1. Time Tracking in PTC

2. First order Time Tracking A high order integrator in the so-called “ s ” variable can be turned into a first order time tracker provided it has some information about the 3d environment PTC has a full 3d information around each of its steps of integrations of type integration_node

3. The Probe TYPE PROBE REAL(DP) X(6) TYPE(SPINOR) S LOGICAL U TYPE(INTEGRATION_NODE),POINTER :: LOST_NODE END TYPE PROBE The probe can track position X(6) and spin. Here we only will care about position. Notice also that the logical U is true if the particle is unstable.

4. Type Temporal_probe TYPE TEMPORAL_PROBE TYPE(PROBE) XS TYPE(INTEGRATION_NODE), POINTER :: NODE REAL(DP) DS,POS(6) TYPE(INTERNAL_STATE) STATE END TYPE TEMPORAL_PROBE The probe XS contains the coordinates as explained in the previous slide. The NODE points to the actual integration where the particle is at. The distance DS is the very approximate distance inside the node the particle traveled assuming a drift. POS(6) is the actual position of the particle in an absolute 3D frame STATE is the tracking state of that beam/machine

5. Temporal Beam TYPE TEMPORAL_BEAM TYPE(TEMPORAL_PROBE), POINTER :: TP(:) REAL(DP) A(3),ENT(3,3),P0C,TOTAL_TIME INTEGER N TYPE(INTEGRATION_NODE),POINTER :: C END TYPE TEMPORAL_BEAM This is obviously a collection of Temporal Probes, i.e., of particles. First it must be allocated: CALL ALLOC(TB,N,P0C) P0C is the reference momentum of the beam

6. Initial Conditions in the Layout CALL POSITION_TEMPORAL_BEAM(LAYOUT,TB,STATE) Locates the beam in 3d and stores TB%TP(i)%POS(1:3)=(X,Y,Z); this is in the global frame of the universe! Calls LOCATE_TEMPORAL_BEAM which then calls ORIGINAL_P_TO_PTC. The beam is values of TB%TP(i)%XS%X(1:6) are now in the local PTC coordinates. ORIGINAL_P_TO_PTC adjusts the momenta. This routine will have to be fixed to take into account the vector potential and the type pancake.

7. Pictorial Representation Here we see an exaggerated view on an integration node of PTC of cas=0, i.e., a true step in the body of some magnet. The darker red dot is the actual position in the accurate ‘ s ’ tracking algorithm of PTC. The light pinkish dot is the projected position inside the step. Time tracking is obviously first order at best and non-symplectic in the time domain due to hard edges in ‘ s ’. Unavoidable! In the absence of collective effects, it is the original PTC since the projected position at DS never enters in any calculations.

8. Initialization The lattice must be surveyed make_node_layout must be called fill_survey_data_in_node_layout must be called call alloc(TB,N,p0c) must be done to allocated N particles at momentum p0c TB%ENT=ENT(3,3) initial frame TB%A=A initial origin TB%TOTAL_TIME= Time at beginning

9. Initial Conditions in Space to be filled in by User Particle “ i ” of the initial beam is located in the temporal probe TB%TP(i)%XS%X(1:6) TB%TP(i)%XS%X(1:3)=(x,y,z) TB%TP(i)%XS%X(4:6)=(p x,p y,p z )/ TB%p0c these are the mechanical momenta.

10. Ready for Tracking Call position_temporal_beam(LAYOUT,TB,STATE) TB%C points to the approximate integration node at the centre of the beam (to be determined by position_temporal_beam) State of temporal beam is set The tracking command is: TRACK_TEMPORAL_BEAM(TB,DT,STATE) State is optional The variable TB%TP(i)%POS(1:6)=(X,Y,Z,P x,P y,P z ) contains the position in absolute space