Symplectic Elements: `invert()`

[ax3l]

A key feature of ImpactX is that we implement symplectic maps.

Add a method to every element to inverse the element. Negate ds (thick) or negate the strength (thin), then all compute_constants().

Corner Cases

• PRot: swap phi_in ↔ phi_out
• DipEdge: negate psi / K coefficients, swap entry↔exit
• RFCavity (thick, energy-changing) needs -ds plus negating escale or shifting phase by 180° to reverse the acceleration
• Apertures: nothing for transmitted particles, not inversible for absorbed particles

Applications

Many... in beam dynamics, stability analysis and ML.

[cemitch99]

@ax3l For the thick elements without s-dependent fields, reversibility requires only ds -> -ds. The reversibility of thin elements is usually straightforward. For soft-edge elements and those using symplectic integration -- these cases need to be checked. However, the symplectic integrators chosen are symmetric, so this should not be difficult to clarify.

Several tests already make use of this property, including: here, here , here.

Note: This is part of Issue #794

[cemitch99]

The above comments also apply to elements with spin tracking.