Book2Feb2015

Lie Methods for Nonlinear Dynamicswith Applications to Accelerator Physics

Alex J. DragtUniversity of Maryland, College Park

http://www.physics.umd.edu/dsat/UN

IVERSITY OF

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A R Y L A N

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18 56

2 February 2015

Alex J. DragtDynamical Systems and Accelerator Theory GroupDepartment of PhysicsUniversity of MarylandCollege Park, Maryland 20742

http://www.physics.umd.edu/dsat

Work supported in part by U. S. Department of Energy Grant DE-FG02-96ER40949.

c 1991 and 2014 by Alex J. Dragt.All rights reserved.

Contents

Preface lix

1 Introductory Concepts 11.1 Transfer Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Maps and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Maps and Accelerator Physics . . . . . . . . . . . . . . . . . . . 61.1.3 Maps and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Map Iteration and Other Background Material . . . . . . . . . . . . . . . 71.2.1 Logistic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Complex Logistic Map . . . . . . . . . . . . . . . . . . . . . . . . 151.2.3 Simplest Nonlinear Symplectic Map . . . . . . . . . . . . . . . . 191.2.4 Goal for Use of Maps in Accelerator Physics . . . . . . . . . . . 241.2.5 Maps from Hamiltonian Differential Equations . . . . . . . . . . 28

1.3 Essential Theorems for Differential Equations . . . . . . . . . . . . . . . . 411.4 Transfer Maps Produced by Differential Equations . . . . . . . . . . . . . 47

1.4.1 Map for Simple Harmonic Oscillator . . . . . . . . . . . . . . . . 481.4.2 Maps for Monomial Hamiltonians . . . . . . . . . . . . . . . . . 491.4.3 Stroboscopic Maps and Duffing Equation Example . . . . . . . . 50

1.5 Lagrangian and Hamiltonian Equations . . . . . . . . . . . . . . . . . . . 571.5.1 The Nonsingular Case . . . . . . . . . . . . . . . . . . . . . . . . 571.5.2 A Common Singular Case . . . . . . . . . . . . . . . . . . . . . . 59

1.6 Hamiltons Equations with a Coordinate as an Independent Variable . . . 671.7 Definition of Poisson Bracket . . . . . . . . . . . . . . . . . . . . . . . . . 92

2 Numerical Integration 1152.1 The General Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162.2 A Crude Solution Due to Euler . . . . . . . . . . . . . . . . . . . . . . . . 116

2.2.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162.2.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.3 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.3.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1242.3.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 1252.3.4 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

2.4 Finite-Difference/Multistep/Multivalue Methods . . . . . . . . . . . . . . 144

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2.4.1 Adams Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1492.4.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 1512.4.3 Derivation and Error Analysis . . . . . . . . . . . . . . . . . . . 158

2.5 (Automatic) Choice and Change of Step Size and Order . . . . . . . . . . 1682.5.1 Adaptive Change of Step Size in Runge-Kutta . . . . . . . . . . 1682.5.2 Adaptive Finite-Difference Methods . . . . . . . . . . . . . . . . 1692.5.3 Jet Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1702.5.4 Virtues of Jet Formulation . . . . . . . . . . . . . . . . . . . . . 1762.5.5 Advice to the Novice . . . . . . . . . . . . . . . . . . . . . . . . 178

2.6 Extrapolation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1812.6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1812.6.2 Making a Meso Step . . . . . . . . . . . . . . . . . . . . . . . . . 1812.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1852.6.4 Again, Advice to the Novice . . . . . . . . . . . . . . . . . . . . 185

2.7 Things Not Covered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1882.7.1 Strmer-Cowell and Nystrom Methods . . . . . . . . . . . . . . . 1892.7.2 Other Starting Procedures . . . . . . . . . . . . . . . . . . . . . 1892.7.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1892.7.4 Regularization, Etc. . . . . . . . . . . . . . . . . . . . . . . . . . 1892.7.5 Solutions with Few Derivatives . . . . . . . . . . . . . . . . . . . 1902.7.6 Symplectic and Geometric/Structure-Preserving Integrators . . . 1902.7.7 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1912.7.8 Backward Error Analysis . . . . . . . . . . . . . . . . . . . . . . 1922.7.9 Comparison of Methods . . . . . . . . . . . . . . . . . . . . . . . 193

3 Symplectic Matrices and Lie Algebras/Groups 2013.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2023.2 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043.3 Simple Symplectic Restrictions and Symplectic Factorization . . . . . . . 207

3.3.1 Large-Block Formulation . . . . . . . . . . . . . . . . . . . . . . 2073.3.2 Symplectic Block Factorization . . . . . . . . . . . . . . . . . . . 2083.3.3 Symplectic Matrices Have Determinant +1 . . . . . . . . . . . . 2113.3.4 Small-Block Formulation . . . . . . . . . . . . . . . . . . . . . . 211

3.4 Eigenvalue Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2123.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2123.4.2 The 2 2 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2143.4.3 The 4 4 and 2n 2n Cases . . . . . . . . . . . . . . . . . . . . 2153.4.4 Further Symplectic Restrictions . . . . . . . . . . . . . . . . . . 2193.4.5 In Praise of and Gratitude for the Symplectic Condition . . . . . 221

3.5 Eigenvector Structure, Normal Forms, and Stability . . . . . . . . . . . . 2253.5.1 Eigenvector Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 2253.5.2 J-Based Angular Inner Product . . . . . . . . . . . . . . . . . . 2253.5.3 Use of Angular Inner Product . . . . . . . . . . . . . . . . . . . 2253.5.4 Definition and Use of Signature . . . . . . . . . . . . . . . . . . . 2273.5.5 Definition of Phase Advances and Tunes . . . . . . . . . . . . . . 229

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3.5.6 The Krein-Moser Theorem and Krein Collisions . . . . . . . . . . 2293.5.7 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2313.5.8 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

3.6 Group Properties, Dyadic and Gram Matrices, and Bases . . . . . . . . . 2383.6.1 Group Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 2393.6.2 Dyadic and Gram Matrices, Bases and Reciprocal Bases . . . . . 2413.6.3 Orthonormal and Symplectic Bases . . . . . . . . . . . . . . . . 2443.6.4 Construction of Orthonormal Bases . . . . . . . . . . . . . . . . 2463.6.5 Construction of Symplectic Bases . . . . . . . . . . . . . . . . . 252

3.7 Lie Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 2583.7.1 Matrix Exponential and Logarithm . . . . . . . . . . . . . . . . . 2583.7.2 Application to Symplectic Matrices . . . . . . . . . . . . . . . . 2613.7.3 Matrix Lie Algebra and Lie Group: The BCH Theorem . . . . . 2633.7.4 Abstract Definition of a Lie Algebra . . . . . . . . . . . . . . . . 2653.7.5 Abstract Definition of a Lie Group . . . . . . . . . . . . . . . . . 2673.7.6 Classification of Lie Algebras . . . . . . . . . . . . . . . . . . . . 2683.7.7 Adjoint Representation of a Lie Algebra . . . . . . . . . . . . . . 272

3.8 Exponential Representations of Group Elements . . . . . . . . . . . . . . 2963.8.1 Exponential Representation of Orthogonal and Unitary Matrices 2973.8.2 Exponential Representation of Symplectic Matrices . . . . . . . . 297

3.9 Unitary Subgroup Structure . . . . . . . . . . . . . . . . . . . . . . . . . 3093.10 Other Subgroup Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 3203.11 Other Factorizations/Decompositions . . . . . . . . . . . . . . . . . . . . 3233.12 Cayley Representation of Symplectic Matrices . . . . . . . . . . . . . . . 3233.13 General Symplectic Forms, Darboux Transformations, etc. . . . . . . . . . 332

3.13.1 General Symplectic Forms . . . . . . . . . . . . . . . . . . . . . . 3323.13.2 Darboux Transformations . . . . . . . . . . . . . . . . . . . . . . 3353.13.3 Symplectic Forms and Pfaffians . . . . . . . . . . . . . . . . . . . 3393.13.4 Variant Symplectic Groups . . . . . . . . . . . . . . . . . . . . . 340

4 Matrix Exponentiation and Symplectification 3514.1 Exponentiation by Scaling and Squaring . . . . . . . . . . . . . . . . . . 352

4.1.1 The Ordinary Exponential Function . . . . . . . . . . . . . . . . 3524.1.2 The Matrix Exponential Function . . . . . . . . . . . . . . . . . 357

4.2 (Orthogonal) Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . 3604.3 Symplectic Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . 365

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3654.3.2 Properties of J-Symmetric Matrices . . . . . . . . . . . . . . . . 3664.3.3 Initial Result on Symplectic Polar Decomposition . . . . . . . . . 3694.3.4 Extended Result on Symplectic Polar Decomposition . . . . . . . 3704.3.5 Symplectic Polar Decomposition Not Globally Possible . . . . . . 3734.3.6 Uniqueness of Symplectic Polar Decomposition . . . . . . . . . . 3764.3.7 Concluding Summary . . . . . . . . . . . . . . . . . . . . . . . . 377

4.4 Finding the Closest Symplectic Matrix . . . . . . . . . . . . . . . . . . . 3934.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

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4.4.2 Use of Euclidean Norm . . . . . . . . . . . . . . . . . . . . . . . 3944.4.3 Geometric Interpretation of Symplectic Polar Decomposition . . 396

4.5 Symplectification Using Symplectic Polar Decomposition . . . . . . . . . 4044.5.1 Properties of Symplectification Using Symplectic Polar Decompo-

sition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4044.5.2 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

4.6 Modified Darboux Symplectification . . . . . . . . . . . . . . . . . . . . . 4124.7 Exponential and Cayley Symplectifications . . . . . . . . . . . . . . . . . 415

