Richard Talman

Geometric Mechanics

Toward a Unification of Classical Physics

Second, Revised and Enlarged Edition

WILEY-VCH Verlag GmbH & Co. KGaA

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Richard Talman Geometric Mechanics

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Richard Talman

Geometric Mechanics

Toward a Unification of Classical Physics

Second, Revised and Enlarged Edition

WILEY-VCH Verlag GmbH & Co. KGaA

The Author Prof. Richard Talman Cornell University Laboratory of Elementary Physics Ithaca, NY 14853 USA

talman@mail.lns.cornell.edu

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V

Contents

Preface XV

Introduction 1Bibliography 9

1 Review of Classical Mechanics and String Field Theory 111.1 Preview and Rationale 111.2 Review of Lagrangians and Hamiltonians 131.2.1 Hamilton’s Equations in Multiple Dimensions 141.3 Derivation of the Lagrange Equation from Hamilton’s Principle 161.4 Linear, Multiparticle Systems 181.4.1 The Laplace Transform Method 231.4.2 Damped and Driven Simple Harmonic Motion 241.4.3 Conservation of Momentum and Energy 261.5 Effective Potential and the Kepler Problem 261.6 Multiparticle Systems 291.7 Longitudinal Oscillation of a Beaded String 321.7.1 Monofrequency Excitation 331.7.2 The Continuum Limit 341.8 Field Theoretical Treatment and Lagrangian Density 361.9 Hamiltonian Density for Transverse String Motion 391.10 String Motion Expressed as Propagating and Reflecting Waves 401.11 Problems 42

Bibliography 44

2 Geometry of Mechanics, I, Linear 452.1 Pairs of Planes as Covariant Vectors 472.2 Differential Forms 532.2.1 Geometric Interpretation 532.2.2 Calculus of Differential Forms 572.2.3 Familiar Physics Equations Expressed Using Differential Forms 61

VI Contents

2.3 Algebraic Tensors 662.3.1 Vectors and Their Duals 662.3.2 Transformation of Coordinates 682.3.3 Transformation of Distributions 722.3.4 Multi-index Tensors and their Contraction 732.3.5 Representation of a Vector as a Differential Operator 762.4 (Possibly Complex) Cartesian Vectors in Metric Geometry 792.4.1 Euclidean Vectors 792.4.2 Skew Coordinate Frames 812.4.3 Reduction of a Quadratic Form to a Sum or Difference of

Squares 812.4.4 Introduction of Covariant Components 832.4.5 The Reciprocal Basis 84

Bibliography 86

3 Geometry of Mechanics, II, Curvilinear 893.1 (Real) Curvilinear Coordinates in n-Dimensions 903.1.1 The Metric Tensor 903.1.2 Relating Coordinate Systems at Different Points in Space 923.1.3 The Covariant (or Absolute) Differential 973.2 Derivation of the Lagrange Equations from the Absolute

Differential 1023.2.1 Practical Evaluation of the Christoffel Symbols 1083.3 Intrinsic Derivatives and the Bilinear Covariant 1093.4 The Lie Derivative – Coordinate Approach 1113.4.1 Lie-Dragged Coordinate Systems 1113.4.2 Lie Derivatives of Scalars and Vectors 1153.5 The Lie Derivative – Lie Algebraic Approach 1203.5.1 Exponential Representation of Parameterized Curves 1203.6 Identification of Vector Fields with Differential Operators 1213.6.1 Loop Defect 1223.7 Coordinate Congruences 1233.8 Lie-Dragged Congruences and the Lie Derivative 1253.9 Commutators of Quasi-Basis-Vectors 130

Bibliography 132

4 Geometry of Mechanics, III, Multilinear 1334.1 Generalized Euclidean Rotations and Reflections 1334.1.1 Reflections 1344.1.2 Expressing a Rotation as a Product of Reflections 1354.1.3 The Lie Group of Rotations 1364.2 Multivectors 138

