Table Of ContentBeam Dynamics
THE PHYSICS AND TECHNOLOGY OF PARTICLE
AND PHOTON BEAMS (formerly ACCELERATORS
AND STORAGE RINGS)
a series of monographs edited by Swapan Chattopadhyay, Lawrence
Berkeley Laboratory, California, USA
VOLUME 1
The Microtron^ S.P. Kapitza and V.N. Melekhin
VOLUME 2
Collective Methods of Acceleration, N. Rostoker and M. Reiser
VOLUME 3
Recirculating Electron Accelerators, Roy E. Rand
VOLUME 4
Particle Accelerators and Their Uses, Waldemar Scharf
VOLUME 5
Theory of Resonance Linear Accelerators, I.M. Kapchinskiy
VOLUME 6
The Optics of Charged Particle Beams, David C. Carey
VOLUME 7
Getter and Getter-Ion Vacuum Pumps, G.L. Saksaganskii
VOLUME 8
Beam Dynamics: A New Attitude and Framework
Etienne Forest
This book is part of a series. The publisher will accept continuation
orders which may be cancelled at any time and which provide for
automatic billing and shipping of each title in the series upon
publication. Please write for details.
Beam Dynamics
A New Attitude and Framework
Etienne Forest
High Energy Accelerator Research Organization
Iharaki, Japan
Graduate University for Advanced Studies
Kanagawa, Japan
and
Ernest Orlando Lawrence
Berkeley National Laboratory
California, USA
harwood academic publishers
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Copyright © 1998 OPA (Overseas Publishers Association) Amsterdam B.V.
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British Library Cataloguing in Publication Data
Forest, Etienne
Beam dynamics : a new attitude and framework. - (The
physics and technology of particle and photon beams ; v. 8)
1. Beam dynamics
I. Title
539.73
ISBN 90-5702-558-2
À ma famille
Masami, Guillaume et Natalie
CONTENTS
About the Series XV
List of Figures and Tables xvii
List of Exercises xix
Preface: the Power of Metaphors xxiii
1 Introduction 1
1.1 Dichotomy . 2
1.2 Global Dynamics 4
1.3 Local Dynamics . 5
1.4 "Layout of this book!" 10
I A Pictorial View in One Degree of Freedom 13
2 From the Map to the Hamiltonian 15
2.1 Stability: Existence of a Closed Orbit 15
2.2 Map around the Fixed Point . . . . . 17
2.3 Lie Maps: Search for Invariants . . . . 20
2.3.1 Definition of the Compositional (Lie) Map . 20
2.3.2 Representing Compositional Maps: Lie Operators 22
Interlude: Differential Equation for Compositional Maps 25
Il Interlude: Proof of the Miracles . . . . . . . . . . 27
2.4 Purpose of Lie Operators . . . . . . . . . . 30
2.4.1 Resonance Eigenbasis: the Invariant 30
Ill Interlude: Elegance . . . . . . . . . . . . . . . . 32
2.4.2 The Canonical Transformation A: Normalization 32
2.4.3 From Linear A to Nonlinear A . . . . . . . . 35
IV Interlude: Action Angle Coordinates . . . . . . . . . . . . . 38
2.4.4 Freedom in Choosing A: Gauge Dependence 39
2.4.5 Measurable Quantities: Gauge Independence 41
vii
viii CONTENTS
2.5 The Ring . . . . . . . . . . . . . . . . . . 45
2.5.1 Why More than One Map? . . . . 46
2.5.2 Phase Advance: the Floquet Ring 49
2.6 Phase Advance and Hamiltonians . . . 53
2.6.1 The Hamiltonian . . . . . . . . . . 54
2.6.2 The Evolution of the Invariant . . 56
2.6.3 Computation of the Phase Advance 59
2.6.4 Connection to Standard Floquet Theory . 61
3 From the Hamiltonian to the Map 65