4.7.1 Exponential Symplectification . . . . . . . . . . . . . . . . . . . 4154.7.2 Cayley Symplectification . . . . . . . . . . . . . . . . . . . . . . 4164.7.3 Cayley Symplectification Near the Identity . . . . . . . . . . . . 417

4.8 Transformation (Generating) Function Symplectification . . . . . . . . . . 418

5 Lie Algebraic Structure of Classical Mechanics and Other Delights 4255.1 Properties of the Poisson Bracket . . . . . . . . . . . . . . . . . . . . . . 4255.2 Equations, Constants, and Integrals of Motion . . . . . . . . . . . . . . . 4275.3 Lie Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4295.4 Lie Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

5.4.1 Definition and Some Properties . . . . . . . . . . . . . . . . . . . 4345.4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

5.5 Realization of the sp(2n) Lie Algebra . . . . . . . . . . . . . . . . . . . . 4395.6 Basis for sp(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4425.7 Basis for sp(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4455.8 Basis for sp(6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

5.8.1 U(3) Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 4515.8.2 Polynomials for u(3) . . . . . . . . . . . . . . . . . . . . . . . . . 4535.8.3 Plan for the Remaining Polynomials . . . . . . . . . . . . . . . . 4535.8.4 Cartan Basis for su(3) . . . . . . . . . . . . . . . . . . . . . . . . 4545.8.5 Representations of su(3) . . . . . . . . . . . . . . . . . . . . . . 4565.8.6 Weight Diagrams for the First Few su(3) Representations . . . . 4585.8.7 Weight Diagram for the General su(3) Representation . . . . . . 4615.8.8 Remaining Polynomials . . . . . . . . . . . . . . . . . . . . . . . 462

5.9 Some Topological Questions . . . . . . . . . . . . . . . . . . . . . . . . . 4765.9.1 Nature and Connectivity of Sp(2n,R) . . . . . . . . . . . . . . . 4765.9.2 Where Are the Stable Elements? . . . . . . . . . . . . . . . . . . 4785.9.3 Covering/Circumnavigating U(n) . . . . . . . . . . . . . . . . . . 480

5.10 Notational Pitfalls and Quaternions . . . . . . . . . . . . . . . . . . . . . 4825.10.1 The Lie Algebras sp(2n,R) and usp(2n) . . . . . . . . . . . . . . 4825.10.2 USp(2n) and Quaternions . . . . . . . . . . . . . . . . . . . . . . 4835.10.3 Quaternion Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 4845.10.4 Properties of Quaternion Matrices . . . . . . . . . . . . . . . . . 4855.10.5 Quaternion Matrices and USp(2n) . . . . . . . . . . . . . . . . . 4875.10.6 Quaternion Inner Product and Its Preservation . . . . . . . . . . 4885.10.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

5.11 Moebius Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

CONTENTS v

5.11.1 Definition in the Context of Complex Variables . . . . . . . . . . 4945.11.2 Matrix Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 4955.11.3 Invertibility Conditions . . . . . . . . . . . . . . . . . . . . . . . 4965.11.4 Transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

5.12 Symplectic Transformations and Siegel Space . . . . . . . . . . . . . . . . 5005.12.1 Action of Sp(2n,C) on Space of Complex Symmetric Matrices . 5005.12.2 Siegel Space and Sp(2n,R) . . . . . . . . . . . . . . . . . . . . . 5015.12.3 Group Action on Homogeneous Space . . . . . . . . . . . . . . . 5015.12.4 Homogeneous Spaces and Cosets . . . . . . . . . . . . . . . . . . 5035.12.5 Group Action on Cosets Equals Group Action on Homogeneous

Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5055.12.6 Application of Results to Action of Sp(2n,R) on Siegel Space . . 5055.12.7 Action of Sp(2n,R) on the Generalized Real Axis . . . . . . . . 5075.12.8 Symplectic Modular Groups . . . . . . . . . . . . . . . . . . . . 508

5.13 Moebius Transformations Relating Symplectic and Symmetric Matrices . 5135.13.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5135.13.2 The Cayley Moebius Transformation . . . . . . . . . . . . . . . . 5145.13.3 Two Symplectic Forms and Their Relation by a Darboux Trans-

formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5155.13.4 The Infinite Family of Darboux Transformations . . . . . . . . . 5155.13.5 Isotropic Vectors and Lagrangian Planes . . . . . . . . . . . . . . 5175.13.6 Connection between Symplectic Matrices and Lagrangian Planes 5185.13.7 Connection between Symmetric Matrices and Lagrangian Planes 5195.13.8 Relation between Symplectic and Symmetric Matrices and the

Role of Moebius Transformations . . . . . . . . . . . . . . . . . . 5215.13.9 Completion of Tasks . . . . . . . . . . . . . . . . . . . . . . . . . 524

5.14 Matrix Symplectification Revisited . . . . . . . . . . . . . . . . . . . . . . 5315.15 Uniqueness of Cayley Moebius Transformation . . . . . . . . . . . . . . . 534

6 Symplectic Maps 5456.1 Preliminaries and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 545

6.1.1 Gradient Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5466.1.2 Symplectic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

6.2 Group Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5516.3 Preservation of General Poisson Brackets . . . . . . . . . . . . . . . . . . 5646.4 Relation to Hamiltonian Flows . . . . . . . . . . . . . . . . . . . . . . . . 567

6.4.1 Hamiltonian Flows Generate Symplectic Maps . . . . . . . . . . 5686.4.2 Any Family of Symplectic Maps Is Hamiltonian Generated . . . 5706.4.3 Almost All Symplectic Maps Are Hamiltonian Generated . . . . 5746.4.4 Transformation of a Hamiltonian Under the Action of a Symplectic

Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5756.5 Mixed-Variable Transformation (Generating) Functions . . . . . . . . . . 582

6.5.1 Transformation Functions Produce Symplectic Maps . . . . . . . 5836.5.2 Finding the Generating Hamiltonian . . . . . . . . . . . . . . . . 5856.5.3 Finding a Transformation Function for a Map . . . . . . . . . . . 588

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6.6 Transformation Functions Come from an Exact Differential . . . . . . . . 5956.6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5956.6.2 A Democratic Differential Form . . . . . . . . . . . . . . . . . . 5956.6.3 Information about M Carried by the Democratic Form . . . . . 5976.6.4 Breaking the Degeneracy . . . . . . . . . . . . . . . . . . . . . . 598

6.7 Plethora of Transformation Functions . . . . . . . . . . . . . . . . . . . . 6016.7.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016.7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6076.7.3 Hamilton-Jacobi Theory and Equations . . . . . . . . . . . . . . 611

6.8 Symplectic Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6246.8.1 Liouvilles Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 6246.8.2 Gromovs Nonsqueezing Theorem and the Symplectic Camel . . 6256.8.3 Poincare Integral Invariants . . . . . . . . . . . . . . . . . . . . . 6306.8.4 Connection between Surface and Line Integrals . . . . . . . . . . 6326.8.5 Poincare-Cartan Integral Invariant . . . . . . . . . . . . . . . . . 636

6.9 Poincare Surface of Section and Poincare Return Maps . . . . . . . . . . 6426.9.1 Poincare Surface of Section Maps . . . . . . . . . . . . . . . . . . 6436.9.2 Poincare Return Maps . . . . . . . . . . . . . . . . . . . . . . . . 645

6.10 Overview and Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646

7 Lie Transformations and Symplectic Maps 6537.1 Production of Symplectic Maps . . . . . . . . . . . . . . . . . . . . . . . 6537.2 Realization of the GroupSp(2n) and Its Subgroups . . . . . . . . . . . . . 6577.3 Invariant Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 6657.4 Symplectic Map for Flow of Time-Independent Hamiltonian . . . . . . . . 6767.5 Taylor Maps and Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6807.6 Factorization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6827.7 Inclusion of Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6897.8 Other Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6957.9 Coordinates and Connectivity . . . . . . . . . . . . . . . . . . . . . . . . 6957.10 Storage Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

8 A Calculus for Lie Transformations and Noncommuting Operators 7058.1 Adjoint Lie Operators and the Adjoint Lie Algebra . . . . . . . . . . . . 7058.2 Formulas Involving Adjoint Lie Operators . . . . . . . . . . . . . . . . . . 7078.3 Questions of Order and other Miscellaneous Mysteries . . . . . . . . . . . 726

8.3.1 Questions of Order and Map Multiplication . . . . . . . . . . . . 7268.3.2 Questions of Order in the Linear Case . . . . . . . . . . . . . . . 7308.3.3 Application to General Operators and General Monomials to Con-

struct Matrix Representations . . . . . . . . . . . . . . . . . . . 7318.3.4 Application to Linear Transformations of Phase Space . . . . . . 7338.3.5 Dual role of the Phase-Space Coordinates za . . . . . . . . . . . 7348.3.6 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7348.3.7 Sign Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . 735

8.4 Lie Concatenation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 738

CONTENTS vii

8.5 Map Inversion and Reverse Factorization . . . . . . . . . . . . . . . . . . 7468.6 Taylor and Hybrid Taylor-Lie Concatenation and Inversion . . . . . . . . 7488.7 Working with Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 756

8.7.1 Formulas for Combining Exponents . . . . . . . . . . . . . . . . 7568.7.2 Nature of Single Exponent Form . . . . . . . . . . . . . . . . . . 759

8.8 Zassenhaus or Factorization Formulas . . . . . . . . . . . . . . . . . . . . 7638.9 Ideals, Quotients, and Gradings . . . . . . . . . . . . . . . . . . . . . . . 765

9 Inclusion of Translations in the Calculus 7899.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7899.2 The Inhomogeneous Symplectic Group ISp(2n,R) . . . . . . . . . . . . . 7909.3 Lie Concatenation in the General Nonlinear Case . . . . . . . . . . . . . . 7989.4 Canonical Treatment of Translations . . . . . . . . . . . . . . . . . . . . . 807

