Moser theory

Post-publication activity

Curator: Luigi Chierchia

Contributors:

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Benjamin Bronner

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John N. Mather

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James Meiss

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Eugene M. Izhikevich

Alessandra Celletti

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.

Contents

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 $$\mathbb{T}^d=http://www.scholarpedia.org/article/Kolmogorov-Arnold-Moser_theory/\mathbb{R}^d/(2\pi \mathbb{Z})^d\ ;$$ 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.

Figure 1: Linear translation on a 2-torus (animation by Corrado Falcolini)

Figure 2: A periodic case (animation by Corrado Falcolini)

Figure 3: An orbit on a 2-dimensional KAM torus in a 3-dimensional energy level (animation by Corrado Falcolini)

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: $\tag{3} H\circ\phi(y,x)=K(y,x):=E+\omega\cdot y + Q(y,x)$