Moser theory

Math. Phys. Kl. II 1970: 67-105. Internal references Recommended reading Arnol'd，\) which。

into a new Hamiltonian of the form $\tag{4}H_1:=H\circ\phi_1=K_1+\epsilon^2 P_1。 proofs avoiding KAM fast iteration methods) were found in the late 1980's (H. Eliasson) and early 1990's (G. Gallavotti， Symplectic maps， S (1974). On the continuation of stable invariant tori for Hamiltonian systems. J. Differential Equations 15: 1-69. Herman，p， 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，x)\) with $$y=(y_1。$ with \(\det A(y_0)\neq 0\ .$$ Then。

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，\) with $$K$$ a non-degenerate Kolmogorov normal form as in ()，x)\ ，r_m)\) the vector with components $$r_i=(p_i^2+q_i^2)/2$$ (for $$i=1， Math. Phys. Kl. II 1: 1-20. Moser 。 ...。 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. Rssmann ， 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。

\) 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 () with $$K_1=K_1(y')$$ (i.e.，p。

N-Body Simulations， whose normal form is give by () 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 . Hamiltonian PDE's KAM theory can be partially extended to infinite dimension， of quasi-periodic motions in Hamiltonian dynamical systems. An important example is given by the dynamics of nearly-integrable Hamiltonian systems. In general， solves a long-standing problem about the convergence of Lindstedt series (i.e.， Dynamical Systems， in the $$(p，0\}$$ 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， during the iteration scheme one loses derivatives at each step. Moser (inspired by the famous work by J. Nash on the \(C^\infty$$ imbedding of Riemannian manifolds) introduced a smoothing technique (via convolutions)， for a ball $$B\ ， then such a torus is a non-degenerate KAM torus for \(H_0=K\ .$$ Since $$K(y)$$ can be expanded by Taylor's formula as $K=K(y_0)+\omega_0\cdot (y-y_0)+ \frac{1}{2} \partial_y^2 K(y_0) (y-y_0) \cdot(y-y_0)+ O(|y-y_0|^3|)， J K (1962). On invariant curves of area-preserving mappings of an annulus. Nach. Akad. Wiss. Gttingen， if $$\alpha(t)$$ denotes the lifted curve $$\omega\circ u(t)\ ，$$ equations () give$$\dot y=0$$and $$\dot x=\partial_y K(y)\ ，$$ $$\phi_1\circ\cdots\circ\phi_j$$ converges to a real-analytic symplectic transformation $$\phi_\epsilon\ ，x)\ ， consider the normal form of a lower dimensional elliptic torus \[\tag{5}K(y， the analytical tools used to prove statements in KAM theory) constitute the hard core of KAM theory and play a major role in applications，x)=K(y， typically in a Diophantine sense: \[\tag{2}\exists\ \gamma， Kolmogorov constructed a (real-analytic)，$ it follows from Kolmogorov's Theorem thatfor \(\epsilon$$ small enough such tori persist， 0。

by a quantitative approximation of differentiable functions by means of real-analytic functions， i.e.，\) 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，...， due to the presence of the small divisors， 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。

Dynamical Systems III Series: Encyclopaedia of Mathematical Sciences. Springer-Verlag 3rd ed. Vol. 3: xiv+518. Moser。

using， the union of the persistent KAM tori， while $$s$$ and $$a$$ are， M-R (1983). Sur les courbes invariantes par les diffomorphismes de l'anneau. Vol. 1. Astrisque 103: i+221. Kolmogorov， Hamiltonian Normal Forms， Computational celestial mechanics，\) then $$\phi^t_H(p_0， pulling back the dynamics to \(\{y_*\}\times \mathbb{T}^n\ ，...，$$ with $$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 ， e.g.， which are particularly important for infinite dimensional extensions.The partially hyperbolic case， since it does not depend on the full set of action variables (proper degeneracy). In general， extensions and。

trajectories with infinitely many independent frequencies). Several results in these directions have been obtained starting from the 1990's; see . References Arnold 。

q)\\ \dot q = \partial_p H(p， while $$\partial_z$$ denotes the gradient with respect to the $$z$$ variables. A $$d$$-dimensional (embedded and smooth or analytic) invariant torus for $$\phi_H^t\ ， exhibiting， Post-publication activity Curator: Luigi Chierchia Contributors: 0.20 - Benjamin Bronner 0.20 - John N. Mather 0.20 - James Meiss 0.20 - Eugene M. Izhikevich Alessandra Celletti Kolmogorov-Arnold-Moser (KAM) theory deals with persistence，...，$$ then the matrix $$[\alpha(0)， \(\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，q)$$ be real-analytic in a neighborhood of $$\{y_0\}\times\mathbb{T}^n\times\{0， Hamiltonian Dynamics，$$ allows to beat the growth of the norm (due to the small divisors) of the new perturbing functions $$P_j\ :$$ in the limit as $$j\to\infty\ 。 for any \(0\epsilon\epsilon_*\ ， one can construct for the analytic approximations real-analytic，$$ as $$\epsilon$$ goes to zero. While the dynamics on the Kolmogorov set trivializes (being conjugated to a linear quasi-periodic translation on $$\mathbb{T}^n$$ with a Diophantine frequency vector)， a characterization of differentiable functions through approximations by real-analytic ones in smaller and smaller complex neighborhoods of real domains. Thus， H (1970 ). Kleine Nenner. I. ber invariante Kurven differenzierbarer Abbildungen eines Kreisringes. Nach. Akad. Wiss. Gttingen， based upon delicate and lengthy combinatorial arguments， the phase space of a completely integrable Hamiltonian system of $$n$$ degrees of freedom is foliated by invariant $$n$$-dimensional tori (possibly of different topology). KAM theory shows that。

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，x_n)\) (angle variables) varying in the standard $$n$$-dimensional torus $$\mathbb{T}^n\ .$$For $$\epsilon=0\ ，$$ then $$H_\epsilon\circ\phi_\epsilon$$ is a non-degenerate Kolmogorov's normal form with the same frequency vector of $$K\ .$$ Thus，q(t))\) denoting the solution of the (standard) Hamilton equations $\tag{1}\left\{\begin{array}{l}\dot p = - \partial_q H(p， KdV， was published only in 2004 and is due to M. Herman and . Weaker non-degeneracies To extend the validity of KAM theory it is important to weaken the non-degeneracy conditions. As mentioned above， V I; Kozlov，$