9.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8079.4.2 Case of Maps with No Nonlinear Part . . . . . . . . . . . . . . . 8129.4.3 Case of General Maps . . . . . . . . . . . . . . . . . . . . . . . . 816

9.5 Map Inversion and Reverse and Mixed Factorizations . . . . . . . . . . . 8279.6 Taylor and Hybrid Taylor-Lie Concatenation and Inversion . . . . . . . . 8309.7 The Lie Algebra of the Group of all Symplectic Maps Is Simple . . . . . . 835

10 Computation of Transfer Maps 83910.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83910.2 Series (Dyson) Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84110.3 Exponential (Magnus) Solution . . . . . . . . . . . . . . . . . . . . . . . . 84210.4 Factored Product Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 846

10.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84610.4.2 Term by Term Procedure . . . . . . . . . . . . . . . . . . . . . . 84810.4.3 Accelerated Procedure . . . . . . . . . . . . . . . . . . . . . . . . 854

10.5 Forward Factorization and Lie Concatenation Revisited . . . . . . . . . . 85510.5.1 Preliminary Discussion . . . . . . . . . . . . . . . . . . . . . . . 85510.5.2 Forward Factorization . . . . . . . . . . . . . . . . . . . . . . . . 85510.5.3 Alternate Derivation of Lie Concatenation Formulas . . . . . . . 857

10.6 Direct Taylor Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . 85910.7 Scaling, Splitting, and Squaring . . . . . . . . . . . . . . . . . . . . . . . 86510.8 Canonical Treatment of Errors . . . . . . . . . . . . . . . . . . . . . . . . 87410.9 Symplectic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87910.10 Taylor Methods and the Complete Variational Equations . . . . . . . . . 879

10.10.1 Case of No or Ignored Parameter Dependence . . . . . . . . . . . 88110.10.2 Inclusion of Parameter Dependence . . . . . . . . . . . . . . . . 88210.10.3 Solution of Complete Variational Equations Using Forward Inte-

gration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88310.10.4 Application of Forward Integration to the Two-Variable Case . . 88510.10.5 Solution of Complete Variational Equations Using Backward Inte-

gration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88810.10.6 The Two-Variable Case Revisited . . . . . . . . . . . . . . . . . 890

viii CONTENTS

10.10.7 Application to Duffings Equation . . . . . . . . . . . . . . . . . 89210.10.8 Expanding in Parameters as Well . . . . . . . . . . . . . . . . . . 89410.10.9 Taylor Methods for the Hamiltonian Case . . . . . . . . . . . . . 900

11 Geometric/Structure-Preserving Integration: Integration on Manifolds 90711.1 Numerical Integration on Manifolds: Rigid-Body Motion . . . . . . . . . 908

11.1.1 Angular Velocity and Rigid-Body Kinematics . . . . . . . . . . . 90811.1.2 Angular Velocity and Rigid-Body Dynamics . . . . . . . . . . . . 91011.1.3 Problem of Integrating the Combined Kinematic and Dynamic

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91011.1.4 Solution by Projection . . . . . . . . . . . . . . . . . . . . . . . . 91111.1.5 Solution by Parameterization: Euler Angles . . . . . . . . . . . . 91111.1.6 Problem of Kinematic Singularities . . . . . . . . . . . . . . . . . 91211.1.7 Quaternions to the Rescue . . . . . . . . . . . . . . . . . . . . . 91311.1.8 Modification of the Quaternion Kinematic Equations of Motion . 91411.1.9 Local Coordinate Patches . . . . . . . . . . . . . . . . . . . . . . 91511.1.10 Canonical Coordinates of the Second Kind: Tait-Bryan Angles . 91611.1.11 Canonical Coordinates of the First Kind: Angle-Axis Parameters 91611.1.12 Cayley Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 91711.1.13 Summary of Integration Using Local Coordinates . . . . . . . . . 91811.1.14 Integration in the Lie Algebra: Exponential Representation . . . 91911.1.15 Integration in the Lie Algebra: Cayley Representation . . . . . . 92111.1.16 Parameterization of G and L(G) . . . . . . . . . . . . . . . . . . 92311.1.17 Quaternions Revisited . . . . . . . . . . . . . . . . . . . . . . . . 923

11.2 Numerical Integration on Manifolds: Spin and Qubits . . . . . . . . . . . 95311.2.1 Constrained Cartesian Coordinates Are Not Global . . . . . . . . 95411.2.2 Polar-Angle Coordinates Are Not Global . . . . . . . . . . . . . 95411.2.3 Local Tangent-Space Coordinates . . . . . . . . . . . . . . . . . 95511.2.4 Exploiting Connection with Rigid-Body Kinematics . . . . . . . 95711.2.5 What Just Happened? Generalizations . . . . . . . . . . . . . . 95811.2.6 Exploiting an Important Simplification: Lie Taylor Factorization

and Lie Taylor Runge Kutta . . . . . . . . . . . . . . . . . . . . 95911.2.7 Factored Lie Runge Kutta . . . . . . . . . . . . . . . . . . . . . 96511.2.8 Magnus Lie Runge Kutta . . . . . . . . . . . . . . . . . . . . . . 97211.2.9 Integration in the Lie Algebra Revisited . . . . . . . . . . . . . . 978

11.3 Numerical Integration on Manifolds: Charged Particle Motion in a StaticMagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100211.3.1 Exploitation of Previous Results . . . . . . . . . . . . . . . . . . 100211.3.2 Splitting: Exploitation of Future Results . . . . . . . . . . . . . 1004

12 Geometric/Structure-Preserving Integration: Symplectic Integration 100912.1 Splitting, T + V Splitting, and Zassenhaus Formulas . . . . . . . . . . . . 101012.2 Symplectic Runge-Kutta Methods for T + V Split Hamiltonians: Parti-

tioned Runge Kutta and Nystrom Runge Kutta . . . . . . . . . . . . . . 101612.3 Symplectic Runge-Kutta Methods for General Hamiltonians . . . . . . . . 1017

CONTENTS ix

12.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017

12.3.2 Condition for Symplecticity . . . . . . . . . . . . . . . . . . . . . 1019

12.3.3 The Single-Stage Case . . . . . . . . . . . . . . . . . . . . . . . . 1019

12.3.4 Two-, Three-, and More-Stage Methods . . . . . . . . . . . . . . 1021

12.4 Study of Single-Stage Method . . . . . . . . . . . . . . . . . . . . . . . . 1023

12.5 Study of Two-Stage Method . . . . . . . . . . . . . . . . . . . . . . . . . 1028

12.6 Numerical Examples for One- and Two-Stage Methods . . . . . . . . . . 1030

12.7 Proof of Condition for Symplecticity . . . . . . . . . . . . . . . . . . . . . 1030

12.8 Symplectic Integration of General Hamiltonians Using Transformation Func-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031

12.9 Special Symplectic Integrator for Motion in General Electromagnetic Fields 1032

12.10 Zassenhaus Formulas and Map Computation . . . . . . . . . . . . . . . . 1037

12.10.1 Case of T + V or General Electromagnetic Field Hamiltonians . 1037

12.10.2 Case of Hamiltonians Expanded in Homogeneous Polynomials . . 1038

12.11 Other Zassenhaus Formulas and Their Use . . . . . . . . . . . . . . . . . 1042

13 Transfer Maps for Idealized Straight Beam-Line Elements 1055

13.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055

13.1.1 Computation of Transfer Map . . . . . . . . . . . . . . . . . . . 1055

13.2 Axial Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

13.3 Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

13.4 Solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

13.5 Quadrupole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

13.6 Combined Function Quadrupole . . . . . . . . . . . . . . . . . . . . . . . 1060

13.7 Sextupole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

13.8 Octupole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

13.9 Higher-Order Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

13.10 Thin Lens Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

13.11 Radio Frequency Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

14 Transfer Maps for Idealized Curved Beam-Line Elements 1063

14.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

14.2 Sector Bend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

14.3 Parallel (Rectangular) Faced Bend . . . . . . . . . . . . . . . . . . . . . . 1063

14.4 Hard-Edge Fringe Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

14.5 Pole Face Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

14.6 General Bend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

14.7 Combined Function Bend . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

15 Taylor and Spherical and Cylindrical Harmonic Expansions 1065

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065

15.2 Spherical Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066

15.2.1 Harmonic Functions and Absolute and Expansion Coordinates . 1066

15.2.2 Spherical and Cylindrical Coordinates . . . . . . . . . . . . . . . 1067

x CONTENTS

15.2.3 Harmonic Polynomials, Harmonic Polynomial Expansions, and Gen-eral Spherical Polynomials . . . . . . . . . . . . . . . . . . . . . 1068

15.2.4 Spherical Polynomial Vector Fields . . . . . . . . . . . . . . . . . 107015.2.5 Determination of Minimum Vector Potential: the Poincare-Coulomb

Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107115.2.6 Uniqueness of Poincare-Coulomb Gauge . . . . . . . . . . . . . . 107715.2.7 Direct Construction of Poincare-Coulomb Gauge Vector Potential 1078

15.3 Cylindrical Harmonic Expansion . . . . . . . . . . . . . . . . . . . . . . . 108415.3.1 Complex Cylindrical Harmonic Expansion . . . . . . . . . . . . . 108415.3.2 Real Cylindrical Harmonic Expansion . . . . . . . . . . . . . . . 108615.3.3 Some Simple Examples: m = 0, 1, 2 . . . . . . . . . . . . . . . . 109015.3.4 Magnetic Field Expansions for the General Case . . . . . . . . . 109215.3.5 Symmetry and Allowed and Forbidden Multipoles . . . . . . . . 109515.3.6 Relation between Harmonic Polynomials in Spherical and Cylin-

drical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 109615.4 Determination of the Vector Potential: Azimuthal-Free Gauge . . . . . . 1100

