Etienne Forest. From Tracking Code. to Analysis. Generalised Courant-Snyder Theory for Any Accelerator Model. 4 } Springer

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1 Etienne Forest From Tracking Code to Analysis Generalised Courant-Snyder Theory for Any Accelerator Model 4 } Springer

2 Contents 1 Introduction Dichotomous Approach Derived from Complexity The Modern Way to Implement the Dichotomous Approach The Induced Hierarchy Inherited from the Tracking Code Give His Dues to Caesar: Dragt, Talman, and Berz The Necessary Properties 1.6 Integrable Systems Are Sitting of the "Exact Code" 6 on Tori The Example Code PTC: A Minimal Tutorial The Propagator Producing the One-Turn Taylor Map Propagator, Propagata and Propaganda 15 References 17 2 The Linear Transverse Normal Form: One Degree of Freedom Conversion Table Between Linear and Nonlinear Why Phasors and Normal Forms? The Average of an Arbitrary Function: Need for Phasors Linear Lattice Functions from de Moivre's Formula Lattice Functions as Coefficients of the Invariant Lattice Functions as Coefficients of the Moments 24 (zizj) The Program one_turn_orbital_map_normal_form_2d Construction of the Matrix A The Computation of the Map Numerical Computation of the Canonical Transformation 32 XV

3 xvi Contents 2.4 The Phase Advance and the Invariant Phase Advance in a Code: Finite Map Theory Numerical Computation of the Invariant and the Phase Advance Computation of the Phase: Some Theory Tracking of the Invariant: Some Hamiltonian Theory 41 References 43 3 The Nonlinear Transverse Normal Form: One Degree of Freedom The Pendulum with Exact Methods Preliminary The Hamiltonian of the Pendulum The Exact Period of the Pendulum The Frequency and the Hamiltonian as a Function of / The Action J as a Function of 0 and p The Pendulum with the Numerical Methods of Cosy-Infinity Phase Space Maps and Lie Maps Vector Field of the Pendulum Hamiltonian Normal Form of the Pendulum Illustration of Normal Form Theory with FPP Tools Normal Form of the Standard Map: The Algorithm for Maps Linear Part The Nonlinear Algorithm: Theory Standard Map: Normalisation in Software Creation of Standard Map and Using a "Canned" Normal Form Normalisation of the Map with Vector Fields Normalisation of the Map... with Poisson Brackets Using a Pseudo-Hamiltonian: Normalising the Logarithm 77 References 80 4 Classification of Linear Normal Forms The Full Harmonically Sinking Phase Space Radiation: The Multidimensional Drain De Moivre Representation. of the One-Turn Matrix Ripken Lattice Functions Representation: Invariants and Moments 84

4 Contents xvii De Moivre-Ripken Ha Matrices in the Symplectic Case: The Dispersions r\ and Kinematic Invariants of Linacs Coming Out of H' and Bl No Cavities: Jordan Normal Form The Reasons for a No-Cavity Normal Form A Glance at the Nonlinear Jordan Normal Form The Linear Part with No Cavity The Slip Factor and the Longitudinal Tune Normal Form for the "AC" Fluctuation of a Magnet Property Adding Pseudo Clocks to Phase Space A Simple Calculation with FPP The Way It Can Be Done in a Code: PTC A Linear Complication Due to the Case of Sect The Stochastic Normal Form: Radiation Theory The Stochastic Map of Moments Ill The Normal Form of the Quadratic Moment... Map Numerical Example 114 References Nonlinear Normal Forms What Do I Mean by Nonlinear? The (Damped) Pseudo-Harmonic Oscillator Case Derivation of Eq. (4.74): Magnet Modulation Revisited One Resonance Orbital Normal Form The Naive Dragt Approach: Raising the Map to a Power General Approach: Leaving One Resonance The Co-moving Map The Instructive Resonant Case Ji = ^v=i+sexp(: I*3 + *fx4 :) A Map with a Limit Cycle: Akin to a Resonance The Computation of Limit Cycles Computation of 0t of Eq. (5.62): Co-sinking Map Example Program for the Limit Cycle of Fig. 5.5b References Spin Normal Form Introductory Verbiage on Spin in the Code PTC The Normal Form for Spin on the Closed Orbit: n0 153