Contents VII

4.2.1 Volume Determined by 3- and by n-Vectors 1384.2.2 Bivectors 1404.2.3 Multivectors and Generalization to Higher Dimensionality 1414.2.4 Local Radius of Curvature of a Particle Orbit 1434.2.5 “Supplementary” Multivectors 1444.2.6 Sums of p-Vectors 1454.2.7 Bivectors and Infinitesimal Rotations 1454.3 Curvilinear Coordinates in Euclidean Geometry (Continued) 1484.3.1 Repeated Exterior Derivatives 1484.3.2 The Gradient Formula of Vector Analysis 1494.3.3 Vector Calculus Expressed by Differential Forms 1514.3.4 Derivation of Vector Integral Formulas 1544.3.5 Generalized Divergence and Gauss’s Theorem 1574.3.6 Metric-Free Definition of the “Divergence” of a Vector 1594.4 Spinors in Three-Dimensional Space 1614.4.1 Definition of Spinors 1624.4.2 Demonstration that a Spinor is a Euclidean Tensor 1624.4.3 Associating a 2 × 2 Reflection (Rotation) Matrix with a Vector

(Bivector) 1634.4.4 Associating a Matrix with a Trivector (Triple Product) 1644.4.5 Representations of Reflections 1644.4.6 Representations of Rotations 1654.4.7 Operations on Spinors 1664.4.8 Real Euclidean Space 1674.4.9 Real Pseudo-Euclidean Space 167

Bibliography 167

5 Lagrange–Poincaré Description of Mechanics 1695.1 The Poincaré Equation 1695.1.1 Some Features of the Poincaré Equations 1795.1.2 Invariance of the Poincaré Equation 1805.1.3 Translation into the Language of Forms and Vector Fields 1825.1.4 Example: Free Motion of a Rigid Body with One Point Fixed 1835.2 Variational Derivation of the Poincaré Equation 1865.3 Restricting the Poincaré Equation With Group Theory 1895.3.1 Continuous Transformation Groups 1895.3.2 Use of Infinitesimal Group Parameters as Quasicoordinates 1935.3.3 Infinitesimal Group Operators 1955.3.4 Commutation Relations and Structure Constants of the Group 1995.3.5 Qualitative Aspects of Infinitesimal Generators 2015.3.6 The Poincaré Equation in Terms of Group Generators 2045.3.7 The Rigid Body Subject to Force and Torque 206

Bibliography 217

VIII Contents

6 Newtonian/Gauge Invariant Mechanics 2196.1 Vector Mechanics 2196.1.1 Vector Description in Curvilinear Coordinates 2196.1.2 The Frenet–Serret Formulas 2226.1.3 Vector Description in an Accelerating Coordinate Frame 2266.1.4 Exploiting the Fictitious Force Description 2326.2 Single Particle Equations in Gauge Invariant Form 2386.2.1 Newton’s Force Equation in Gauge Invariant Form 2396.2.2 Active Interpretation of the Transformations 2426.2.3 Newton’s Torque Equation 2466.2.4 The Plumb Bob 2486.3 Gauge Invariant Description of Rigid Body Motion 2526.3.1 Space and Body Frames of Reference 2536.3.2 Review of the Association of 2 × 2 Matrices to Vectors 2566.3.3 “Association” of 3 × 3 Matrices to Vectors 2586.3.4 Derivation of the Rigid Body Equations 2596.3.5 The Euler Equations for a Rigid Body 2616.4 The Foucault Pendulum 2626.4.1 Fictitious Force Solution 2636.4.2 Gauge Invariant Solution 2656.4.3 “Parallel” Translation of Coordinate Axes 2706.5 Tumblers and Divers 274

Bibliography 276

7 Hamiltonian Treatment of Geometric Optics 2777.1 Analogy Between Mechanics and Geometric Optics 2787.1.1 Scalar Wave Equation 2797.1.2 The Eikonal Equation 2817.1.3 Determination of Rays from Wavefronts 2827.1.4 The Ray Equation in Geometric Optics 2837.2 Variational Principles 2857.2.1 The Lagrange Integral Invariant and Snell’s Law 2857.2.2 The Principle of Least Time 2877.3 Paraxial Optics, Gaussian Optics, Matrix Optics 2887.4 Huygens’ Principle 292