15.4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110115.4.2 Some Simple Examples: m = 1, 2 . . . . . . . . . . . . . . . . . . 1103

15.5 Determination of the Vector Potential: Coulomb Gauge . . . . . . . . . . 110715.5.1 The m = 0 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 110715.5.2 The m 1 Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 1111

15.6 Nonuniqueness of Coulomb Gauge . . . . . . . . . . . . . . . . . . . . . . 112015.6.1 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . 112015.6.2 Normal Dipole Example . . . . . . . . . . . . . . . . . . . . . . . 1122

15.7 Determination of the Vector Potential: Poincare-Coulomb Gauge . . . . . 112715.7.1 The m = 0 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 112715.7.2 The m 1 Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 1127

15.8 Magnetic Monopole Doublet Example . . . . . . . . . . . . . . . . . . . . 113215.8.1 Magnetic Scalar Potential and Magnetic Field . . . . . . . . . . . 113315.8.2 Analytic On-Axis Gradients for Monopole Doublet . . . . . . . . 1136

15.9 Minimum Vector Potential for Magnetic Monopole Doublet . . . . . . . . 114515.10 Caveat about Significance of Integrated Multipoles . . . . . . . . . . . . . 114615.11 Need for Generalized Gradients and the Use of Surface Data . . . . . . . 1149

16 Realistic Transfer Maps for Straight Iron-Free Beam-Line Elements 115316.1 Terminating End Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153

16.1.1 Preliminary Observations . . . . . . . . . . . . . . . . . . . . . . 115316.1.2 Matching Conditions . . . . . . . . . . . . . . . . . . . . . . . . 115516.1.3 Changing Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . 1160

16.2 Simple Solenoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116116.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116116.2.2 Simple Air-Core Solenoid . . . . . . . . . . . . . . . . . . . . . . 116316.2.3 Computation of Transfer Map . . . . . . . . . . . . . . . . . . . 116816.2.4 Solenoidal Fringe-Field Effects: Ignoring or Hard-Edge Modeling

Them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1171

CONTENTS xi

16.2.5 Terminating Solenoidal End Fields . . . . . . . . . . . . . . . . . 117116.2.6 More Complicated Solenoids . . . . . . . . . . . . . . . . . . . . 1174

16.3 Iron-Free Dipoles/Wigglers/Undulators . . . . . . . . . . . . . . . . . . . 117616.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117616.3.2 Single Monopole Doublet . . . . . . . . . . . . . . . . . . . . . . 117616.3.3 Line of Monopole Doublets . . . . . . . . . . . . . . . . . . . . . 117916.3.4 Idealized Air-Core Dipole . . . . . . . . . . . . . . . . . . . . . . 117916.3.5 Lambertson-Type Dipoles . . . . . . . . . . . . . . . . . . . . . . 118416.3.6 Simple Air-Core Wiggler/Undulator Model . . . . . . . . . . . . 118416.3.7 Terminating Air-Core Wiggler/Undulator End Fields . . . . . . . 1185

16.4 Realistic Wigglers/Undulators . . . . . . . . . . . . . . . . . . . . . . . . 118616.4.1 Iron-Free Rare Earth Cobalt (REC) Wiggler/Undulator . . . . . 118616.4.2 Terminating Iron-Free Rare Earth Cobalt (REC) Wiggler/Undulator

End Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118616.5 Iron-Free Quadrupoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189

16.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118916.5.2 Single Monopole Quartet . . . . . . . . . . . . . . . . . . . . . . 119016.5.3 Line of Monopole Quartets . . . . . . . . . . . . . . . . . . . . . 119216.5.4 Idealized Air-Core Quadrupole . . . . . . . . . . . . . . . . . . . 119316.5.5 Lambertson-Type Quadrupoles . . . . . . . . . . . . . . . . . . . 119616.5.6 Terminating Air-Core Quadrupole End Fields . . . . . . . . . . . 119616.5.7 Rare Earth Cobalt (REC) Quadrupoles . . . . . . . . . . . . . . 119916.5.8 Overlapping Fringe Fields . . . . . . . . . . . . . . . . . . . . . . 120216.5.9 Terminating Rare Earth Cobalt (REC) Quadrupole End Fields . 1202

16.6 Sextupoles and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120216.7 Lithium Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202

17 Surface Methods for General Straight Beam-Line Elements 120517.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120517.2 Use of Potential Data on Surface of Circular Cylinder . . . . . . . . . . . 120917.3 Use of Field Data on Surface of Circular Cylinder . . . . . . . . . . . . . 121217.4 Use of Field Data on Surface of Elliptical Cylinder . . . . . . . . . . . . . 1214

17.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121417.4.2 Elliptic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 121617.4.3 Mathieu Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 121817.4.4 Periodic Mathieu Functions and Separation Constants . . . . . . 121917.4.5 Modified Mathieu Functions . . . . . . . . . . . . . . . . . . . . 123417.4.6 Analyticity in x and y . . . . . . . . . . . . . . . . . . . . . . . . 123817.4.7 Elliptic Cylinder Harmonic Expansion and On-Axis Gradients . . 1238

17.5 Use of Field Data on Surface of Rectangular Cylinder . . . . . . . . . . . 124217.5.1 Finding the Magnetic Scalar Potential (x, y, z) . . . . . . . . . 124217.5.2 Finding the On-Axis Gradients . . . . . . . . . . . . . . . . . . . 124917.5.3 Fourier-Bessel Connection Coefficients . . . . . . . . . . . . . . . 1251

17.6 Attempted Use of Nearly On-Axis and Midplane Field Data . . . . . . . . 125917.6.1 Use of Nearly On-Axis Data . . . . . . . . . . . . . . . . . . . . 1260

xii CONTENTS

17.6.2 Use of Midplane Field Data . . . . . . . . . . . . . . . . . . . . . 126117.7 Terminating End Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264

17.7.1 Preliminary Observations . . . . . . . . . . . . . . . . . . . . . . 126417.7.2 Changing Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . 126517.7.3 Finding the Minimal Vector Potential . . . . . . . . . . . . . . . 126717.7.4 The m = 0 Case: Solenoid Example . . . . . . . . . . . . . . . . 127217.7.5 The m = 1 Case: Magnetic Monopole Doublet and Wiggler Ex-

amples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127417.7.6 The m = 2 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 127717.7.7 The m = 3 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 127817.7.8 More Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278

18 Tools for Numerical Implementation 128518.1 Third-Order Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285

18.1.1 Fitting Over an Interval . . . . . . . . . . . . . . . . . . . . . . . 128518.1.2 Periodic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . 128818.1.3 Error Estimate for Spline Approximation . . . . . . . . . . . . . 1290

18.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129218.2.1 Bicubic Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 129318.2.2 Bicubic Spline Interpolation . . . . . . . . . . . . . . . . . . . . . 1297

18.3 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129818.3.1 Exact Fourier Transform and Its Large |k| Behavior . . . . . . . 129818.3.2 Inverse Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 129918.3.3 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . 130318.3.4 Discrete Inverse Fourier Transform . . . . . . . . . . . . . . . . . 130718.3.5 Spline-Based Fourier Transforms . . . . . . . . . . . . . . . . . . 130718.3.6 Fast Spline-Based Fourier Transforms . . . . . . . . . . . . . . . 1317

18.4 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131818.5 Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318

18.5.1 Calculation of Separation Constants an(q) and bn(q) . . . . . . . 131818.5.2 Calculation of Mathieu Functions . . . . . . . . . . . . . . . . . 131818.5.3 Calculation of Fourier and Mathieu-Bessel Connection Coefficients 1321

19 Numerical Benchmarks 132519.1 Circular Cylinder Numerical Results for Monopole Doublet . . . . . . . . 1325

19.1.1 Testing the Spline-Based Inverse (k z) Fourier Transform . . . 132519.1.2 Testing the Forward (z k) and ( m) Fourier Transforms . 133219.1.3 Test of Interpolation off a Grid . . . . . . . . . . . . . . . . . . . 133619.1.4 Reproduction of Interior Field Values . . . . . . . . . . . . . . . 1338

19.2 Elliptical Cylinder Numerical Results for Monopole Doublet . . . . . . . . 135119.2.1 Finding the Mathieu Coefficients . . . . . . . . . . . . . . . . . . 135119.2.2 Behavior of Kernels . . . . . . . . . . . . . . . . . . . . . . . . . 136019.2.3 Truncation of Series . . . . . . . . . . . . . . . . . . . . . . . . . 136119.2.4 Approximation of Angular Integrals by Riemann Sums . . . . . . 136719.2.5 Further Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376

CONTENTS xiii

19.2.6 Completion of Test . . . . . . . . . . . . . . . . . . . . . . . . . 137619.3 Rectangular Cylinder Numerical Results for Monopole Doublet . . . . . . 1388

20 Smoothing and Insensitivity to Errors 139120.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391

20.1.1 Preliminary Considerations . . . . . . . . . . . . . . . . . . . . . 139120.1.2 Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139120.1.3 Equivalent Spatial Kernel . . . . . . . . . . . . . . . . . . . . . . 139220.1.4 What Work Lies Ahead . . . . . . . . . . . . . . . . . . . . . . . 1398

20.2 Circular Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139820.3 Elliptic Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141520.4 Rectangular Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439

21 Realistic Transfer Maps for General Straight Beam-Line Elements 144321.1 Solenoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443

21.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144321.1.2 Qualitatively Correct Iron-Dominated Solenoid Model . . . . . . 144421.1.3 Improved Model for Iron-Dominated Solenoid . . . . . . . . . . . 144621.1.4 Quantitatively Correct Iron-dominated Solenoid . . . . . . . . . 1449

21.2 Realistic Wigglers/Undulators . . . . . . . . . . . . . . . . . . . . . . . . 144921.2.1 An Iron-Dominated Superconducting Wiggler/Undulator . . . . 1449

