Hamiltonian reduction of Vlasov-Maxwell to a dark slow manifold

Abstract

We show that nonrelativsitic 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 dynamics on the slow manifold naturally inherit 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.