5 xviii Contents 6.3 The Nonlinear Normal Form for the Invariant Spin Field n(z) Short History and Comments on the Mysterious ISFn(z) The Normal Form and Why We Get n(z) from It The Algorithm for the Spin Normal Form A Code Implementation for the Spin Normal Form Leaving One Resonance in the Spin Map Two Cases for the Spectator Spin: Orbital and Spin-Orbit The Abell-Barber Co-moving Map for the Spin Orbit Case No and the Co-moving Spin Map Solution for the ISF n of the One-Resonance Map of Eq. (6.57) Numerical Behaviour of the One-Resonance ISF References The Nonlinear Spin-Orbital Phase Advance: The Mother of All Algorithms Introductory Verbiage on Phase Advance and "Canonisation" A Little Notation Hurdle Due to the Jordan Normal Form My Choice for the Fixed Point Map The Trivial Case The Case of a Jordan Normal Form The Linear Transformation A(s) My Choice for the Nonlinear Transformation b My Choice for the Spin Transformation D The Canonisation in the Example Code of Appendix M Setting Up the Example Finding the One-Turn Map and Normalising It Example of Canonisations in the "Courant-Snyder Loop" The Phase Advance, Its Freedom and the Code of Appendix M The Case of Hamiltonian Perturbation Theory The Case of Map Perturbation Theory: Orbital and Spin Description of the Phase Advance Loop of Appendix M Example of Something Being Computed in the Courant-Snyder Loop 197 References 203

6 Contents xix 8 Deprit-Guignard Perturbation Theory Faithful to the Code How About Hamiltonian Perturbation Theory? Definining a Time-Like Variable Using a Courant-Snyder Type Phase Advance Using a Constant Phase Advance Code Example for Sect. 8.3: Using the Courant-Snyder Phase Advance Code Example for Sect. 8.4: Using a Constant Phase Advance Normalising the 6(s)-Dependent Equations of Motion: Deprit-Guignard Approach Transforming the Equations of Motions The Actual Deprit-Guignard Normal Form Example Code for a Deprit-Guignard Normalisation Numerical Example (jc): Analytical, Guignard and Map The Final Strategy: Be Prepared to Mix Everything! 234 References Here Is the Conclusion of This Book Conclusion Exclusion A Deeper Discussion About Guignard Normalisation Synchro-Betatron Effects Phasors Basis: Why Do I Reject Symplectic Phasors? The Logarithm of a Map 247 Reference Stroboscopic Average for the ISF Vector n 251 References Hierarchy of Analytical Methods Green's Function Method The Rules of Analytical Perturbation Theory with Maps The Actual Calculation with Maps Fourier Mode Calculations with the Hamiltonian Changing the Time-Like Variable into a Phase Advance The Fourier Method Approach: Guignard Numerical Example (jc): Analytical, Guignard and Map 264 References 268

7 xx Contents Appendix A: The Hardwired ALS Lattice 269 Appendix B: Program for one_turn_orbital_map 273 Appendix C: Program one_turn_orbital_map_normal_form_2d 275 Appendix D: Program one_turn_orbital_map_phase_ad 279 Appendix E: Program Pendulum 283 Appendix F: Program standardjmap 287 Appendix G: Program one_turn_cavity_map 293 Appendix H: Program radiation_map 299 Appendix I: Program modulated_map 303 Appendix J: Program modulated_map_jordan 307 Appendix K: Program one_resonance_map 311 Appendix L: Program very_damped_map 317 Appendix M: Program spin_phase_advance_isf 319 Appendix N: Program hamitonian_guignard_cs.f Appendix O: Program hamitonian_guignard.f Appendix P: Program hamiltonian_guignard_ldf.f Appendix Q: Program hamiltonian_guignard_ldf_x.f Index of Links to Useful Concepts and Formulae 347

Accelerator Physics. Tip World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI BANGALORE. Second Edition. S. Y.

Accelerator Physics. Tip World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI BANGALORE. Second Edition. S. Y. Accelerator Physics Second Edition S. Y. Lee Department of Physics, Indiana University Tip World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI BANGALORE Contents Preface Preface