1. Symplectic Tracking Routine Malte Titze, Helmholtz-Zentrum Berlin, 10.05.2014 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

2. 1. Introduction 2. Main Idea 3. Advantages 4. Theory 5. Summary 0. Overview

3. Z X Y Given: (A x, A y, A z, φ) as a Fourier-decomposition with respect to the (longitudinal) Z-axis. 1. Introduction

4. Z Y Given: (A x, A y, A z, φ) as a Fourier-decomposition with respect to the (longitudinal) Z-axis. X 1. Introduction

5. Z X Y z0z0z0z0 zfzfzfzf How to effectively track particles symplectic from z 0 to z f ? 1. Introduction

6. Z X Y z0z0z0z0 zfzfzfzf Find the dependency of the cartesian coordinates to the cyclic ones at the final position z f (time-independent case). 2. Main Idea

7. Z X Y z0z0z0z0 zfzfzfzf Find the dependency of the cartesian coordinates to the cyclic ones at the final position z f (time-independent case). Functions of (x 0, y 0, p x0, p y0 ), z 0 and z f 2. Main Idea

8. Based on the following data: 3. Advantages

9. Based on the following data: 1. The initial coordinates (x 0, y 0, p x0, p y0 ) and z 0, 3. Advantages

10. Based on the following data: 1. The initial coordinates (x 0, y 0, p x0, p y0 ) and z 0, 2. The values of the field at the initial coordinates (Fourier-coefficients), 3. Advantages

11. Based on the following data: 1. The initial coordinates (x 0, y 0, p x0, p y0 ) and z 0, 2. The values of the field at the initial coordinates (Fourier-coefficients), 3. The final position z f. 3. Advantages

12. Based on the following data: 1. The initial coordinates (x 0, y 0, p x0, p y0 ) and z 0, 2. The values of the field at the initial coordinates (Fourier-coefficients), 3. The final position z f. → (x f, y f, p xf, p yf ) can be computed to arbitrary precision without the need of a PDE-solver! 3. Advantages

13. Based on the following data: 1. The initial coordinates (x 0, y 0, p x0, p y0 ) and z 0, 2. The values of the field at the initial coordinates (Fourier-coefficients), 3. The final position z f. → (x f, y f, p xf, p yf ) can be computed to arbitrary precision without the need of a PDE-solver! → The coordinate transformation is symplectic. 3. Advantages

14. Based on the following data: 1. The initial coordinates (x 0, y 0, p x0, p y0 ) and z 0, 2. The values of the field at the initial coordinates (Fourier-coefficients), 3. The final position z f. → (x f, y f, p xf, p yf ) can be computed to arbitrary precision without the need of a PDE-solver! → The coordinate transformation is symplectic. → Fringe fields are included. 3. Advantages

15. Based on the following data: 1. The initial coordinates (x 0, y 0, p x0, p y0 ) and z 0, 2. The values of the field at the initial coordinates (Fourier-coefficients), 3. The final position z f. → (x f, y f, p xf, p yf ) can be computed to arbitrary precision without the need of a PDE-solver! → The coordinate transformation is symplectic. → Fringe fields are included. → There are analytic formulas of the fields in the case of multipoles. 3. Advantages

16. The Hamiltonian of particle with charge e, mass m and energy E in an electromagnetic field (A x, A y, A z, φ ) can be written as 4. Theory

17. The Hamiltonian of particle with charge e, mass m and energy E in an electromagnetic field (A x, A y, A z, φ ) can be written as 4. Theory

18. The Hamiltonian of particle with charge e, mass m and energy E in an electromagnetic field (A x, A y, A z, φ ) can be written as with 4. Theory

19. The Hamiltonian of particle with charge e, mass m and energy E in an electromagnetic field (A x, A y, A z, φ ) can be written as with Note that in this description, the Z-component will play the role as the 'time' and (t, -E) is a new pair of conjugate variables. 4. Theory

20. This means, the equations of motion have the form: 4. Theory

21. This means, the equations of motion have the form: 4. Theory

22. This means, the equations of motion have the form: It follows especially: for the kicks in X- and Y-direction. 4. Theory

23. In the following we assume that 1. all fields are time-independent. 4. Theory

24. In the following we assume that 1. all fields are time-independent. 2. no electric fields. 4. Theory

25. In the following we assume that 1. all fields are time-independent. 2. no electric fields. 3. the kicks are small enough, so that products of order two and higher can be neglected. 4. Theory

26. In the following we assume that 1. all fields are time-independent. 2. no electric fields. 3. the kicks are small enough, so that products of order two and higher can be neglected. Assumption 3 is not necessary in order to make the method work, it merely simplifies the Hamiltonian. Higher orders can be included. 4. Theory

27. In the following we assume that 1. all fields are time-independent. 2. no electric fields. 3. the kicks are small enough, so that products of order two and higher can be neglected. Assumption 3 is not necessary in order to make the method work, it merely simplifies the Hamiltonian. Higher orders can be included. Dropping assumptions 1 and/or 2 will have a deeper impact on the theory. 4. Theory

28. Exclude in the radicand of the Hamiltonian 4. Theory

29. Exclude in the radicand of the Hamiltonian and develop the square root, using the small angular approximation: 4. Theory