Bibliography 294

8 Hamilton–Jacobi Theory 2958.1 Hamilton–Jacobi Theory Derived from Hamilton’s Principle 2958.1.1 The Geometric Picture 2978.1.2 Constant S Wavefronts 2988.2 Trajectory Determination Using the Hamilton–Jacobi Equation 299

Contents IX

8.2.1 Complete Integral 2998.2.2 Finding a Complete Integral by Separation of Variables 3008.2.3 Hamilton–Jacobi Analysis of Projectile Motion 3018.2.4 The Jacobi Method for Exploiting a Complete Integral 3028.2.5 Completion of Projectile Example 3048.2.6 The Time-Independent Hamilton–Jacobi Equation 3058.2.7 Hamilton–Jacobi Treatment of 1D Simple Harmonic Motion 3068.3 The Kepler Problem 3078.3.1 Coordinate Frames 3088.3.2 Orbit Elements 3098.3.3 Hamilton–Jacobi Formulation. 3108.4 Analogies Between Optics and Quantum Mechanics 3148.4.1 Classical Limit of the Schrödinger Equation 314

Bibliography 316

9 Relativistic Mechanics 3179.1 Relativistic Kinematics 3179.1.1 Form Invariance 3179.1.2 World Points and Intervals 3189.1.3 Proper Time 3199.1.4 The Lorentz Transformation 3219.1.5 Transformation of Velocities 3229.1.6 4-Vectors and Tensors 3229.1.7 Three-Index Antisymmetric Tensor 3259.1.8 Antisymmetric 4-Tensors 3259.1.9 The 4-Gradient, 4-Velocity, and 4-Acceleration 3269.2 Relativistic Mechanics 3279.2.1 The Relativistic Principle of Least Action 3279.2.2 Energy and Momentum 3289.2.3 4-Vector Notation 3299.2.4 Forced Motion 3299.2.5 Hamilton–Jacobi Formulation 3309.3 Introduction of Electromagnetic Forces into Relativistic

Mechanics 3329.3.1 Generalization of the Action 3329.3.2 Derivation of the Lorentz Force Law 3349.3.3 Gauge Invariance 335

Bibliography 338

10 Conservation Laws and Symmetry 33910.1 Conservation of Linear Momentum 33910.2 Rate of Change of Angular Momentum: Poincaré Approach 341

X Contents

10.3 Conservation of Angular Momentum: Lagrangian Approach 34210.4 Conservation of Energy 34310.5 Cyclic Coordinates and Routhian Reduction 34410.5.1 Integrability; Generalization of Cyclic Variables 34710.6 Noether’s Theorem 34810.7 Conservation Laws in Field Theory 35210.7.1 Ignorable Coordinates and the Energy Momentum Tensor 35210.8 Transition From Discrete to Continuous Representation 35610.8.1 The 4-Current Density and Charge Conservation 35610.8.2 Energy and Momentum Densities 36010.9 Angular Momentum of a System of Particles 36210.10 Angular Momentum of a Field 363

Bibliography 364

11 Electromagnetic Theory 36511.1 The Electromagnetic Field Tensor 36711.1.1 The Lorentz Force Equation in Tensor Notation 36711.1.2 Lorentz Transformation and Invariants of the Fields 36911.2 The Electromagnetic Field Equations 37011.2.1 The Homogeneous Pair of Maxwell Equations 37011.2.2 The Action for the Field, Particle System 37011.2.3 The Electromagnetic Wave Equation 37211.2.4 The Inhomogeneous Pair of Maxwell Equations 37311.2.5 Energy Density, Energy Flux, and the Maxwell Stress Energy