21.3 Quadrupoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144921.3.1 Validation of Circular Cylinder Surface Method . . . . . . . . . . 144921.3.2 Final Focus Quadrupoles . . . . . . . . . . . . . . . . . . . . . . 1459

21.4 Closely Adjacent Quadrupoles and Sextupoles . . . . . . . . . . . . . . . 145921.5 Application to Radio-Frequency Cavities . . . . . . . . . . . . . . . . . . 1459

22 Realistic Transfer Maps for General Curved Beam-Line Elements 146322.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146322.2 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465

22.2.1 Electric Dirac Strings . . . . . . . . . . . . . . . . . . . . . . . . 146522.2.2 Magnetic Dirac Strings . . . . . . . . . . . . . . . . . . . . . . . 146822.2.3 Full (Two) String Monopole Vector Potential . . . . . . . . . . . 147522.2.4 Monopole Doublet Vector Potentials . . . . . . . . . . . . . . . . 147622.2.5 Helmholtz Decomposition . . . . . . . . . . . . . . . . . . . . . . 1486

22.3 Construction of Kernels Gn and Gt . . . . . . . . . . . . . . . . . . . . . 149322.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149322.3.2 Construction of Gn Using Half-Infinite String Monopoles . . . . 149422.3.3 Construction of Gn Using Fully Infinite String Monopoles . . . . 149522.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149622.3.5 Construction of Gt . . . . . . . . . . . . . . . . . . . . . . . . . . 149722.3.6 Final Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1499

22.4 Expansion of Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150222.4.1 Our Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150222.4.2 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1502

xiv CONTENTS

22.4.3 Expansion of Gt(r, r) . . . . . . . . . . . . . . . . . . . . . . . . 150222.4.4 Expansion of Gn(r, r) . . . . . . . . . . . . . . . . . . . . . . . . 1502

22.5 Numerical Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1502

22.5.1 Exact Field, Design Orbit Selection, and Choice of SurroundingBent Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1502

22.5.2 Bent Box Results: Comparison of Fields . . . . . . . . . . . . . . 1513

22.5.3 Bent Box Results: Comparison of Design Orbits . . . . . . . . . 1525

22.5.4 Bent Box Results: Comparison of Maps . . . . . . . . . . . . . . 1525

22.6 Place Holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525

22.7 Plane Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525

22.7.1 Setting Up Reference Planes . . . . . . . . . . . . . . . . . . . . 1526

22.7.2 Integrating Between Reference Planes . . . . . . . . . . . . . . . 1529

22.7.3 Transition Relations . . . . . . . . . . . . . . . . . . . . . . . . . 1530

22.8 Terminating End Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534

22.9 Smoothing and Insensitivity to Errors . . . . . . . . . . . . . . . . . . . . 1534

22.10 Application to a Storage-Ring Dipole . . . . . . . . . . . . . . . . . . . . 1534

23 Error Effects and the Euclidean Group 1537

24 Representations of sp(2n) and Related Matters 1539

24.1 Structure of sp(2, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1540

24.2 Representations of sp(2, R) . . . . . . . . . . . . . . . . . . . . . . . . . . 1542

24.3 Symplectic Classification of Analytic VectorFields in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546

24.4 Structure of sp(4, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555


24.6 Symplectic Classification of Analytic VectorFields in Four Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567

24.7 Structure of sp(6, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1572


24.9 Symplectic Classification of Analytic VectorFields in Six Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1582

24.10 Scalar Product and Projection Operators forVector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587

24.11 Products and Casimir Operators . . . . . . . . . . . . . . . . . . . . . . . 1596

24.11.1 The Quadratic Casimir Operator . . . . . . . . . . . . . . . . . . 1596

24.11.2 Applications of the Quadratic Casimir Operator . . . . . . . . . 1602

24.11.3 Higher-Order Casimir Operators . . . . . . . . . . . . . . . . . . 1606

24.12 The Killing Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611

24.13 Enveloping Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613

24.14 The Symplectic Lie Algebras sp(8) and Beyond . . . . . . . . . . . . . . . 1619

24.15 Moment (Momentum) Maps and Casimirs . . . . . . . . . . . . . . . . . 1620

24.15.1 Moment (Momentum) Maps and Conservation Laws . . . . . . . 1621

24.15.2 Use of Casimirs . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623

CONTENTS xv

25 Numerical Study of Stroboscopic Duffing Map 163125.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163125.2 Review of Simple Harmonic Oscillator Behavior . . . . . . . . . . . . . . 163225.3 Behavior for Small Driving when Nonlinearity is Included . . . . . . . . . 163525.4 Saddle-Node (Blue-Sky) Bifurcations, Basins,

Symmetry, Amplitude Jumps, and Hysteresis . . . . . . . . . . . . . . . . 163725.4.1 Saddle-Node (Blue-Sky) Bifurcations . . . . . . . . . . . . . . . . 163725.4.2 Basins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163725.4.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164125.4.4 Amplitude Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 164125.4.5 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1643

25.5 Pitchfork Bifurcations and Symmetry . . . . . . . . . . . . . . . . . . . . 164325.6 Period Tripling Bifurcations and Fractal Basin Boundaries . . . . . . . . 164925.7 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165525.8 Period Doubling Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . 165825.9 Strange Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166225.10 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668

26 General Maps 167126.1 Lie Factorization of General Maps . . . . . . . . . . . . . . . . . . . . . . 167126.2 Classification of General Two-Dimensional

Quadratic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167626.3 Lie Factorization of General Two-Dimensional

Quadratic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168126.4 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1689

26.4.1 Attack a Map at its Fixed Points . . . . . . . . . . . . . . . . . . 168926.4.2 Fixed Points are Generally Isolated . . . . . . . . . . . . . . . . 169026.4.3 Finding Fixed Points with Contraction Maps . . . . . . . . . . . 169126.4.4 Persistence of Fixed Points . . . . . . . . . . . . . . . . . . . . . 169226.4.5 Application to Accelerator Physics . . . . . . . . . . . . . . . . . 1694

26.5 Poincare Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169626.6 Manifolds, and Homoclinic Points and Tangles . . . . . . . . . . . . . . . 170826.7 The General Henon Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 172126.8 Preliminary Study of General Henon Map . . . . . . . . . . . . . . . . . . 1729

26.8.1 Location, Expansion About, and Nature of Fixed Points . . . . . 172926.8.2 Lie Factorization About the First (Hyperbolic) Fixed Point . . . 173626.8.3 Location and Nature of Second Fixed Point . . . . . . . . . . . . 173926.8.4 Expansion and Lie Factorization About

Second Fixed Point . . . . . . . . . . . . . . . . . . . . . . . . . 174826.9 Period Doubling and Strange Attractors . . . . . . . . . . . . . . . . . . . 1754

26.9.1 Behavior about Hyperbolic Fixed Point . . . . . . . . . . . . . . 175426.9.2 Behavior about Second Fixed Point . . . . . . . . . . . . . . . . 1754

26.10 Attempts at Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175626.11 Quadratic Maps in Higher Dimensions . . . . . . . . . . . . . . . . . . . . 175626.12 Truncated Taylor Approximations to Stroboscopic Duffing Map . . . . . . 1756

xvi CONTENTS

26.12.1 Saddle-Node Bifurcations . . . . . . . . . . . . . . . . . . . . . . 175626.12.2 Pitchfork Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 177026.12.3 Infinite Period-Doubling Cascade and Strange Attractor . . . . . 177826.12.4 Undoing a Cascade by Successive Mergings . . . . . . . . . . . . 179226.12.5 Convergence of Taylor Maps: Performance of Lower-Order Poly-

nomial Approximations . . . . . . . . . . . . . . . . . . . . . . . 179926.12.6 Concluding Summary and Discussion . . . . . . . . . . . . . . . 180526.12.7 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806

26.13 Analytic Properties of Fixed Points and Eigenvalues . . . . . . . . . . . . 1806

27 Normal Forms for Symplectic Maps and Their Applications 181327.1 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181327.2 Symplectic Conjugacy of Symplectic Maps . . . . . . . . . . . . . . . . . 181427.3 Normal Forms for Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181427.4 Sample Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181627.5 Dynamic Maps Without Translation Factor . . . . . . . . . . . . . . . . . 181727.6 Dynamic Maps With Translation Factor . . . . . . . . . . . . . . . . . . . 181727.7 Static Maps Without Translation Factor . . . . . . . . . . . . . . . . . . 1817

27.7.1 Preparatory Steps . . . . . . . . . . . . . . . . . . . . . . . . . . 181727.8 Static Maps With Translation Factor . . . . . . . . . . . . . . . . . . . . 182327.9 Tunes, Phase Advances and Slips, Momentum Compaction, Chromaticities,

and Anharmonicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182327.10 Courant-Snyder Invariants and Lattice

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182327.11 Analysis of Tracking Data . . . . . . . . . . . . . . . . . . . . . . . . . . 1823

28 Lattice Functions 1827

29 Solved and Unsolved Polynomial Orbit Problems: Invariant Theory 182929.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182929.2 Solved Polynomial Orbit Problems . . . . . . . . . . . . . . . . . . . . . . 1831

29.2.1 First-Order Polynomials . . . . . . . . . . . . . . . . . . . . . . . 183129.2.2 Second-Order Polynomials . . . . . . . . . . . . . . . . . . . . . 1832

29.3 Mostly Unsolved Polynomial Orbit Problems . . . . . . . . . . . . . . . . 186029.3.1 Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 186029.3.2 Quartic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 1861

29.4 Application to Analytic Properties . . . . . . . . . . . . . . . . . . . . . . 1862

30 Beam Description and Moment Transport 187530.1 Beam Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187530.2 Moments and Moment Transport . . . . . . . . . . . . . . . . . . . . . . 187530.3 Emittances and Moment Invariants . . . . . . . . . . . . . . . . . . . . . 187630.4 Some Properties of Second Moments . . . . . . . . . . . . . . . . . . . . . 1877

