiable version

J.K. Moser (followed by H. Rüssmann, J. Pöschel and others) showed that the real-analyticity assumption is not necessary. Indeed, Kolmogorov's Theorem holds under the milder assumption that $$H$$ is a $$C^\ell$$ differentiable function with $$\ell>2n$$ (meaning that $$H$$ is of class $$C^{2n}$$ and that the derivatives of order $$2n$$ are Hölder continuous).

Originally (Moser, 1962), Moser's work focused on $$C^{333}$$ (exact symplectic) perturbations of integrable twist mappings of the annulus (the most famous example being the so-called standard map). In this case, maximal KAM tori correspond to homotopically non-trivial curves intersecting each radius in only one point. The number of derivatives were reduced to 5 by H. Rüssmann (Rüssmann, 1970), and M. Herman (Herman, 1983) showed that the theorem is valid for $$C^k$$ perturbation with $$k\ge 3\ ,$$ but false for $$k<3\ .$$

### Small divisors and classical KAM techniques

KAM techniques (i.e., the analytical tools used to prove statements in KAM theory) constitute the hard core of KAM theory and play a major role in applications, extensions and, in general, in the full comprehension of the results. The main technical problem is related to the appearance of small divisors in the Fourier series of perturbative expansions (averaging methods, series expansions of quasi-periodic motions, etc.).

Small divisors are expressions of the form $$\omega \cdot k=\sum_{i=1}^d \omega_i k_i$$ with $$k\in\mathbb{Z}^d\backslash\{0\}$$ an integer vector, which usually are related to Fourier modes associated to the perturbing function and where the frequency vector $$\omega$$ often depends upon the slow (action) variables. Such expressions appear in the denominator of (formal) Fourier expansions of the object one aims to construct (e.g., a generating function or the formal expansion of a quasi-periodic solution). Since $$\omega\cdot k$$ may became arbitrarily small for any vector $$\omega\in\mathbb{R}^d$$ as $$k$$ varies, the convergence of the perturbative series is in doubt.

#### Kolmogorov's scheme

Two main ideas are needed to overcome the convergence problems related to the appearance of small divisors: (i) keep the frequency of the motion fixed; (ii) use a Newton quadratic method (the name comes form the elementary tangent Newton method for finding roots of real functions) to control the growth of the remainder terms. More specifically, Kolmogorov (Kolmogorov, 1954) constructed a (real-analytic), near-identity, symplectic transformation $$\phi_1$$ transforming a Hamiltonian of the form $$H=K+\epsilon P\ ,$$ with $$K$$ a non-degenerate Kolmogorov normal form as in (3), into a new Hamiltonian of the form $\tag{4} H_1:=H\circ\phi_1=K_1+\epsilon^2 P_1,$

with $$K_1$$ again in non-degenerate Kolmogorov normal form with the same frequency vector $$\omega$$ as $$K\ .$$ Once this is achieved, one can iterate the construction to obtain a sequence of symplectic transformations $$\phi_j$$ so that $H_j:=H\circ\phi_1\circ\cdots\circ\phi_j=K_j+\epsilon^{2^j} P_j$ with $$K_j$$ a non-degenerate Kolmogorov normal form with fixed frequency vector, and $$P_j$$ a real-analytic perturbation. Indeed, the equations leading to the determination of the symplectic transformation $$\phi:(y',x')\to(y,x)$$ may be (essentially uniquely) solved and admit as generating function a (real-analytic) function of the form $g(y',x)=y'\cdot x+\epsilon \Big(b\cdot x+s(x)+ y'\cdot a(x)\Big)$ where, $$b$$ is a constant vector, while $$s$$ and $$a$$ are, respectively, a scalar and a vector-valued multi-periodic functions with vanishing average over $$\mathbb{T}^n\ .$$ In the denominators of the Fourier expansion of $$s$$ and $$a$$ (and in the determination of the constant $$b$$) there appear the small divisors $$\omega\cdot k\ ,$$ which are controlled through the Diophantine inequality (2). The super exponential decrease of $$\epsilon^{2^j}\ ,$$ for small $$\epsilon\ ,$$ allows to beat the growth of the norm (due to the small divisors) of the new perturbing functions $$P_j\ :$$ in the limit as $$j\to\infty\ ,$$ $$\phi_1\circ\cdots\circ\phi_j$$ converges to a real-analytic symplectic transformation $$\phi_\epsilon\ ,$$ $$H_j\to K_\epsilon\ ,$$ with $$K_\epsilon=\lim_{j\to\infty} K_j=H\circ\phi_\epsilon$$ a real-analytic non-degenerate Kolmogorov normal form with frequency $$\omega\ .$$

