As an interested outsider in both mathematics and accelerator physics, I will recklessly attempt to propose a variety of problems in pure mathematics that appear to arise in the dynamics of particle beams – spanning from interactions between electron packets to electron interactions within packets to the Hamiltonian dynamics of the single-particle trajectories ignoring interactions.
Systematic attempts to avoid Hamiltonian chaos in the single-particle trajectories have led to systems where the dynamics is either explicitly or approximately integrable. Can one hope for theorems and/or algorithms for multiparameter tuning to avoid chaos? Will the resulting systems approach integrability? Adding realistic noise to these approximately-integrable trajectories should allow calculations of the beam brightness and particle loss, ignoring the chaos. Can one systematically incorporate both?
The number of particles N in a packet is large. Can one develop a systematic theory for the effects of particle interactions inside a packet in the limit of large N? Some effects (space charge, the head-tail instability? small-angle scattering?) may be captured by partial-differential equations, but others (large-angle scattering, disorder-induced heating) appear to demand explicit particle-based descriptions. In stochastic random walks, the central limit theorem controls typical behavior while extreme value statistics describes the tails. Can some kind of dual mathematical formulation be developed for electromagnetic interactions inside fast-moving packets?
Packets leave echoes behind them – both electromagnetic ‘wake fields’ and through excitations of contaminant gases. These cause instabilities for packets behind them in the train, and can cause self-interaction effects as a packet orbits the accelerator. These have inspired recent work on the stability theory of differential-delay equations. Also, there is an appealing analogy between an infinite pulse train and the transition to turbulence for fluid flow through a cylinder. There the complete lack of a linear instability at any fluid speed led to the mathematical introduction of pseudospectra for non-normal linear stability problems. Can such methods provide insights and improved control for packet trains?