Tensor 374Bibliography 377

12 Relativistic Strings 37912.1 Introduction 37912.1.1 Is String Theory Appropriate? 37912.1.2 Parameterization Invariance 38112.1.3 Postulating a String Lagrangian 38112.2 Area Representation in Terms of the Metric 38312.3 The Lagrangian Density and Action for Strings 38412.3.1 A Revised Metric 38412.3.2 Parameterization of String World Surface by σ and τ 38512.3.3 The Nambu–Goto Action 38512.3.4 String Tension and Mass Density 38712.4 Equations of Motion, Boundary Conditions, and Unexcited

Strings 38912.5 The Action in Terms of Transverse Velocity 39112.6 Orthogonal Parameterization by Energy Content 394

Contents XI

12.7 General Motion of a Free Open String 39612.8 A Rotating Straight String 39812.9 Conserved Momenta of a String 40012.9.1 Angular Momentum of Uniformly Rotating Straight String 40112.10 Light Cone Coordinates 40212.11 Oscillation Modes of a Relativistic String 406

Bibliography 408

13 General Relativity 40913.1 Introduction 40913.2 Transformation to Locally Inertial Coordinates 41213.3 Parallel Transport on a Surface 41313.3.1 Geodesic Curves 41613.4 The Twin Paradox in General Relativity 41713.5 The Curvature Tensor 42213.5.1 Properties of Curvature Tensor, Ricci Tensor, and Scalar

Curvature 42313.6 The Lagrangian of General Relativity and the Energy–Momentum

Tensor 42513.7 “Derivation” of the Einstein Equation 42813.8 Weak, Nonrelativistic Gravity 43013.9 The Schwarzschild Metric 43313.9.1 Orbit of a Particle Subject to the Schwarzschild Metric 43413.10 Gravitational Lensing and Red Shifts 437

Bibliography 440

14 Analytic Bases for Approximation 44114.1 Canonical Transformations 44114.1.1 The Action as a Generator of Canonical Transformations 44114.2 Time-Independent Canonical Transformation 44614.3 Action-Angle Variables 44814.3.1 The Action Variable of a Simple Harmonic Oscillator 44814.3.2 Adiabatic Invariance of the Action I 44914.3.3 Action/Angle Conjugate Variables 45314.3.4 Parametrically Driven Simple Harmonic Motion 45514.4 Examples of Adiabatic Invariance 45714.4.1 Variable Length Pendulum 45714.4.2 Charged Particle in Magnetic Field 45914.4.3 Charged Particle in a Magnetic Trap 46114.5 Accuracy of Conservation of Adiabatic Invariants 46614.6 Conditionally Periodic Motion 46914.6.1 Stäckel’s Theorem 470

XII Contents

14.6.2 Angle Variables 47114.6.3 Action/Angle Coordinates for Keplerian Satellites 474

Bibliography 475

15 Linear Hamiltonian Systems 47715.1 Linear Hamiltonian Systems 47715.1.1 Inhomogeneous Equations 47915.1.2 Exponentiation, Diagonalization, and Logarithm Formation of

Matrices 47915.1.3 Alternate Coordinate Ordering 48115.1.4 Eigensolutions 48115.2 Periodic Linear Systems 48415.2.1 Floquet’s Theorem 48515.2.2 Lyapunov’s Theorem 48715.2.3 Characteristic Multipliers, Characteristic Exponents 48715.2.4 The Variational Equations 489

Bibliography 490

16 Perturbation Theory 49116.1 The Lagrange Planetary Equations 49216.1.1 Derivation of the Equations 49216.1.2 Relation Between Lagrange and Poisson Brackets 49616.2 Advance of Perihelion of Mercury 49716.3 Iterative Analysis of Anharmonic Oscillations 50216.4 The Method of Krylov and Bogoliubov 50816.4.1 First Approximation 50816.4.2 Equivalent Linearization 51216.4.3 Power Balance, Harmonic Balance 51416.4.4 Qualitative Analysis of Autonomous Oscillators 51516.4.5 Higher K–B Approximation 51816.5 Superconvergent Perturbation Theory 52316.5.1 Canonical Perturbation Theory 52316.5.2 Application to Gravity Pendulum 52516.5.3 Superconvergence 527