#### Arnold's scheme

Arnold (who was the first to provide a detailed proof of Kolmogorov's Theorem) followed a different approach (Arnold, 1963a), which, however, shared with Kolmogorov's scheme the two main ingredients. Arnold considered a nearly-integrable Hamiltonian system of the form $$H:=K(y) + \epsilon P(y,x)$$ real analytic in a complex neighborhood $$D_0$$ of $$\{y_0\}\times\mathbb{T}^n\ ,$$ where $$y_0$$ is such that $$\partial_y K(y_0)=\omega$$ is Diophantine and $$\partial^2_y K$$ is invertible on $$D_0\ .$$ One then constructs a near-identity symplectic transformation $$\phi_1: D_1\to D_0$$ transforming $$H$$ as in (4) with $$K_1=K_1(y')$$ (i.e., integrable). The new domain $$D_1$$ is a complex neighborhood of $$\{y_1\}\times\mathbb{T}^n$$ contained in $$D_0\ ,$$ and with the property that $$\partial_{y'} K_1(y_1)=\omega$$ (same frequency) and $$\partial^2_{y'} K_1$$ is invertible on $$D_1\ .$$ This is not difficult to achieve, by classical averaging theory, through a symplectic transformation associated to a near-identity generating function $$g(y',x)=y'\cdot x + \epsilon \tilde g(y',x)\ ,$$ with $$\tilde g$$ a trigonometric polynomial in $$x$$ having degree $$\delta$$ depending on $$\epsilon$$ ($$\delta$$ can be chosen as $$(\log \epsilon^{-1})^p$$ and being related to a cut-off of the high Fourier modes of the perturbation). The iteration leads to a sequence of Hamiltonians $$H_j=K_j+\epsilon^{2^j} P_j$$ closer and closer to integrable but in shrinking domains $$D_j\ .$$ In the limit the projection onto the action variables of $$D_j$$ is a single point $$y_*\ .$$ Nevertheless, one can show that, pulling back the dynamics to $$\{y_*\}\times \mathbb{T}^n\ ,$$ there corresponds a Diophantine KAM torus for the original Hamiltonian $$H\ .$$

#### Moser's differentiable case

In dealing with a finitely differentiable perturbation $$P$$ there appears an extra technical problem. Namely, due to the presence of the small divisors, during the iteration scheme one loses derivatives at each step. Moser (inspired by the famous work by J. Nash on the $$C^\infty$$ imbedding of Riemannian manifolds) introduced a smoothing technique (via convolutions), which re-stores at each step of the Newton iteration a certain number of derivatives. The super-exponential convergence of the Newton scheme is fast enough to compensate also for the smoothing leading to a convergent algorithm. Later, Moser developed different and sharper methods, using, e.g., a characterization of differentiable functions through approximations by real-analytic ones in smaller and smaller complex neighborhoods of real domains. Thus, by a quantitative approximation of differentiable functions by means of real-analytic functions, one can construct for the analytic approximations real-analytic, invariant tori; such approximate solutions are analytic in shrinking domains and in the limit converge to differentiable solutions of the original problem.

The analytical tools needed in KAM proofs are classical and involve, in particular:
- exponential decay of Fourier coefficients of analytic functions
- quantitative versions of the classical implicit function theorem in real-analytic settings
- Cauchy estimates, which allow to bound the sup-norm of derivatives of analytic functions in smaller domains in term of the sup-norm of the function divided by the loss of the extension of the domain
- quantitative analysis of the PDE $$\sum_{j=1}^d \omega_j \partial_{x_j} u= f$$ where $$f$$ is real-analytic function on $$\mathbb{T}^d$$ with vanishing average and $$(\omega_1,...,\omega_d)$$ a Diophantine vector.

