On global variables of action-angle type and the harmonic oscillator type in a neighborhood of an isotropic torus. (Q1432610)

It is supposed that the Hamiltonian system \[ \dot{y}=\widehat{\mathcal{J}}H_y, \qquad \widehat{\mathcal{J}}=\begin{pmatrix} 0, & -E_n\\ E_n,& 0 \end{pmatrix}, \] with a smooth function \(H=H(y)\) in the \(2n\)th-dimensional phase space \(\mathbb{R}_y^{2n}\), \(y= {}^t(p,x)\), has a smooth \(\ell\)-parameter family of \(k\)-dimensional \((l\leq k\leq n)\) invariant isotopic tori \(\Lambda^k(I)=\{y\in \mathbb{R}_y^{2n}\), \(y=Y(\phi,I)\), \(\phi=(\varphi,\psi)\), \(\varphi= {}^t(\varphi_1,\ldots,\varphi_{\ell})\in [0, 2\pi]^\ell\), \(\psi= {}^t(\psi_1,\ldots,\psi_\ell)\in [0,2\pi]^{k-\ell}\), \(I=(I_1,\ldots,I_\ell) \}\) with a conditionally periodic or periodic motion on it at frequences \(\omega= {}^t(\omega_1,\ldots,\omega_k)\). Here, \(I_1,\ldots,I_\ell\) are variable actions associated with the angles \(\varphi_1,\ldots,\varphi_{\ell}\). The purpose of the article is to show that in a neighborhood of a fixed invariant torus \(\Lambda^k(I)\) it is possible to introduce variables of the action-angle type and harmonic oscillator type globally explicitly specialized by some system of quadratic equations.