Dynamical systems and ergodic theory Hamiltonian dynamics Theorem

Nekhoroshev estimates

Builds on Hamiltonian system Action-angle coordinates Integrable system

Statement

Let $n \ge 2$ , let $D \subseteq \R^n$ be a bounded open set and let $\mathbb{T}^n = \R^n / 2\pi \Z^n$ . Use action-angle coordinates $(I, \theta)$ on $D \times \mathbb{T}^n$ . Consider the Hamiltonian system with Hamiltonian $H_\varepsilon(I, \theta) = h(I) + \varepsilon f(I, \theta).$ Assume that $h$ and $f$ are real-analytic on a complex neighbourhood of $\overline{D} \times \mathbb{T}^n$ . Assume that $h$ is steep, for example strictly convex. Then there are constants $a, b, C, \varepsilon_0 > 0$ with the following property. Let $0 < \varepsilon < \varepsilon_0$ , and take a solution with $I(0)$ in $D$ , away from the boundary of $D$ . Then $\abs{I(t) - I(0)} \le C \varepsilon^b \quad \text{for} \quad \abs{t} \le C \exp\left(\varepsilon^{-a}\right).$

Remarks

The proof has two parts. The analytic part builds, near each resonance, a normal form in which the angle dependence that does not belong to the resonance is exponentially small in $1/\varepsilon$ . The geometric part uses steepness to show that an orbit cannot move far along the directions allowed by a resonance before it leaves the resonance zone.

For strictly convex $h$ one can take $a = b = \frac{1}{2n}$ .

Unlike the KAM theorem, the estimate holds for all initial conditions, not only for most of them, but only for a finite, exponentially long time. Together with the existence of Arnold diffusion, it shows that instability can occur in nearly integrable systems but only on exponentially long time scales.

Some steepness condition is needed. For $n = 2$ , $h(I) = I_1^2 - I_2^2$ , which is not steep, and $f = \sin(\theta_1 + \theta_2)$ , the equations give $(\theta_1 + \theta_2)' = 2(I_1 - I_2)$ and $I_1' = I_2' = -\varepsilon \cos(\theta_1 + \theta_2)$ . The solution with $I_1(0) = I_2(0)$ and $\theta_1(0) + \theta_2(0) = 0$ has $I_1(t) = I_1(0) - \varepsilon t$ and $I_2(t) = I_2(0) - \varepsilon t$ , so the actions drift by order one in time of order $1/\varepsilon$ .

Going deeper

For a student

The Nekhoroshev estimates complement KAM theory. KAM gives stability forever on a large set of invariant tori, but says nothing about the orbits between them. Nekhoroshev gives stability for all orbits, over times that are exponentially long in a power of $1/\varepsilon$ . They are used in celestial mechanics, for example to discuss the long-time stability of asteroids near the Lagrange points.

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Tag	`PYEN6SFS9P`
URI	`m2://PYEN6SFS9P`
Kind	statement, theorem
Writing	W0 stub
MSC	37J40, 70H08
Status	This copy of the site has no build data. Run `m2 status PYEN6SFS9P`.
About	Integrable system
Uses	Hamiltonian system, Action-angle coordinates, Integrable system
Same as	Wikidata Q6989812 Wikipedia Nekhoroshev estimates