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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $D \\subseteq \\R^n$ be open 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 family of [[Hamiltonian systems|0EDKGA2FAK]] with Hamiltonians\n$$H_\\varepsilon(I, \\theta, t) = H_0(I) + \\varepsilon H_1(I, \\theta, t),$$\nwhere $H_1$ is smooth and may depend periodically on time $t$.\nThe family exhibits *Arnold diffusion* if there is a constant $\\delta \u003e 0$ with the following property.\nFor every sufficiently small $\\varepsilon \u003e 0$ there are a solution $(I(t), \\theta(t))$ of the system of $H_\\varepsilon$ and a time $T \u003e 0$ with\n$$\\abs{I(T) - I(0)} \\ge \\delta.$$\n\n## Remarks\n\nThe point is that $\\delta$ does not depend on $\\varepsilon$: the actions drift by an amount of order one, while for $\\varepsilon = 0$ they are constant.\nThe time $T$ needed tends to infinity as $\\varepsilon \\to 0$.\n\nExample: Arnold's 1964 example is the time-periodic Hamiltonian\n$$H(I, \\theta, t) = \\tfrac{1}{2}(I_1^2 + I_2^2) + \\varepsilon (\\cos \\theta_1 - 1)\\big(1 + \\mu (\\sin \\theta_2 + \\cos t)\\big)$$\nwith two angles and periodic time.\nArnold proved that for suitable small $\\mu$ depending on $\\varepsilon$ there are orbits along which $I_2$ changes by an amount of order one.\n\nNon-example: an autonomous real-analytic system with $n = 2$ degrees of freedom and isoenergetically nondegenerate $H_0$ has no Arnold diffusion.\nBy the [[KAM theorem|3ZZSXKM16Z]], each $3$-dimensional energy level contains invariant $2$-tori that cut it into regions.\nThe action of an orbit is trapped between two such tori, so the drift tends to $0$ with $\\varepsilon$.\n\nFor real-analytic $H_\\varepsilon$ with steep $H_0$, the [[Nekhoroshev estimates|PYEN6SFS9P]] show that the drift time $T$ grows at least exponentially in a power of $1/\\varepsilon$.\n",
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