[WIP] Linear Transport: Centroid Tracking

[ax3l]

Add a centroid 6D phase vector to our envelope tracking logic.

The goal is to support off-axis envelope propagation and feed-down effects in linear optics from misalignment (translation & rotation) and off-axis reference particle.

This lets transfer maps, and currently developed map traces and Twiss optics see the same feed-down effects that particles experience through the element.

Reusable helper methods are added to the mixin::Alignment class that we can use in other elements as well.

Multipole (Illustrative Example)

Higher-order thin multipoles contribute the correct linear focusing when the reference orbit is off axis or the element is misaligned, including the normal/skew mixing from transverse roll.

Adds regression coverage for sextupole feed-down (e.g. Quads kicks) and periodic lattice optics.

[ax3l]

While implementing the feed-down logic, I think I stumbled on a convention-caused issue that we might want to check in ImpactX. Correct me if I got something wrong in my assumptions.

1. We want to use one (ideal?) reference particle update implementation for all three tracking modes, otherwise this gets very confusing.
2. By convention so far, we model all element alignment/rotation errors on the beam/map, not modifying the reference particle.

For the current implementation in this PR regarding feed-down effects in the linear maps from misalignment (something that MAD-X and Xsuite and others can model), we thus lack the capability to model ideal dipole kicks from an off-axis higher-order multipoles (or anything that depends on propagating the constant orbit offset): The current 6x6 map only carries linear transport maps (Jacobians), not orbit offsets.

• we do capture the linear focusing/coupling effect around the current reference orbit
• we do not propagate the orbit offset caused by the constant kick to later elements

With this PR we can model:

• Local feed-down at the current reference orbit.
• Normal/skew mixing from roll.
• Misalignment/rotation effects in the element’s own linearized map.
• Composition of those local linear maps through the lattice.

There are probably a few ways forward, but we need to discuss them:

a) accept & document the limitations of what feed-down effects we can model
b) flip the alignment & rotation generally in the reference particle: possible but not always what people want when they search for an ideal reference orbit -- also the most intrusive change
c) evolve the reference particle (using the ideal kicks) for envelope tracking

[codspeed-hq]

Merging this PR will not alter performance

✅ 37 untouched benchmarks

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_{Comparing ax3l:linmap-multipole (7ac6b9c) with development (de02c77)}