Include electric field for time-dependent multipole kicks

[mattsignorelli]

We should modify multipole_kick! so that if the multipoles are of type Function, then a corresponding electric field kick is also included. also the unpacking step should make it so that DKD is therefore used whenever any multipole is time-dependent

[DavidSagan]

@mattsignorelli If you start worrying about the electric field here then you should also worry about the variation of the time dependent fields while a particle transverses the element. Additionally, to be symplectic, you cannot simply solve for the electric field given the magnetic field since the time variation of the electric field modifies the magnetic field. And the edge fields become important here. Bottom line: This is not a simple problem.

[mattsignorelli]

Ouch... ok let's put this on the backburner then for now

[mattsignorelli]

@DavidSagan do I remember you saying that the electric field only matters if you have to worry about the variation while it traverses the element?

[DavidSagan]

Yes that is the hand waving argument. From the Bmad manual:

Note: The calculated field will only obey Maxwell’s equations in the limit that the time variation of the
field is “slow”:
ω≪ c/r
where ω is the characteristic frequency of the field variation, c is the speed of light, and r is the charac-
teristic size of the ac_kicker element. That is, the fields at opposite ends of the element must be able
to “communicate” (which happens at the speed of light) in a time scale short compared to the time scale
of the change in the field.

[mattsignorelli]

but over some small time dt while the magnetic field is changing, there will be some electric field ~constant over the timeframe dt . why would that be negligble? Or are you saying the total effect of the electric field cancels itself out?

[mattsignorelli]

Oh nvm I think I understand because if that is true then the B field must be changing so much that you have to worry about variations inside the element. is that correct?

[DavidSagan]

Maxwell's equations essentially demands that if the B field is oscillating at some frequency omega then the electric field will also oscillate at that frequency.  Of course you can manufacture a situation where the B field is linearly ramping and the E field is constant but in the general case both B and E will vary on the same time scale.

[mattsignorelli]

So then how can the electric field be neglected?

[DavidSagan]

If the variation of the magnetic field is slow enough, the electric field will be small enough to be negligible. I would propose that for now the AC_Kicker element the electric field is neglected. In the future, we can contemplate something more accurate but just throwing in an electric field component without thought as to the factors I mentioned above is not valid.