system, environment

So physics provides us with interpretive models. We have ways of translating things into mathematical structures. Let's take this example of the electron storage ring.
We have magnets. These are big, heavy metallic objects with current running through them, shaped in ways to produce magnetic fields. So we line these up and put them in some configuration. Now, there's a certain region of space that maps out a doughnut-type shape inside all these magnets. Physics gives us the model of a magnetic field at all places inside this doughnut.
We have devices that mesh well with this picture. They measure the field and we basically assume that at a given time, the field has some value everywhere, and one can repeat measurements and get the same value. Then, to this, we throw some matter in there. We interpret that matter in terms of point charges with various masses and charges, such as electrons or air molecules.

The magnets and magnetic field is the environment or the background. The charges now move in this background. Now, depending on the needs, one can use different descriptions of the dynamics of the electrons. One can use classical electrodynamics to describe the motion of the charges, and the electric and magnetic fields they produce that may then also act back on those charges. I'm not entirely sure the status of the self-force and consistency within classical E&M. But I think its basically understood how to deal with it.

But actually, we need a little more than classical E&M. We need a bit of quantum mechanics. The radiation the electrons give off comes in lumps, and the lumpiness actually has an important effect that we can't ignore. Without the quantum lumpiness, for an appropriately set-up storage ring, the electrons would all end up at the center of the potential. Classical E&M says there is a damping mechanism that causes this to happen. Now, the interaction between electrons would limit the size of the resulting beam to a very small, finite size. But it turns out that the quantum lumpiness causes the beam to be much larger, and together with the damping mechanism, sets the size of the electron beam.

How do we treat the lumpiness? We use quantum mechanics (is it really full-blown QED? Or some semiclassical approximation given the emission spectrum of the electron?) to provide the diffusion coefficient. This turns the Lorentz equation into a stochastic differential equation. In the case of linear dynamics and constant damping and diffusion, the result is a Gaussian probability distribution, which when considered for an ensemble of electrons results in an actual Gaussian charge distribution.

Once the magnetic field has been set, and one is considering charged particles, there are other formulations for describing the classical dynamics besides the Lorentz force law. In particular one can use Lagrangian or Hamiltonian mechanics. Let us take the lead of Michelotti in describing this framework. He begins with the pendulum to introduce model systems that have the properties we need that the maps around the storage ring will have. He emphasizes with the pendulum that the phase space may not be R^n, but is a manifold. In chapter two he introduces linear and nonlinear models. He discusses the Hopf map and the Henon map, and gives the ideas of ergodicity and some other probability concepts such as partitions. So in general, we are actually in the realm of dynamical systems. And where does Michelotti end up? By chapter 5, he is discussing perturabtion theories for Hamiltonian dynamics and tries to give description of the Forest, Berz, Irwin normal form algorithm, which may contain isolated resonances.

So this is a long path stretching away from the magnets we see and the measured fields. It provides tools. A path to walk on. But there has to be a pulling from the other end. We have to know where we want to go. I would say that one usually wants to go to questions of stability. One wants to know whether a given bunch of electrons moving through this doughnut will last very long or not. And unfortunately, the elaborate normal form perturbation theories don't tell us this. And the same goes for the numerical implementation of these perturbation theories. One can compute resonance strengths and tune shift with amplitude to arbitrary order, for a machine with all the appropriate misallignments and field errors, and the full Hamiltonian, and one still doesn't answer the stability question by the perturbation theories.

But these are nice paths. And the tools are good tools. But without some pulling from the other side, the use of these tools gets lost. One doesn't know that sometimes one needs to develop new tools, or maybe give up on full understanding and just track the particles and see what happens.
Between the end point of injection efficiency and Touschek lifetime (momentum aperture), and the beginning or magnets leading to particular paths through classical mechanics with brief borrowings/harvestings from the quantum, one will simply get lost on these paths.