30.4.1 Classical Uncertainty Principle . . . . . . . . . . . . . . . . . . . 187830.4.2 Minimum Emittance Theorem . . . . . . . . . . . . . . . . . . . 1880

CONTENTS xvii

31 Optimal Evaluation of Symplectic Maps 188531.1 Overview of Symplectic Map Approximation . . . . . . . . . . . . . . . . 188531.2 Symplectic Completion of Symplectic Jets . . . . . . . . . . . . . . . . . . 1891

31.2.1 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189131.2.2 Monomial Approximation . . . . . . . . . . . . . . . . . . . . . . 189131.2.3 Transformation Function Approximation . . . . . . . . . . . . . 189131.2.4 Cremona Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1891

31.3 Connection Between Transformation Functionsand Lie Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189131.3.1 Method of Calculation . . . . . . . . . . . . . . . . . . . . . . . . 189231.3.2 Computing g2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189431.3.3 Low Order Results: Computing g3 and g4 . . . . . . . . . . . . . 189531.3.4 Two Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189831.3.5 Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189931.3.6 Comments and Comparisons . . . . . . . . . . . . . . . . . . . . 1909

31.4 Use of Poincare Transformation Function . . . . . . . . . . . . . . . . . . 191231.4.1 Determination of Poincare Transformation Function

in Terms of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191231.4.2 Application to Quadratic Hamiltonian . . . . . . . . . . . . . . . 191331.4.3 Application to Symplectic Approximation . . . . . . . . . . . . . 1914

31.5 Use of Other Transformation Functions . . . . . . . . . . . . . . . . . . . 191631.6 Cremona Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916

32 Orbit Stability, Long-Term Behavior, and Dynamic Aperture 1919

33 Reversal Symmetry 192133.1 Reversal Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192133.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192733.3 General Consequences for Straight and Circular Machines . . . . . . . . . 193533.4 Consequences for some Special Cases . . . . . . . . . . . . . . . . . . . . 194033.5 Consequences for Closed Orbit in a Circular Machine . . . . . . . . . . . 194133.6 Consequences for Courant-Snyder Functions in a Circular Machine . . . . 194633.7 Some Nonlinear Consequences . . . . . . . . . . . . . . . . . . . . . . . . 1952

34 Standard First- and Higher-Order Optical Modules 1961

35 Analyticity and Convergence 196335.1 Analyticity in One Complex Variable . . . . . . . . . . . . . . . . . . . . 196335.2 Analyticity in Several Complex Variables . . . . . . . . . . . . . . . . . . 196735.3 Convergence of Homogeneous Polynomial Series . . . . . . . . . . . . . . 198035.4 Application to Potentials and Fields . . . . . . . . . . . . . . . . . . . . . 198835.5 Application to Taylor Maps: The Anharmonic Oscillator . . . . . . . . . 198835.6 Application to Taylor Maps: The Pendulum . . . . . . . . . . . . . . . . 198835.7 Convergence of the BCH Series . . . . . . . . . . . . . . . . . . . . . . . . 198835.8 Convergence of Lie Transformations and the Factored Product Representation1988

xviii CONTENTS

36 Truncated Power Series Algebra 199336.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199336.2 Monomial Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1994

36.2.1 An Obvious but Memory Intensive Method . . . . . . . . . . . . 199436.2.2 Polynomial Grading . . . . . . . . . . . . . . . . . . . . . . . . . 199536.2.3 Monomial Ordering . . . . . . . . . . . . . . . . . . . . . . . . . 199536.2.4 Labeling Based on Ordering . . . . . . . . . . . . . . . . . . . . 199736.2.5 Formulas for Lowest and Highest Indices . . . . . . . . . . . . . 199936.2.6 The Giorgilli Formula . . . . . . . . . . . . . . . . . . . . . . . . 200036.2.7 Finding the Required Binomial Coefficients . . . . . . . . . . . . 200036.2.8 Computation of the Index i Given the Exponent Array j . . . . 200236.2.9 Preparing a Look-Up Table for the Exponent Array j Given the

Index i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200336.2.10 Verification of the Giorgilli Formula . . . . . . . . . . . . . . . . 2006

36.3 Scalar Multiplication and Polynomial Addition . . . . . . . . . . . . . . . 201136.4 Polynomial Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . 201236.5 Look-Up Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201336.6 Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201936.7 Look-Back Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202636.8 Poisson Bracketing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203436.9 Linear Map Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204236.10 General Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204536.11 Expanding Functions of Polynomials . . . . . . . . . . . . . . . . . . . . . 204736.12 Differential Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204736.13 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2047

A Strmer-Cowell and Nystrom Integration Methods 2051A.1 Preliminary Derivation of Strmer-Cowell

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2051A.2 Summed Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2053

A.2.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2053A.2.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2054

A.3 Computation of First Derivative . . . . . . . . . . . . . . . . . . . . . . . 2056A.4 Example Program and Numerical Results . . . . . . . . . . . . . . . . . . 2057

A.4.1 Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2057A.4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 2060

A.5 Nystrom Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . 2061

B Computer Programs for Numerical Integration 2067B.1 A 3rd Order Runge-Kutta Routine . . . . . . . . . . . . . . . . . . . . . . 2068

B.1.1 Butcher Tableau for RK3 . . . . . . . . . . . . . . . . . . . . . . 2068B.1.2 The Routine RK3 . . . . . . . . . . . . . . . . . . . . . . . . . . 2068

B.2 A 4th Order Runge-Kutta Routine . . . . . . . . . . . . . . . . . . . . . . 2069B.2.1 Butcher Tableau for RK4 . . . . . . . . . . . . . . . . . . . . . . 2069B.2.2 The Routine RK4 . . . . . . . . . . . . . . . . . . . . . . . . . . 2069

CONTENTS xix

B.3 A Subroutine to Compute f . . . . . . . . . . . . . . . . . . . . . . . . . 2070B.4 A Partial Double-Precision Version of RK3 . . . . . . . . . . . . . . . . . 2071B.5 A 6th Order 8 Stage Runge-Kutta Routine . . . . . . . . . . . . . . . . . 2073

B.5.1 Butcher Tableau for RK6 . . . . . . . . . . . . . . . . . . . . . . 2073B.5.2 The Routine RK6 . . . . . . . . . . . . . . . . . . . . . . . . . . 2073

B.6 Embedded Runge-Kutta Pairs . . . . . . . . . . . . . . . . . . . . . . . . 2075B.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2075B.6.2 Fehlberg 4(5) Pair . . . . . . . . . . . . . . . . . . . . . . . . . . 2076B.6.3 Dormand-Prince 5(4) Pair . . . . . . . . . . . . . . . . . . . . . . 2078

B.7 A 5th Order PECEC Adams Routine . . . . . . . . . . . . . . . . . . . . 2080B.8 A 10th Order PECEC Adams Routine . . . . . . . . . . . . . . . . . . . . 2082

C Baker-Campbell-Hausdorff and Zassenhaus Formulas, Bases, and Paths 2087C.1 Differentiating the Exponential Function . . . . . . . . . . . . . . . . . . 2087C.2 The Baker-Campbell-Hausdorff Formula . . . . . . . . . . . . . . . . . . . 2087C.3 The Baker-Campbell-Hausdorff Series . . . . . . . . . . . . . . . . . . . . 2087C.4 Zassenhaus Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2091C.5 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2091C.6 Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2091

C.6.1 Paths in the Group Yield Paths in the Lie Algebra . . . . . . . . 2091C.6.2 Paths in the Lie Algebra Yield Paths in the Group . . . . . . . . 2091C.6.3 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 2091

D Canonical Transformations 2095

E Mathematica Notebooks 2097

F Analyticity, Aberration Expansions, and Smoothing 2099F.1 The Static Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2099F.2 The Time Dependent Case . . . . . . . . . . . . . . . . . . . . . . . . . . 2108F.3 Smoothing Properties of the Laplacian Kernel . . . . . . . . . . . . . . . 2110

G Invariant Scalar Products 2115

H Harmonic Functions 2117H.1 Representation of Gradients . . . . . . . . . . . . . . . . . . . . . . . . . 2117

H.1.1 Low-Order Results . . . . . . . . . . . . . . . . . . . . . . . . . . 2117H.1.2 Results to All Orders . . . . . . . . . . . . . . . . . . . . . . . . 2119

H.2 Range of Transverse Gradient Operators . . . . . . . . . . . . . . . . . . 2128H.2.1 Solution of x = . . . . . . . . . . . . . . . . . . . . . . . . . 2128H.2.2 Solution of y = . . . . . . . . . . . . . . . . . . . . . . . . . 2132

H.3 Harmonic Functions in Two Variables and Their Associated Fields . . . . 2135H.3.1 Harmonic Functions in x, z . . . . . . . . . . . . . . . . . . . . . 2136H.3.2 Harmonic Functions in y, z . . . . . . . . . . . . . . . . . . . . . 2139H.3.3 More About Bod(y, z) and Another Application of Analytic Func-

tion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2140

xx CONTENTS

I Poisson Bracket Relations 2145I.1 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2145I.2 Preparatory Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2147I.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2148

J Feigenbaum Cascade Denied/Achieved 2151J.1 Simple Map and Its Initial Bifurcations . . . . . . . . . . . . . . . . . . . 2151J.2 Complete Cascade Denied . . . . . . . . . . . . . . . . . . . . . . . . . . 2152J.3 Complete Cascade Achieved . . . . . . . . . . . . . . . . . . . . . . . . . 2154

K Supplement to Chapter 17 2159K.1 Computation of Generalized Gradients from Spinning Coil Data . . . . . 2159K.2 Computation of Generalized Gradients from Current Data . . . . . . . . 2161

