Open AccessHamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers
Author(s) -
Lorenzo Valvo,
Ugo Locatelli
Publication year - 2022
Publication title -
journal of computational dynamics
Language(s) - English
Resource type - Journals
eISSN - 2158-2505
pISSN - 2158-2491
DOI - 10.3934/jcd.2022002
Subject(s) - kolmogorov–arnold–moser theorem , hamiltonian system , integrable system , mathematical proof , computer science , phase space , hamiltonian (control theory) , torus , novelty , software , magnetic field , invariant (physics) , theoretical computer science , mathematics , physics , pure mathematics , mathematical analysis , mathematical optimization , mathematical physics , quantum mechanics , geometry , programming language , philosophy , theology
We reconsider a control theory for Hamiltonian systems, that was introduced on the basis of KAM theory and applied to a model of magnetic field in previous articles. By a combination of Frequency Analysis and of a rigorous (Computer Assisted) KAM algorithm we 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. Our analysis, which is common (but often also limited) to Celestial Mechanics, is based on a normal form approach; it is also quite general and can be applied to quasi-integrable Hamiltonian systems satisfying a few additional mild assumptions. As a novelty with respect to the works that in the last two decades applied Computer Assisted Proofs into the framework of KAM theory, we provide all the codes allowing to produce our results. They are collected in a software package that is publicly available from the Mendeley Data repository. All these codes are designed in such a way to be easy-to-use, also for what concerns eventual adaptations for applications to similar problems.