1. Hong Qin and Ronald C. Davidson Plasma Physics Laboratory, Princeton University US Heavy Ion Fusion Science Virtual National Laboratory www.princeton.edu/~hongqin/ Non-Abelian Courant-Snyder Theory for Coupled Transverse Dynamics

2. How to make a smooth round beam?  Solenoid – Final focus (NDCX) – How to match quadrupole with solenoid (NDCX-III)?  Skew-quadrupole

3. Möbius Accelerator  Round beam, one tune, one chromaticity  How? – Solenoid or skew-quadrupole  What is going on during the flip? [Talman, PRL 95]

4. Coupled transverse dynamics (2 degree of freedom) solenoidal, quadrupole, & skew-quadrupole

5. L. Spitzer suggested R. Kulsrud and M. Kruskal to look at a simpler problem first (1950s). Similar 2D problem – adiabatic invariant of gyromotion

6. Particles dynamics in accelerators ( uncoupled, 1 degree of freedom) +

7. Courant-Snyder theory for uncoupled dynamics Courant (1958) Envelope eq. Courant-Snyder invariant Phase advance

8. Courant-Snyder theory is the best parameterization  Provides the physics concepts of envelope, phase advance, emittance, C-S invariant, KV beam, … K. Takayama [82,83,92]

9. Higher dimensions? 2D coupled transverse dynamics? solenoidal, quadrupole, & skew-quadrupole 10 free parameters

10. Many ways [Teng, 71] to parameterize the transfer matrix Symplectic rotation form [Edward-Teng, 73]: Have to define beta function from particle trajectories [ Ripken, 70], [Wiedemann, 99] uncoupled No apparent physical meaning uncoupled CS transfer matrix Lee Teng

11. Can we do better? A hint from 1 DOF C-S theory

12. Original Courant-Snyder theory Non-Abelian generalization scalar Transfer matrix

13. Original Courant-Snyder theory Non-Abelian generalization Envelope equation scalar

14. Original Courant-Snyder theory Non-Abelian generalization Phase advance rate

15. Original Courant-Snyder theory Non-Abelian generalization Phase advance

16. Original Courant-Snyder theory Non-Abelian generalization Courant-Snyder Invariant

17. How did we do it? General problem

18. Time-dependent canonical transformation Target Hamiltonian symplectic group

19. Non-Abelian Courant-Snyder theory for coupled transverse dynamics Step I: envelope

21. Step II: phase advance

22. Application: Strongly coupled system Stability completely determined by phase advance Suggested by K. Takayama one turn map

23. Application: Weakly coupled system Stability determined by uncoupled phase advance

25. Numerical example – mis-aligned FODO lattice [Barnard, 96] mis-alignment angle

29. Other appliations – exact invariant of magnetic moment [K. Takayama, 92] [H. Qin and R. C. Davidson, 06]

30. Other applications  Globally strongly coupled beams. (many-fold Möbius accelerator).  4D emittance [Barnard, 96].  4D KV beams.  …

31. Other applications – symmetry group of 1D time-dependent oscillator Courant-Snyder symmetry (3D) Wronskian (2D) Scaling (1D) ?! (2D) [H. Qin and R. C. Davidson, 06 ]