L Spline Routines 2165

M Routines for Mathieu Separation Constants an(q) and bn(q) 2171

N Mathieu-Bessel Connection Coefficients 2179

O Quadratic Forms 2181O.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2181O.2 Effect of Small Perturbations in the Definite Case . . . . . . . . . . . . . 2182

P Parameterization of the Coset Space GL(2n,R)/Sp(2n,R) 2185P.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2185P.2 M Must Have Positive Determinant . . . . . . . . . . . . . . . . . . . . . 2185P.3 It is Sufficient to Consider SL(2n,R)/Sp(2n,R) . . . . . . . . . . . . . . 2186P.4 Some Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2186P.5 Connection between Symmetries and Being J-Symmetric . . . . . . . . . 2188P.6 Relation to Darboux Matrices . . . . . . . . . . . . . . . . . . . . . . . . 2189P.7 Some Observations on SL(2n,R)/Sp(2n,R) . . . . . . . . . . . . . . . . 2190P.8 Action of on s`(2n,R) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2190P.9 Lie Triple System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2191P.10 A Factorization Theorem (Theorem 1.1 of Goodman) . . . . . . . . . . . 2192

P.10.1 A Particular Mapping from Real Symmetric Matrices to Positive-Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 2192

P.10.2 The Map Is Real Analytic . . . . . . . . . . . . . . . . . . . . . . 2192P.10.3 Trace and Determinant Properties . . . . . . . . . . . . . . . . . 2193P.10.4 Study of the Inverse of the Map . . . . . . . . . . . . . . . . . . 2193P.10.5 Formula for Sa in terms of Z . . . . . . . . . . . . . . . . . . . . 2193P.10.6 Uniqueness of Solution for Sa . . . . . . . . . . . . . . . . . . . . 2194P.10.7 Verification of Expected Symmetry for Sa . . . . . . . . . . . . . 2195P.10.8 Formula for Sc in Terms of Z . . . . . . . . . . . . . . . . . . . . 2195P.10.9 Verification of Expected Symmetry for Sc . . . . . . . . . . . . . 2196P.10.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2196

CONTENTS xxi

P.10.11 Motivation for Mapping . . . . . . . . . . . . . . . . . . . . . . . 2196P.11 Theorem 1.2 of Goodman Due to Mostow . . . . . . . . . . . . . . . . . . 2197P.12 Goodmans Work on Symplectic Polar Decomposition . . . . . . . . . . . 2199

P.12.1 Some More Symmetry Operations . . . . . . . . . . . . . . . . . 2199P.12.2 Fixed-Point Subgroups Associated with Symmetry Operations . 2202

P.13 Decomposition of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . 2204P.14 Preparation for Lemma 2.1 of Goodman . . . . . . . . . . . . . . . . . . . 2208P.15 Lemma 2.1 of Goodman . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209P.16 Preparation for Theorem 2.1 of Goodman . . . . . . . . . . . . . . . . . . 2211P.17 Theorem 2.1 of Goodman . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212P.18 Search for Counter Examples . . . . . . . . . . . . . . . . . . . . . . . . . 2215

Q Improving Convergence of Fourier Representation 2219Q.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2219Q.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2221

R Abstract Lie Group Theory 2225

S Mathematica Realization of TPSAand Taylor Map Computation 2229S.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2229S.2 AD Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2230

S.2.1 Labeling Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 2230S.2.2 Implementation of Labeling Scheme . . . . . . . . . . . . . . . . 2233S.2.3 Pyramid Operations: General Procedure . . . . . . . . . . . . . . 2236S.2.4 Pyramid Operations: Scalar Multiplication and Addition . . . . 2236S.2.5 Pyramid Operations: Background for Polynomial Multiplication 2237S.2.6 Pyramid Operations: Implementation of Multiplication . . . . . 2240S.2.7 Pyramid Operations: Implementation of Powers . . . . . . . . . 2248S.2.8 Replacement Rule and Automatic Differentiation . . . . . . . . . 2248S.2.9 Taylor Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2251

S.3 Numerical Integration and Replacement Rule . . . . . . . . . . . . . . . . 2254S.3.1 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . 2254S.3.2 Replacement Rule, Single Equation/Variable Case . . . . . . . . 2255S.3.3 Multi Equation/Variable Case . . . . . . . . . . . . . . . . . . . 2258

S.4 Duffing Equation Application . . . . . . . . . . . . . . . . . . . . . . . . . 2261S.5 Relation to the Complete Variational Equations . . . . . . . . . . . . . . 2264S.6 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2267

T Quadrature and Cubature Formulas 2271T.1 Quadrature Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2271

T.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2271T.1.2 Newton Cotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2273T.1.3 Legendre Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274T.1.4 Clenshaw Curtis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2276

xxii CONTENTS

T.1.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2277

T.2 Cubature Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2281

T.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2281

T.2.2 Cubature on a Square . . . . . . . . . . . . . . . . . . . . . . . . 2282

T.2.3 Cubature on a Rectangle . . . . . . . . . . . . . . . . . . . . . . 2287

T.2.4 Cubature on the Two-Sphere . . . . . . . . . . . . . . . . . . . . 2291

U Rotational Classification and Properties of Polynomials andAnalytic/Polynomial Vector Fields 2295

U.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2295

U.2 Polynomials and Spherical Polynomials . . . . . . . . . . . . . . . . . . . 2295

U.2.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2295

U.2.2 Spherical Polar Coordinates and Harmonic Polynomials . . . . . 2296

U.2.3 Examples of Harmonic Polynomials and Missing HomogeneousPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2297

U.2.4 Spherical Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 2297

U.3 Analytic/Polynomial Vector Fields and Spherical Polynomial Vector Fields 2298

U.3.1 Vector Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . 2298

U.3.2 Spherical Polynomial Vector Fields . . . . . . . . . . . . . . . . . 2300

U.3.3 Examples of and Counting Spherical Polynomial Vector Fields . 2300

U.4 Independence/Orthogonality/Integral Properties of Spherical Polynomialsand Spherical Polynomial Vector Fields . . . . . . . . . . . . . . . . . . . 2304

U.5 Differential Properties of Spherical Polynomials and Spherical PolynomialVector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2304

U.5.1 Gradient Action on Spherical Polynomials . . . . . . . . . . . . . 2305

U.5.2 Divergence Action on Spherical Polynomial Vector Fields . . . . 2305

U.5.3 Curl Action on Spherical Polynomial Vector Fields . . . . . . . . 2306

U.6 Multiplicative Properties of Spherical Polynomials and Spherical Polyno-mial Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2307

U.6.1 Ordinary Multiplication . . . . . . . . . . . . . . . . . . . . . . . 2307

U.6.2 Dot Product Multiplication . . . . . . . . . . . . . . . . . . . . . 2308

U.6.3 Cross Product Multiplication . . . . . . . . . . . . . . . . . . . . 2308

V PROT in the Presence of a Magnetic Field 2315

V.1 The Constant Magnetic Field Case: Preliminaries . . . . . . . . . . . . . 2315

V.2 Dimensionless Variables and Limiting Hamiltonian . . . . . . . . . . . . . 2316

V.3 Design Trajectory and Deviation Variables . . . . . . . . . . . . . . . . . 2317

V.4 Deviation Variable Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 2318

V.5 Computation of Transfer Map . . . . . . . . . . . . . . . . . . . . . . . . 2319

V.6 The Inhomogeneous Field Case . . . . . . . . . . . . . . . . . . . . . . . . 2320

V.6.1 Vector Potential for Magnetic Monopole Doublet . . . . . . . . . 2320

V.6.2 Transition to Cylindrical Coordinates . . . . . . . . . . . . . . . 2321

V.6.3 Limiting Vector Potential . . . . . . . . . . . . . . . . . . . . . . 2322

V.6.4 Computation of Limiting Hamiltonian in Dimensionless Variables 2322

CONTENTS xxiii

V.6.5 Design Trajectory, Deviation Variables, and Deviation VariableHamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2323

V.6.6 Expansion of Deviation Variable Hamiltonian and Computationof Transfer Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 2323

List of Figures

1.1.1 In Dynamics the future can be determined by performing a certain op-eration, called a mapping M, on the present. . . . . . . . . . . . . . . . 4

1.2.1 The insect populations in successive years are related by a map M. . . 81.2.2 The values xm as a function m for the case = 2.8. . . . . . . . . . . . 101.2.3 The values xm as a function m for the case = 3.01. . . . . . . . . . . 101.2.4 Feigenbaum diagram showing limiting values x as a function of for

the logistic map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.5 An enlargement of Figure 2.4 exhibiting how sucessive bifurcations scale. 141.2.6 Douadys rabbit, the dynamic aperture in the mapping plane z for the

case = 2.55268 0.959456i. . . . . . . . . . . . . . . . . . . . . . . . 171.2.7 The Mandelbrot set M in the control plane . . . . . . . . . . . . . . . 171.2.8 Douadys rabbit in color. The white points lie in the basin of under

the action of M. The origin is a repelling fixed point of M. The otherrepelling fixed point has the location zf = .656747 .129015i. Under theaction of M3, red points lie in the basin of z1, green points lie in thebasin of z2, and yellow points lie in the basin of z3. . . . . . . . . . . . 19

1.2.9 Schematic representation of the map (2.50). . . . . . . . . . . . . . . . 211.2.10 The dynamic aperture of the Henon map for the case /2pi = 0.22. . . . 221.2.11 Stereographic view of the dynamic aperture of the Henon map as a func-

tion of the parameter . The region shown is q [.8, .8], p [.7.7],/2pi [0, .5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2.12 The Mandelbrot set in the plane. The plate has been somewhatoverexposed compared to Figure 2.7 to bring out the island chains. . 35

1.2.13 The analog of Figure 2.4 for real and the variable w. . . . . . . . . . 351.3.1 An illustration of Theorem 3.1 in the case that y space is two dimen-

sional. The solution y exists, is unique, and is continuous in t as long asit remains within the large cylinder of base R where f is continuous andthe f/yj are continuous. If the point y