### Remarks

• In a nearly-integrable analytic Hamiltonian system with $$2n$$ degrees of freedom, the Kolmogorov set, i.e., the union of the persistent KAM tori, fills locally a region in phase space of density $$1-O(\sqrt{\epsilon})\ ,$$ as $$\epsilon$$ goes to zero. While the dynamics on the Kolmogorov set trivializes (being conjugated to a linear quasi-periodic translation on $$\mathbb{T}^n$$ with a Diophantine frequency vector), in its complement (which asymptotically represents a small region of measure $$O(\sqrt{\epsilon})$$) the dynamics can be very complicated, exhibiting, in many cases, "random motions" or "Arnold diffusion".
• In nearly-integrable Hamiltonian systems, Kolmogorov's non-degeneracy condition is equivalent to require that $$\det \partial_y^2 K(y_0)\neq 0\ ,$$ which, in turn, means that the frequency map $$y\to \omega(y):=\partial_y K(y)$$ is a local diffeomorphism in a neighborhood of $$y_0\ .$$
• The global geometry of the Kolmogorov set is simple: the fibers of the set (i.e., the individual KAM tori) are level sets of a global $$C^\infty$$ symplectic map $$\phi_*(\eta,x)$$ as the $$n$$ vector $$\eta$$ varies in a Cantor-like $$n$$-disk of almost full density. This phenomenon may be interpreted by saying that nearly-integrable Hamiltonian systems are integrable over Cantor sets (Pöschel, 1982, Chierchia and Gallavotti, 1982).
• The Kolmogorov symplectic map $$\phi_\epsilon$$ and the Kolmogorov normal form $$K_\epsilon$$ (see above) depend analytically upon the perturbative parameter $$\epsilon\ .$$ Therefore quasi-periodic trajectories taking place on KAM tori admit a convergent series expansions in $$\epsilon$$. This fact, which was first observed by Moser (1967), solves a long-standing problem about the convergence of Lindstedt series (i.e., $$\epsilon$$-power series expansions of formal quasi-periodic solution with Diophantine frequencies). Direct proofs, based upon delicate and lengthy combinatorial arguments, of the convergence of Lindstedt series (i.e., proofs avoiding KAM fast iteration methods) were found in the late 1980's (H. Eliasson) and early 1990's (G. Gallavotti, L. Chierchia and C. Falcolini).

## Applications and extensions

### Iso-energetic tori and perpetual stability Figure 4: Motion trapped by two KAM tori (from A. Celletti and L. Chierchia KAM tori for N-body problems (a brief history) Celestial Mechanics & Dynamical Astronomy, 95 , 2006 117-139

The tori found through Kolmogorov's (or Arnold's) scheme have, as $$\epsilon$$ varies, the same frequencies, but different energies. Arnold noticed that, instead, one could keep fixed the ratios of the frequencies and the energy so as to analytically continue KAM tori on a fixed energy surface. The analytical non-degeneracy condition to achieve this (in the nearly-integrable setting) is $\det \left(\begin{matrix}\partial^2_y K & \partial_y K \\ \partial_y K & 0\end{matrix}\right) \neq 0$ (this is a $$(n+1)\times(n+1)$$ matrix having as last column and as last row the gradient of $$K$$ and a 0).

Iso-energetic non-degeneracy leads, in low dimensional nearly-integrable systems, to perpetual stability: an energy level for a system with two degrees of freedom is a 3-dimensional surface and, for small perturbation, an iso-energetically non-degenerate, nearly-integrable systems admits a positive measure set of invariant two dimensional tori (which are graphs over the angle variables); thus such tori separate the energy level, and a generic trajectory either lies on an invariant torus or is trapped among two of them. In both cases no escape is possible, and the action variables stay forever close to their initial values ("perpetual stability").

