Slow manifold reduction as a systematic tool for revealing the geometry of phase space

Classically, the phase space geometry underlying an ideal model was revealed through identification of canonical coordinates, in which Hamilton's equations take their most elementary form, but where simple observables like velocity and scaling laws may become obscure. The pioneering work of Littlejohn, Morrison, and Greene established a new state-of-the-art known as Bracketology that began to uncover the geometry of phase space for plasma models using noncanonical variables. This enabled novel approaches to the study of stability, as embodied by dynamically-accessible free-energy principles, and new paradigms for asymptotic model reduction, as exemplified by variational gyrokinetic theory. In this talk, I will describe a new perspective on the phase space geometry of non-dissipative plasma physics that represents a step beyond Bracketology in terms of power and systematism. After highlighting the ubiquitous connection between reduced plasma models and approximate invariant manifolds, I will explain how each submanifold in an ambient phase space inherits a natural (pre) symplectic geometry of its own, much as submanifolds in Euclidean space inherit metric tensors by restricting the usual dot product. This submanifold geometry may be computed systematically using perturbation theory and provides a general, uniform explanation for a variety of Hamiltonian structures in plasma physics, both new and old. To illustrate the power of this perspective, I will describe how it has lead to (a) discovery of the Hamiltonian structure underlying the 70-year-old kinetic MHD model, (b) a Hamiltonian formulation of the nonlinear WKB method for Eulerian fluid models, and (c) derivation of the first post-Darwin kinetic plasma model, along with its Hamiltonian structure.