30. Exclude in the radicand of the Hamiltonian and develop the square root, using the small angular approximation: where we introduced the normalized quantities 4. Theory

31. Note: By introducing this new Hamiltonian, the equations of motions for the X- and Y-coordinates will not change in these approximations: 4. Theory

32. Note: By introducing this new Hamiltonian, the equations of motions for the X- and Y-coordinates will not change in these approximations: 4. Theory

33. Note: By introducing this new Hamiltonian, the equations of motions for the X- and Y-coordinates will not change in these approximations: In the following we will drop all tilde symbols again. 4. Theory

34. A canonical transformation to the cyclic coordinates can be obtained by a generating function F of the variables (x, y, v x, v y, z) satisfying 4. Theory

35. A canonical transformation to the cyclic coordinates can be obtained by a generating function F of the variables (x, y, v x, v y, z) satisfying 4. Theory

36. A canonical transformation to the cyclic coordinates can be obtained by a generating function F of the variables (x, y, v x, v y, z) satisfying and 4. Theory

37. A canonical transformation to the cyclic coordinates can be obtained by a generating function F of the variables (x, y, v x, v y, z) satisfying and inserting p x and p y into this last equation gives the partial differential equation for F we are going to solve. 4. Theory

38. Hence, the partial differential equation for F has the form: 4. Theory

39. Hence, the partial differential equation for F has the form: where we redefined the magnetic potentials by a parameter epsilon to provide a measure of the field strength. 4. Theory

40. Hence, the partial differential equation for F has the form: where we redefined the magnetic potentials by a parameter epsilon to provide a measure of the field strength. In the absence of any fields, this differential equation can be solved directly: in which is a constant. 4. Theory

41. This leads to 4. Theory

42. This leads to and from the last equation we get Similar equations hold for the Y-component. This corresponds to a free drift. 4. Theory

43. This leads to and from the last equation we get Similar equations hold for the Y-component. This corresponds to a free drift. The canonical momenta p x and p y can be converted to the kinetic momenta using the vector potentials and the normalization factor introduced earlier. 4. Theory

44. In the general case, we make the following perturbative ansatz: 4. Theory

45. In the general case, we make the following perturbative ansatz: where f ijk are functions of x, y and z. 4. Theory

46. In the general case, we make the following perturbative ansatz: where f ijk are functions of x, y and z. We enter the general differential equation with this ansatz and compare coefficients, using its special nature: 4. Theory

47. This yields the following system of equations for the f ijk 's: 4. Theory

48. This yields the following system of equations for the f ijk 's: with The semicolon indicates a partial derivative with respect to the corresponding coordinate. 4. Theory

49. This yields the following system of equations for the f ijk 's: with The semicolon indicates a partial derivative with respect to the corresponding coordinate. Note that the left-hand side of the above equation is determined by functions of lower total order i + j + k and the potentials up to a function h ijk of x and y, the 'integration constant'. 4. Theory

50. The equations of the smallest total orders have the form 4. Theory

51. Let us see the implication of this ansatz for the momenta: 4. Theory

52. Let us see the implication of this ansatz for the momenta: 4. Theory

53. Let us see the implication of this ansatz for the momenta: and similar for the Y-component: 4. Theory

54. Let us see the implication of this ansatz for the momenta: and similar for the Y-component: This means: If we fix the functions h ijk (x, y) by the condition f ijk (x, y, z f ) ≡ 0, then we get and we can invert the above system of equations (for p x and p y ) at ε = 1 by a Newton-iteration to get p xf and p yf. 4. Theory

55. Once we have computed the values p xf and p yf, we can determine the offset x f by 4. Theory

56. Once we have computed the values p xf and p yf, we can determine the offset x f by 4. Theory

57. Once we have computed the values p xf and p yf, we can determine the offset x f by and similarly for the Y-component. Again we have set ε = 1. 4. Theory

58. Once we have computed the values p xf and p yf, we can determine the offset x f by and similarly for the Y-component. Again we have set ε = 1. The kicks at the final position are computed by 4. Theory

59. We have shown how to construct a symplectic mapping routine through time-independent magnetic fields in the approximation of small kicks. 5. Summary

60. We have shown how to construct a symplectic mapping routine through time-independent magnetic fields in the approximation of small kicks. Generalizations to higher orders in the kicks are possible without changing the theory, if we develop everything up - and including - to even order. 5. Summary

61. We have shown how to construct a symplectic mapping routine through time-independent magnetic fields in the approximation of small kicks. Generalizations to higher orders in the kicks are possible without changing the theory, if we develop everything up - and including - to even order. The time-dependent case and the inclusion of electric fields will however alter the differential equation. It is an open question of how to implement a perturbative generating function in these cases. 5. Summary

62. We have shown how to construct a symplectic mapping routine through time-independent magnetic fields in the approximation of small kicks. Generalizations to higher orders in the kicks are possible without changing the theory, if we develop everything up - and including - to even order. The time-dependent case and the inclusion of electric fields will however alter the differential equation. It is an open question of how to implement a perturbative generating function in these cases. Another interesting subject are the generalization of the method to density distributions. Thank you for your attention! 5. Summary