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Hamiltonian reduction of Vlasov–Maxwell to a dark slow manifold

Published online by Cambridge University Press:  08 June 2021

George Miloshevich*
École Normale Supérieure de Lyon, Laboratoire de Physique, 46, allée d'Italie, F-69364Lyon CEDEX 07, France
Joshua W. Burby
Los Alamos National Laboratory, Los Alamos, NM87545, USA
Email address for correspondence:


We show that non-relativistic scaling of the collisionless Vlasov–Maxwell system implies the existence of a formal invariant slow manifold in the infinite-dimensional Vlasov–Maxwell phase space. Vlasov–Maxwell dynamics restricted to the slow manifold recovers the Vlasov–Poisson and Vlasov–Darwin models as low-order approximations, and provides higher-order corrections to the Vlasov–Darwin model more generally. The slow manifold may be interpreted to all orders in perturbation theory as a collection of formal Vlasov–Maxwell solutions that do not excite light waves, and are therefore ‘dark’. We provide a heuristic lower bound for the time interval over which Vlasov–Maxwell solutions initialized optimally near the slow manifold remain dark. We also show how the dynamics on the slow manifold naturally inherits a Hamiltonian structure from the underlying system. After expressing this structure in a simple form, we use it to identify a manifestly Hamiltonian correction to the Vlasov–Darwin model. The derivation of higher-order terms is reduced to computing the corrections of the system Hamiltonian restricted to the slow manifold.

Research Article
Copyright © The Author(s), 2021. Published by Cambridge University Press

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