0 is varied slightly, the solutionalso changes only slightly so that nearby solutions form a bundle. . . . 44

1.4.1 The transfer mapM sends the initial conditions yi to the final conditionsyf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.5.1 Illustration of the , y, cylindrical coordinate system and a sample unit-vector pair e and e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

1.6.1 Typical choice of a Cartesian coordinate system for the description ofcharged-particle trajectories in a magnet. . . . . . . . . . . . . . . . . . 68

xxv

xxvi LIST OF FIGURES

1.6.2 Top view of a particle trajectory in a rectangular magnet. . . . . . . . . 68

1.6.3 Top view of a particle trajectory in a wedge magnet. The trajectoryis conveniently described using the cylindrical coordinates , y, . SeeFigure 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.1.1 The Time Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.3.1 The result of integrating with RK3 the set (2.7) through (2.9) to t = 1.5with several different step sizes to illustrate how the cumulative errordepends on h. The error is measured by y(1.5) ye(1.5) where yeis the exact solution. The dashed line on the right has a slope of +3showing that the global truncation error at first decreases as h3. Thedashed line on the left has a slope of 1 showing that in this examplethe global round-off error increases as the number of steps N . Thesecalculations were made on a computer that had an accuracy of about 81/2 significant figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.4.1 Possible cases for the eigenvalues of a 2 2 real symplectic matrix. . . . 2163.4.2 Possible eigenvalue configurations for a 44 real symplectic matrix. The

mirror image of each configuration is also a possible configuration, andtherefore is not shown in order to save space. Various authors have giventhese configurations various names. Notably, Case 1 is commonly calleda Krein quartet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

3.4.3 Eigenvalues of a 22 real symplectic matrix M as a function of A = tr (M). 2203.4.4 Eigenvalues of a 4 4 real symplectic matrix M as a function of the

coefficients A and B in its characteristic polynomial. . . . . . . . . . . . 222

3.5.1 Illustration of eigenvalues colliding and then leaving the unit circle toform what is called a Krein quartet. . . . . . . . . . . . . . . . . . . . . 230

4.3.1 Schematic depiction of matrix space showing the zero matrix, the identitymatrix I, the ray N(M), and the unit ball around the identity matrix. 372

4.4.1 The matrices R and M are connected by a path that is both an affinegeodesic and is perpendicular to the subspace of symplectic matrices atthe point R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

5.4.1 a) The summation points in m,n space for the sum (4.7) indicating thatthe inner sum is over m followed by a sum over n. b) The summationpoints for the sum (4.8) illustrating that the points are the same, but theinner sum is now over n followed by a sum over m. . . . . . . . . . . . 436

5.8.1 Root diagram showing the root vectors for su(3). . . . . . . . . . . . . 455

5.8.2 Fundamental weights 1 and 2 for su(3). The root vectors are also shown. 457

5.8.3 Weight diagram for the representation 1 = (0, 0). . . . . . . . . . . . . 458





LIST OF FIGURES xxvii

5.8.8 Weight diagram for the adjoint representation 8 = (1, 1). The 6 weightsat the hexagonal vertices lie at the tips of the root vectors , , shown in Figure 8.1. The highest weight lies at the tip of the vector .There are two eigenvectors corresponding to the weight at the origin. . 461

5.8.9 General form of the weight diagram for the representation (m,n). Shownhere is the case (m,n) = (7, 3). All eigenvectors |w corresponding toweights w on a given layer have the same multiplicity. Those correspond-ing to boundary sites have multiplicity 1. Those corresponding to siteson the next two layers have multiplicities 2 and 3, respectively. Thosecorresponding to sites on the two triangular layers have multiplicity 4. . 462

5.9.1 Stability diagram for Sp(2, R) showing the quantity rmax as a function of0. All elements with r < rmax are stable, and all elements with r > rmaxare unstable. In accord with toroidal topology, corresponding points atthe top and bottom of the figure (0 = pi) are to be identified. . . . . 480

6.1.1 The map M sends z to z(z, t). . . . . . . . . . . . . . . . . . . . . . . . 5466.1.2 The action of a symplectic mapM on phase space. The general point z0

is mapped to the point z0, and the small vectors dz and z are mapped tothe small vectors dz and z. The figure is only schematic since in generalphase space has a large number of dimensions. . . . . . . . . . . . . . . 548

6.4.1 A trajectory in augmented phase space. Under the Hamiltonian flowspecified by a Hamiltonian H, the general phase-space point zi is mappedinto the phase-space point zf . The mapping M is symplectic for anyHamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

6.4.2 The symplectic map family N (t) in augmented symplectic map space. . 5716.5.1 A trajectory of H(, t) in the augmented , phase space having initial

coordinates q and final momenta P . . . . . . . . . . . . . . . . . . . . . 591

6.7.1 A trajectory of H(, ) in the augmented = (, ) phase space. Given aDarboux matrix , an initial time ti, a final time t, and the 2n-vector u,the initial condition (ti) = z is to be selected such that CZ +Dz = uwhere (t) = Z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616

6.8.1 The domain 2 in , space. Also shown is its subdivision into rectanglesof sides d, d and its boundary 1. . . . . . . . . . . . . . . . . . . . . 631

6.8.2 The closed paths Ci and Cf in augmented phase space and the trajecto-ries that join them. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

6.8.3 The t, parameter space. The left and right boundaries are the curvesti() and tf (), and their augmented phase-space images are the paths Ci

and Cf . Also shown as dashed lines are pairs of parameter-space pathstraversed in opposite directions whose images are augmented phase-spacetrajectories traversed in opposite directions. Note that the lines = 0and = 1 have the same image in augmented phase space. . . . . . . . 638

6.8.4 Two adjacent loops in parameter space. . . . . . . . . . . . . . . . . . . 639

xxviii LIST OF FIGURES

6.8.5 The loops in augmented phase space corresponding to the two parameter-space loops of Figure 8.4. Note that the long sides of the loops aretrajectories for the Hamiltonian H, and the short sides are pieces of Ci

and Cf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

6.8.6 The integral over a loop is the sum of integrals over top and bottom halves. 639

6.8.7 The integral over a half loop is the integral over a trajectory of H or itsreverse plus the change in the integral resulting from deforming this path. 640

6.8.8 Initial phase-space distribution for Exercise 8.1. . . . . . . . . . . . . . 642

6.9.1 Two surfaces of section in augmented phase space. Trajectories leavingSg are assumed to eventually enter and cross Sh, perhaps at different times. 644

8.3.1 The composite action of two maps Mf and Mg. . . . . . . . . . . . . . 7278.3.2 Successive passage of a trajectory with initial condition z through beam

line elements f and g resulting in the intermediate condition z and final

condition=z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

8.6.1 Various possibilities for the representation of maps in the operation ofconcatenation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748

8.6.2 Product of a map in Lie form with a map in Taylor form. . . . . . . . . 750

9.4.1 Concatenation of origin-preserving maps in an enlarged phase space tofind equivalent results for maps, including translations, in the originalphase space. The concatenator depicted at the top of the figure workswith the usual phase space. When translations are taken into account,it involves the use of complicated feed-down formulae as illustrated inSection 9.3. The concatenator at the bottom of the figure works in anenlarged phase space, and employs the far-simpler concatenation rulesfor origin preserving maps. . . . . . . . . . . . . . . . . . . . . . . . . . 815

9.4.2 A recursive step that takes a map r1Nh and a pair of indices `(r) andm(r) as input, and produces a map rNh and polynomial hm` as output. 823

15.8.1 A monopole doublet consisting of two magnetic monopoles of equal andopposite sign placed on the y axis and centered on the origin. Also shown,for future reference, is a cylinder with circular cross section placed in theinterior field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134

15.8.2 The interior field of a monopole doublet in the z = 0 plane. Also shownis an ellipse whose purpose will become clear in Sections 17.4 and 19.2. 1135

15.8.3 The on-axis field component By(x = 0, y = 0, z) for the monopole doubletin the case that a = 2.5 cm and g = 1 Tesla-(cm)2. The coordinate z isgiven in centimeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135

15.8.4 The field component Bx on the line = 1/2 cm, = pi/4, z [,]for the monopole doublet in the case that a = 2.5 cm and g = 1 Tesla-(cm)2. In Cartesian coordinates, this is the line x = y ' .353 cm,z [,]. The coordinate z is given in centimeters. . . . . . . . . . 1136

LIST OF FIGURES xxix

15.8.5 The field componentBz on the line = 1/2 cm, = pi/4, z [,] forthe monopole doublet in the case that a = 2.5 cm and g = 1 Tesla-(cm)2.In Cartesian coordinates, this is the line x = y ' .353 cm, z [,].The coordinate z is given in centimeters. . . . . . . . . . . . . . . . . . 1137

15.8.6 The quantity B(R, , z = 0) for the monopole doublet in the case thatR = 2 cm, a = 2.5 cm, and g = 1 Tesla-(cm)2. . . . . . . . . . . . . . . 1137

15.8.7 The quantity B(R, = pi/2, z) for the monopole doublet in the case thatR = 2 cm, a = 2.5 cm, and g = 1 Tesla-(cm)2. The coordinate z is givenin centimeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138

15.8.8 The on-axis gradient function C[0]1,s for the monopole doublet in the case

that a = 2.5 cm and g = 1 Tesla-(cm)2. . . . . . . . . . . . . . . . . . . 1141


that a = 2.5 cm and g = 1 Tesla-(cm)2. . . . . . . . . . . . . . . . . . . 114115.8.10 An enlargement of a portion of Figure 8.9 showing a zero hidden in a tail.114215.8.11 The on-axis gradient function C

[0]3,s for the monopole doublet in the case







that a = 2.5 cm and g =

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