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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $D \\subseteq \\R^n$ be a bounded open set and let $\\mathbb{T}^n = \\R^n / 2\\pi \\Z^n$.\nUse [[action-angle coordinates|M3T81YNSKC]] $(I, \\theta)$ on $D \\times \\mathbb{T}^n$.\nConsider the [[Hamiltonian system|0EDKGA2FAK]] with Hamiltonian\n$$H_\\varepsilon(I, \\theta) = h(I) + \\varepsilon f(I, \\theta).$$\nAssume that $h$ and $f$ are real-analytic on a complex neighbourhood of $\\overline{D} \\times \\mathbb{T}^n$.\nAssume that $h$ is steep, for example strictly convex.\nThen there are constants $a, b, C, \\varepsilon_0 \u003e 0$ with the following property.\nLet $0 \u003c \\varepsilon \u003c \\varepsilon_0$, and take a solution with $I(0)$ in $D$, away from the boundary of $D$.\nThen\n$$\\abs{I(t) - I(0)} \\le C \\varepsilon^b \\quad \\text{for} \\quad \\abs{t} \\le C \\exp\\left(\\varepsilon^{-a}\\right).$$\n\n## Remarks\n\nThe proof has two parts.\nThe 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 $1/\\varepsilon$.\nThe 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.\n\nFor strictly convex $h$ one can take $a = b = \\frac{1}{2n}$.\n\nUnlike the [[KAM theorem|3ZZSXKM16Z]], the estimate holds for all initial conditions, not only for most of them, but only for a finite, exponentially long time.\nTogether with the existence of [[Arnold diffusion|ZVY046R4XB]], it shows that instability can occur in nearly integrable systems but only on exponentially long time scales.\n\nSome steepness condition is needed.\nFor $n = 2$, $h(I) = I_1^2 - I_2^2$, which is not steep, and $f = \\sin(\\theta_1 + \\theta_2)$, the equations give $(\\theta_1 + \\theta_2)' = 2(I_1 - I_2)$ and $I_1' = I_2' = -\\varepsilon \\cos(\\theta_1 + \\theta_2)$.\nThe 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 $1/\\varepsilon$.\n\n## Explanations\n\n### L1 human\n\nTake an [[integrable system|N0N7CMRCMH]], where the actions never change, and add a small perturbation of size $\\varepsilon$.\nThe theorem says that the actions then change only by a small amount for an extremely long time, longer than any power of $1/\\varepsilon$.\nSo even when a system is not stable forever, it can be stable for times much longer than the age of the universe.\n\n### L2 human\n\nThe Nekhoroshev estimates complement KAM theory.\nKAM gives stability forever on a large set of invariant tori, but says nothing about the orbits between them.\nNekhoroshev gives stability for all orbits, over times that are exponentially long in a power of $1/\\varepsilon$.\nThey are used in celestial mechanics, for example to discuss the long-time stability of asteroids near the Lagrange points.\n",
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