= Exact quadratic normal form for a perturbed linear rotor
{title2=$H\circ\Phi=E+\omega\cdot y'+\tfrac12Q+\epsilon^2\widetilde V$}
For $H=\omega\cdot y+\frac12|y|^2+\epsilon V(x)$ with a <Diophantine frequency vector>, solve $\mathcal D_\omega u=\langle V\rangle-V$ and apply <translated conjugacy of a Diophantine vector field> to $w=\epsilon\nabla u$. Set $\eta=\epsilon\nabla u+a$ and use the <symplectic cotangent lift with a closed momentum shift>. The transformed <Hamiltonian> is exactly $E+\omega\cdot y'+\frac12y'^T D\chi^{-1}D\chi^{-T}y'+\frac12|\eta\circ\chi|^2$, where $E=\epsilon\langle V\rangle+\omega\cdot a$. Since $a=O(\epsilon^2)$, the last term is an angular potential of order $\epsilon^2$ and the quadratic coefficient is identity plus $O(\epsilon)$. The construction is on a smaller angular strip and momentum ball.
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