### Properly degenerate KAM theory

One of the original motivations for KAM theory was to find bounded motions in the planetary many body problem (i.e., a mechanical system formed by $$1+N$$ point-masses, one of which is much larger than the other, interacting only through gravity). It is a classical fact that such a system may be seen as a perturbation of $$N$$ decoupled two-body systems (star-planet). However, the limiting unperturbed Hamiltonian is highly degenerate, since it does not depend on the full set of action variables (proper degeneracy). In general, perturbations of properly degenerate Hamiltonian systems may admit no KAM tori. However, under suitable assumptions on the (average over the fast angles of the) perturbation KAM tori do exist:

Theorem (Arnold, 1963b) Let $$(y,x)\in \mathbb{R}^n\times\mathbb{T}^n$$ and $$(p,q)\in\mathbb{R}^{2m}$$ be couples of conjugate symplectic variables and let the Hamiltonian $$H=K(y)+\epsilon P(y,x,p,q)$$ be real-analytic in a neighborhood of $$\{y_0\}\times\mathbb{T}^n\times\{0,0\}\ .$$ Denote by $$\bar P$$ the secular perturbation (i.e., the average over the fast angles $$x$$ of $$P$$) $$\bar P(y,p,q)=\int_{\mathbb{T}^n} P(y,x,p,q) dx/(2\pi)^n$$ and by $$r=(r_1,...,r_m)$$ the vector with components $$r_i=(p_i^2+q_i^2)/2$$ (for $$i=1,...,m$$). Assume that $$\det \partial_y^2 K(y_0)\neq 0\ .$$ Assume also that the secular perturbation has an elliptic equilibrium$\bar P= \bar P_0(y)+ \Omega(y)\cdot r + \frac12 A(y) r\cdot r + O(|r|^3)\ ,$ with $$\det A(y_0)\neq 0\ .$$ Then, if $$\epsilon$$ is small enough, in a neighborhood of $$\{y_0\}\times\mathbb{T}^n\times\{0,0\}$$ there exists a positive measure set of initial data whose evolution lies on $$(n+m)$$-dimensional tori close to $$\{y_0\}\times\mathbb{T}^n\times\{r_k=\epsilon^a, \ \forall\ k\}$$ for a suitable $$a>0\ .$$

This theorem, or refinements of it, is at the basis of the application of KAM theory to the planetary many body problem; a complete proof of such result, however, was published only in 2004 and is due to M. Herman and J. Fejóz.

### Weaker non-degeneracies

To extend the validity of KAM theory it is important to weaken the non-degeneracy conditions. As mentioned above, Kolmogorov's non-degeneracy condition for nearly-integrable systems with Hamiltonian $$H_\epsilon=K(y)+\epsilon P(y,x)$$ means that the frequency map $$\omega=\partial_y K$$ is a local diffeomorphism. Rüssmann pointed out (Rüssmann, 1989) that it is sufficient (and in a suitable sense also necessary) to assume that the image of the frequency map $$y\to \omega(y)$$ does not lie in any hyperplane, (more precisely, for a ball $$B\ ,$$ $$\omega(B)$$ does not lie in any hyperplane passing through the origin). A similar condition (that suites better differentiable settings) due to Arnold and Pyartli is to require that the frequency map $$\omega$$ is skew at some point $$y_0\ .$$ This means that there exists a smooth curve $$t\in(-1,1)\to u(t)\in\mathbb{R}^n$$ passing through $$y_0=u(0)$$ such that, if $$\alpha(t)$$ denotes the lifted curve $$\omega\circ u(t)\ ,$$ then the matrix $$[\alpha(0),\alpha'(0),...,\alpha^{(n-1)}(0)]$$ is invertible. Under these types of non-degeneracy conditions one can guarantee that, under small enough perturbations, there exists a positive measure set of initial data evolving on maximal KAM tori for $$H_\epsilon\ .$$

