Adiabatic action drift bound: order-$\epsilon$ change of the action on the $O(1\mathrm{/}\epsilon )$ time scale

Let $\epsilon >0$, $A>0$, ${\omega }_0>0$, let $\widehat{J},\widehat{S}\subseteq \mathbb{R}$ be open intervals, and let $J\subseteq \widehat{J}$ and $S\subseteq \widehat{S}$ be intervals with nonempty interiors (A one-frequency parameter-dependent action-angle branch, its transformed Hamiltonian and its drift generator, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). Let $h:\widehat{J}\times \widehat{S}\to \mathbb{R}$ and $k:\widehat{J}\times \mathbb{R}\times \widehat{S}\to \mathbb{R}$ be $C^2$, with $k$ $2\pi$-periodic in $\theta$ on $J\times \mathbb{R}\times S$. Let $K\subseteq J$ and $S_0\subseteq S$ be compact intervals, let $s_0\in S_0$, and suppose

$\mathrm{\mid }{\mathrm{\partial }}_{I}h(I,s)\mathrm{\mid }\ge {\omega }_0((I,s)\in K\times S_0)\mathrm{.}$

Let $(I,\theta ,s):[0,A\mathrm{/}\epsilon ]\to J\times \mathbb{R}\times S$ be a branch solution of the branch vector field of the datum (A one-frequency parameter-dependent action-angle branch, its transformed Hamiltonian and its drift generator) with $s(0)=s_0$, and suppose that for every $t\in [0,A\mathrm{/}\epsilon ]$ one has $I(t)\in K$ and $s(t)\in S_0$. Then for every $t\in [0,A\mathrm{/}\epsilon ]$,

$\mathrm{\mid }I(t)-I(0)\mathrm{\mid }\le \epsilon [2{\sup }_{K\times [0,2\pi ]\times S_0}\mathrm{\mid }g\mathrm{\mid }+A{\sup }_{K\times [0,2\pi ]\times S_0}\mid {\mathrm{\partial }}_{I}g{\mathrm{\partial }}_{\theta }k-{\mathrm{\partial }}_{\theta }g{\mathrm{\partial }}_{I}k-{\mathrm{\partial }}_{s}g\mid ],$

where $g=(k-⟨k⟩)\mathrm{/}{\mathrm{\partial }}_{I}h$ is the drift generator of the datum.

• Conditional character. The bound is conditional on the branch solution staying in the compact sets $K\times S_0$, i.e. on the action coordinate not leaving the chart along the motion; the trapping is an explicit hypothesis, not a consequence. The chart itself — the existence of a smooth parameter-dependent action-angle family with $\mathrm{\mid }\omega \mathrm{\mid }\ge {\omega }_0$ — is likewise a hypothesis of the definition, not proved here. No claim of exact invariance is made: the right-hand side is a bound of order $\epsilon$ and grows linearly in the rescaled time $A$.
• Comparison with the sources. Both cited treatments derive the order-$\epsilon$ (averaged) statement $⟨\dot{I}⟩=0$; Tong's Section 4.6 explicitly remarks that stricter proofs are deferred to the literature, and Likharev's averaged identity likewise treats the parameter as frozen over one period. The present theorem replaces averaging by the exact chain-rule identity $(I+\epsilon g{)}^{\mathrm{\prime }}=-{\epsilon }^2(\dots )$ and hence yields a uniform-in-time bound on the stated horizon under the explicit trapping and frequency hypotheses. No exact conservation is asserted.