Hamiltonian control of magnetic field lines: computer assisted results proving the existence of KAM barriers (Q2093487)

The authors consider a control theory for Hamiltonian systems in the framework of KAM theory and apply their results to a model of magnetic fields. They start their work by introducing the physical model of interest. The so-called poloidal flux is governed by the Hamiltonian \[ H(\psi)=\psi -\frac{3}{4}\psi^{2}+\frac{1}{3}\psi^{3}-\frac{1}{16}\psi^{4}. \] Here the toroidal flux \(\psi \in [0, 1]\) is the conjugated action variable. The toroidal angle \(\varphi\in [0, 2\pi)\) has the role of time and the poloidal angle \(\theta\in [0, 2\pi)\) is an angle variable. The perturbed Hamiltonian is given by \[ \tilde{H}=\tilde{\omega}\psi + P +h(\theta, \varphi, \psi) +v(\theta, \varphi), \qquad h(\theta, \varphi, \psi)=\sum_{l=2}^{l_{\max}}h_{l}. \] For this Hamiltonian, the control term is \[ f=\epsilon^{2} \left(-\frac{9}{2}\cos(6\theta-4\varphi)+6(\cos(\theta-\varphi)-\cos(5\theta-3\varphi))+2\cos(4\theta-2\varphi)\right)\partial^{2}_{\psi\psi}h_{2}. \] The authors provide qualitative analysis of the perturbed dynamics. They apply a frequency analysis to study how the control term affects the dynamics induced by the Hamiltonian \(H\). The authors discuss the scheme of a Computer Assisted Proof (CAP) based on KAM theory to show the existence of an invariant torus. As a novelty with respect to the works that in the last two decades applied CAPs in the framework of KAM theory, the authors provide all the codes, thus allowing reproducibility of their results. All these codes are designed in such a way to be easy-to-use, also for what concerns applications to similar problems. They are collected in a software package that is available from the Mendeley Data repository. At the \(r\)-th step of the algorithm the Hamiltonian displays like \[ H^{(r-1)}=\omega\psi +P+\sum_{l=2}^{l_{\max}}\sum_{s\geq 0} F_{l}^{(r-1,s)} + \sum_{s\geq 0} (F_{0}^{(r-1,s)}+F_{1}^{(r-1,s)}). \] By a combination of frequency analysis and of a rigorous CAP based KAM algorithm, the authors prove that in the phase space of the magnetic field, due to the control term, a set of invariant tori appear, and it acts as a transport barrier. Their analysis, which is common in the literature but often limited to celestial mechanics, is based on a normal form approach; it is also quite general and can be applied to quasi-integrable Hamiltonians whose Taylor-Fourier expansion in the action-angle coordinates is finite. Further, the authors compute the iteration of the estimates so that they describe bounds on the generating functions and bounds on the transformed Hamiltonian. Finally, the authors explain rigorously the application of their general results to the problem of controlled magnetic fields.