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For $H=\omega \cdot y+\frac{1}{2}|y{|}^2+ϵV(x)$ with a Diophantine frequency vector, solve ${\mathcal{D}}_{\omega }u=⟨V⟩-V$ and apply translated conjugacy of a Diophantine vector field to $w=ϵ\nabla u$. Set $\eta =ϵ\nabla u+a$ and use the symplectic cotangent lift with a closed momentum shift. The transformed Hamiltonian is exactly $E+\omega \cdot y^{\prime }+\frac{1}{2}{y}^{\prime T}D{\chi }^{-1}D{\chi }^{-T}{y}^{\prime }+\frac{1}{2}|\eta \circ \chi {|}^2$, where $E=ϵ⟨V⟩+\omega \cdot a$. Since $a=O({ϵ}^2)$, the last term is an angular potential of order ${ϵ}^2$ and the quadratic coefficient is identity plus $O(ϵ)$. The construction is on a smaller angular strip and momentum ball.

Fix a smaller angular strip and a momentum ball $B^{\prime }⋐B$; the canonical transformation will be defined there and map into the original domain for small $ϵ$. This domain restriction is natural for a near-identity change of coordinates with a nonzero momentum shift.

First solve the scalar cohomological equation on a Diophantine torusThe preceding result gives a real-analytic periodic $u$ on every smaller strip. We need a near-identity angular diffeomorphism $\chi$ and a constant vector $a$ such thatHere is a complete analytic construction of this translated conjugacy of a Diophantine vector field.

Write $w=ϵ\nabla u$ and, for a current approximation, putAssume $A$ is close to the identity. For a correction $\Delta \chi =Av$, the linearized defect isThe identity follows by differentiating the definition of $e$, so the term $(De)v$ already contains the current error. ChooseThe matrix average is invertible near the identity, and the right-hand side has zero mean. The preceding torus small-divisor estimate therefore solves this vector equation componentwise. Update ${\chi }_{+}=\chi +Av$, $a_{+}=a+\Delta a$. The exact new error isIt is quadratic in the current defect.

For convergence, reserve a fixed outer strip on which $w$ is analytic and work on shrinking inner strips whose total loss is less than half the starting width. Let $s=n+\tau$ and choose loss parameters ${\delta }_j=d{2}^{-j}$ with $0<d<1$ small enough that the fixed multiples of these losses used at each stage fit in the reserved total width. The torus small-divisor estimate and Cauchy estimates give, with constants uniform while $A$ stays close to the identity and $\chi$ remains in the reserved strip,andIndeed, the first remainder is bounded using $\Vert D{e}_j\Vert \le C{\delta }_j^{-1}\Vert e_j\Vert$, and the second using the uniformly bounded second derivatives of $w$ on the reserved strip. Taking a slightly larger exponent $\mu$ covers all these losses. Start with ${\chi }_0=\mathrm{id}$ and $a_0=0$, so $e_0=-w=O(ϵ)$. If a small constant $K$ satisfies $C{d}^{-\mu }K{2}^{2\mu }\le 1$ and $\Vert e_0\Vert \le K$, induction gives $\Vert e_j\Vert \le K{2}^{-2\mu j}$. The estimates on $\Delta {\chi }_j$ and $\Delta a_j$ are summable. Choosing $ϵ$ still smaller keeps the derivatives close to the identity and the images inside the reserved strip, closing the induction. The real periodic analytic limits $\chi ,a$ solve the conjugacy equation. Since $⟨w⟩=0$, the very first constant correction is zero; the first new defect is $O({ϵ}^2)$, and the remaining summed constant corrections are $O({ϵ}^2)$. Also the summed change of $\chi$ is $O(ϵ)$. This proves the claimed estimates as well as existence, without leaving an unsolved linear remainder.

Now set $\eta (x)=ϵ\nabla u(x)+a$ and define the symplectic cotangent lift with a closed momentum shiftThe cotangent lift of a diffeomorphism is symplectic. Translation by $\eta$ is also symplectic because the one-form $\eta \cdot dx=ϵdu+a\cdot dx$ is closed. The constant term need not be exact on the flat torus; closedness is sufficient. Thus their composition is the required near-identity canonical transformation.

Expanding the quadratic kinetic term gives a transformed linear coefficient $D{\chi }^{-1}(\omega +\eta \circ \chi )$, which is exactly $\omega$ by the conjugacy equation. The constant-in-$y^{\prime }$ part issince ${\mathcal{D}}_{\omega }u=⟨V⟩-V$. Consequently defineand, for $ϵ\ne 0$,In the exact identities, squared Euclidean norms mean the real polynomial ${\sum }_j{v}_j^2$, continued bilinearly on complex strips, without complex conjugation. The latter is uniformly bounded and analytic on the retained strip because $a=O({ϵ}^2)$. At $ϵ=0$ use the identity map, $E_0=0$, and $Q_0=|y^{\prime }{|}^2$. We have the exact requested normal formHere $E_{ϵ}=O(ϵ)$, $Q_{ϵ}$ is genuinely homogeneous quadratic in $y^{\prime }$, and its coefficient matrix differs from the identity by $O(ϵ)$. Thus $|Q_{ϵ}-|y^{\prime }{|}^2|\le C|ϵ||y^{\prime }{|}^2$. The remainder depends only on the angular coordinate. If the printed final $\tilde{V}(x)$ uses the old angle $x$, the same term is $\frac{1}{2}|\nabla u(x)+a/ϵ{|}^2$ evaluated at $x=\chi (x^{\prime })$; this is just the corresponding coordinate convention.