### Lower dimensional tori

Orbits of great interest for KAM theory are also quasi-periodic trajectories spanning lower dimensional tori, i.e., orbits $$z(t)=\phi_H^t(z_0)$$ such that the closure of the set $$\{z(t): t\in\mathbb{R}\}$$ is diffeomorphic to $$\mathbb{T}^d$$ with $$1< d<n\ ,$$ $$n$$ being the number of degrees of freedom (i.e., half of the dimension of the phase space). At variance with maximal KAM tori, the union of lower dimensional tori form a set of Lebesgue measure zero in phase space; nevertheless they are important in order to understand the dynamics and for extensions of KAM theory to PDEs. To fix ideas, consider the normal form of a lower dimensional elliptic torus $\tag{5} K(y,x,p,q;\xi):=E(\xi) + \omega(\xi)\cdot y + \frac12 \sum_{j=1}^m \Omega_j(\xi)(p_j^2+q_j^2)$

where $$(y,x)\in\mathbb{R}^d\times \mathbb{T}^d$$ are (partial) action-angle variables; $$(p,q)\in\mathbb{R}^{2m}$$ are conjugated variables; $$\Omega_j(\xi)>0$$ and $$\xi$$ is a real $$d$$-dimensional parameter (for example, $$\xi$$ might be a fixed action $$y_0$$ around which one is making a Taylor expansion). The set $$\mathcal{T}^d_0:=\{y=0\}\times\mathbb{T}^d\times\{p=0=q\}$$ is an invariant $$d$$-dimensional torus for $$\phi_K^t\ :$$ $$\phi_K^t(0,x_0,0,0)=(0,x_0+\omega(\xi)t, 0,0)\ .$$ Such a torus is linearly stable (elliptic), and the dynamics close to it, in the $$(p,q)$$-variables is just given by harmonic oscillations with frequencies $$\Omega_j(\xi)$$ (tangential frequencies). Under suitable regularity and non-degeneracy assumptions (on the inner and tangential frequencies) such tori are persistent.

For example, let $$\xi$$ vary in a closed set $$\Pi$$ of positive $$d$$-dimensional Lebesgue measure; let $$\xi\to\omega(\xi)$$ be a Lipschitz homeomorphism and let $$K$$ and $$P(y,x,p,q;\xi)$$ be real-analytic in the symplectic variables $$(y,x,p,q)$$ and Lipschitz continuous in $$\xi\ .$$ Assume that $$\Omega_j(\xi)\neq\Omega_i(\xi)>0$$ for all $$i\neq j$$ and $$\xi\in \Pi\ .$$ Assume also the following (Melnikov -Pöschel) condition ${\rm meas}\,\Big(\{\xi\in\Pi: \omega(\xi)\cdot k + \Omega(\xi)\cdot \ell=0\}\Big)=0 \ ,\quad \forall\ k\in\mathbb{Z}^d\backslash\{0\}\ ,\forall\ \ell\in\mathbb{Z}^m\ {\rm with}\ \sum_{j=1}^m|\ell_j|\le 2\ .$ Then, there exists $$\epsilon_*>0$$ and a Cantor set $$\Pi_*\subset\Pi$$ of positive measure such that to each $$\xi\in\Pi_*$$ there corresponds, for any $$0<\epsilon<\epsilon_*\ ,$$ a torus $$\mathcal{T}_\epsilon^d(\xi)$$ invariant for $$\phi_{K+\epsilon P}^t\ .$$

We remark that this kind of result admits many generalizations, which are particularly important for infinite dimensional extensions. The partially hyperbolic case, whose normal form is give by (5) with $$(p_j^2+q_j^2)$$ replaced by $$(p_j^2-q_j^2)$$ is much simpler (as in this case the tangential frequencies do not resonate with the inner ones); see (Graff, 1974).

### Hamiltonian PDE's

KAM theory can be partially extended to infinite dimension, i.e., to partial differential equations (PDEs) carrying a Hamiltonian structure. Examples of such equations are: the wave equation, the (stationary) Schrödinger equation, KdV, etc. Under suitable hypotheses, nonlinear perturbations of these equations may be reduced to infinitely many coupled dynamical (ordinary differential) equations (e.g., for the wave equation one obtains infinitely many coupled harmonic oscillators). It is then possible to find quasi-periodic solutions corresponding to the embedding of a linear quasi-periodic flow on a finite dimensional torus into the infinite dimensional phase space associated to the equation. Also almost-periodic motions have been considered (i.e., trajectories with infinitely many independent frequencies). Several results in these directions have been obtained starting from the 1990's; see (Kuksin, 2004).