Tracking with RF modulation

In the example of this manual, there are no synchrotron oscillation and the 5-6 plane (Energy-like and time-like) is dormant completely. But of course all that follows could have synchrotron oscilations and spin. It could also have a coasting beam normal form: Jordan normal form.

    state=default0+only_4d0
 CALL FIND_ORBIT_x(als,X,state,1.0e-5_dp,fibre1=1)
   Ray=x
   ray%ac%om=0.01d0
   ray%ac%x(1)=1.d0    ! fake phasor 
   ray%ac%x(2)=0.d0    !   fake phasor 
   
    state=state+modulation0+only_4d0
    write(6,*) "Tracking data in modulation.dat"
    call kanalnummer(mf,"modulation.dat")
   write(6,*) ray%x
   write(6,*) ray%ac%x
          WRITE(mf,'(4(1x,E15.8))') ray%x(1:4)
   do i=1,1000
    call TRACK_PROBE(als,ray,state, FIBRE1=1)
       WRITE(mf,'(4(1x,E15.8))') ray%x(1:4)
  enddo
  close(mf)

First the closed orbit is found without RF-modulation. In effect, the RF modulation is proportional to the ``fake'' phasor ray%ac%x(1:2). Thus the closed orbit is obviously the solution when this phasor is at the origin since the modulation, in our approximation, is not influenced by the orbital motion. (We use a common ``fake'' time proportional to some $s$ variable stored in c%DS_AC of the integration node c.)

Notice that modulation is possible if and only if the internal state contains modulation: state=state+modulation0.

The above graph shows the plot of the for three different frequencies. Clearly for a slower modulation, the trajectory becomes the local closed orbit since we have an adiabatic change.