3.1 Hamiltonian of a Section of Gutter 66
3.1.1 Lagrangian and Hamiltonian 66
V Interlude: From Lagrange to Hamilton . . . 66
3.1.2 A Coordinate as the Independent Variable . 69
VI Interlude: Changing the Independent Variable . . . . . . . 74
3.2 Approximations of Individual Sections . . . . . . . 75
3.2.1 Types of Elements: Gutter and Magnets . . 75
3.2.2 Connecting Gutters or Magnets: LEGO Blocks 78
3.2.3 Misplacing Segments: the Euclidean Group 80
3.3 Symplectic Integration . . . . . . . . 88
3.3.1 Integration or Modeling? . . . 88
3.3.2 Explicit Symplectic Integration 93
3.4 Map Extraction . . . . . . . . . . . . 96
3.4.1 Tracking . . . . . . . . . . . . 97
3.4.2 Automatic Differentiation: Taylor Series Maps 98
3.4.3 Algebra of k-jets: Differential and Lie Algebras 100
11 Global Description: Purpose 105
4 Classification of One-Turn Maps 107
4.1 The Quest for Invariants . . . . . 108
4.2 Existence of a Central Closed Orbit 110
4.3 The k-jets around the Closed Orbit . 111
4.4 Perturbation Theory: Lie Methods . 114
4.4.1 The Linear Part of the Map . 115
4.4.2 The Nonlinear Part .... . 117
4.4.3 Information in k-jets: Zero ... What?! 121
4.5 Linear Properties around a Fixed Point 123
4.5.1 The Symplectic Linear Map . 124
4.5.2 The Dissipative Linear Map . 130
VII Interlude: Extension of the State Variables . 132
CONTENTS ix
5 From Linear to Nonlinear Maps 137
5.1 Standard Solvable Problems in Rings . . . . . . . . . . 138
5.1.1 Amplitude Dependent Rotations . . . . . . . . 139
5.1.2 Amplitude Dependent Rotations for a Coasting
Beam . . . . . . . . . . . 140
5.1.3 One-Resonance Map . . . . . . . . . . . . . . 141
5.2 About Normalization Algorithms . . . . . . . . . . . 149
5.2.1 Bringing the Map into a One-Resonance Map 153
5.2.1.1 Lumping the perturbations 153
5.2.1.2 Going into resonance basis . 154
5.2.1.3 Canonical transformation 155
5.2.1.4 Let us pause and think . . 157
5.2.1.5 The eo-moving map 159
5.2.1.6 Computation of C1;. and he 160
5.2.1.7 Discussion: limiting cases 161
5.2.2 Other normalizations 163
6 Vector Fields and Canonical Transformations 165
6.1 Vector Fields ........ . 166
6.1.1 The Homomorphisms 166
6.1.2 The BCH Theorem .. 168
6.2 Lie 'fransformations on Vector Fields . 171
6.2.1 The Mixed Generating Function: if· pf + w 171
6.2.2 Canonical 'fransformation on H: a Map Per-
spective . . . . . . . . . . . . . . 173
6.3 Revisiting Hamiltonian Normalization . 176
6.3.1 Computation of w a la Guignard 177
6.3.2 "Isolated Resonance Theory" . . 179
6.3.3 Hamiltonian and Map: Equivalence 181
6.3.3.1 The Green's function: map-wise . . . 182
6.3.3.2 The Green's function: Hamiltonian-wise 183
6.4 Resonance Basis for Vector Fields. . . . . . . 186
6.4.1 'fransforming into the Resonance Basis 186
VIII Interlude: Transforming into the Resonance Basis: Real Axis Tunes 188
6.4.2 Eigenvector Fields . . . . . . . . . . . . 189
6.4.3 Section 4.4.2 Revisited . . . . . . . . . 190
IX Interlude: Normal Form for the Stochastic Moment Map . 192
7 The Ring 197
7.1 Rings and Sub-rings . . . . . . . . . . . . . . . . 198
7.2 'fransforming the Ring: A [Ring"K] . . . . . . . 200
X Interlude: Coordinate Dependent or Independent Description 202
