IFS Seminar with Bradley Shadwick

While the importance of gauge invariance is quantum field theories is well understood, it is generally assumed that gauge invariance is of little practical importance in the classical setting.  In classical electrodynamics, we often work in terms of Maxwell’s equation for the fields (which are naturally gauge invariant).  The situation changes significantly when one considers variational principles since the Lagrangian (or canonical Hamiltonian) must be expressed in terms of potentials.  For example, choosing a gauge prior to varying the action can be surprisingly difficult to do correctly.  When solving the dynamical equations for charged-particle systems numerically, obtaining reasonable computational performance necessitates using a grid to represent the potentials (and fields).  The introduction of a grid can break gauge invariance leading to an unusual structure of the variational principle.  Starting from Maxwell’s equations, I will systematically introduce potentials and outline how gauge fixing conditions interact with variational principles and the corresponding implications for building numerical models of charged particle system.  I will discuss how this analysis affects variational macro-particle models and provide various numerical examples.