Bibliography 527

17 Symplectic Mechanics 52917.1 The Symplectic Properties of Phase Space 53017.1.1 The Canonical Momentum 1-Form 53017.1.2 The Symplectic 2-Form ω̃ωω 53317.1.3 Invariance of the Symplectic 2-Form 53717.1.4 Use of ω̃ωω to Associate Vectors and 1-Forms 538

Contents XIII

17.1.5 Explicit Evaluation of Some Inner Products 539

17.1.6 The Vector Field Associated with d̃H 54017.1.7 Hamilton’s Equations in Matrix Form 54117.2 Symplectic Geometry 54317.2.1 Symplectic Products and Symplectic Bases 54317.2.2 Symplectic Transformations 54517.2.3 Properties of Symplectic Matrices 54617.3 Poisson Brackets of Scalar Functions 55417.3.1 The Poisson Bracket of Two Scalar Functions 55417.3.2 Properties of Poisson Brackets 55517.3.3 The Poisson Bracket and Quantum Mechanics 55517.4 Integral Invariants 55717.4.1 Integral Invariants in Electricity and Magnetism 55717.4.2 The Poincaré–Cartan Integral Invariant 56017.5 Invariance of the Poincaré–Cartan Integral Invariant I.I. 56217.5.1 The Extended Phase Space 2-Form and its Special Eigenvector 56317.5.2 Proof of Invariance of the Poincaré Relative Integral Invariant 56517.6 Symplectic System Evolution 56617.6.1 Liouville’s Theorem and Generalizations 568

Bibliography 570

Index 571

XV

Preface

This text is designed to accompany a junior/senior or beginning graduatestudent course in mechanics for students who have already encountered La-grange’s equations. As the title Geometric Mechanics indicates, the content isclassical mechanics, with emphasis on geometric methods, such as differentialgeometry, tensor analysis, and group theory. Courses for which the materialin the text has been used and is appropriate are discussed in the Introduc-tion. To reflect a substantial new emphasis in this second edition, comparedto the first, the subtitle “Toward a Unification of Classical Physics” has beenadded. Instead of just laying the groundwork for follow-on, geometry-based,physics subjects, especially general relativity and string theory, this editioncontains substantial introductions to both of those topics. To support this,introductory material on classical field theory, including electrodynamic the-ory (also formulated as mechanics) has been included. The purpose of these“other physics” chapters is to show how, based on Hamilton’s principle ofleast action, all, or at least most, of classical physics is naturally subsumedinto classical mechanics.

Communications pointing out errors, or making comments or suggestionswill be appreciated; E-mail address; talman@mail.lepp.cornell.edu. Becauseof its complete reorganization, there are undoubtedly more minor errors anddangling references than might be expected for a second edition.

The institutions contributing (in equal parts) to this text have been the pub-lic schools of London, Ontario, and universities U.W.O., Caltech, and Cornell.I have profited, initially as a student, and later from my students, at these insti-tutions, and from my colleagues there and at accelerator laboratories world-wide. I have also been fortunate of family; parents, brother, children, and,especially my wife, Myrna.

Ithaca, New YorkMay, 2007

Richard Talman

1

Introduction

The first edition of this text was envisaged as a kind of Mathematical Methodsof Classical Mechanics for Pedestrians, with geometry playing a more importantrole than in the traditional pedagogy of classical mechanics. Part of the ra-tionale was to prepare the student for subsequent geometry-intensive physicssubjects, especially general relativity. Subsequently I have found that, as atext for physics courses, this emphasis was somewhat misplaced. (Almost bydefinition) students of physics want to learn “physics” more than they wantto learn “applied mathematics.” Consistent with this, there has been a ten-dency for classical mechanics to be squeezed out of physics curricula in favorof general relativity or, more recently, string theory. This second edition hasbeen revised accordingly. Instead of just laying the groundwork for subjectssuch as electromagnetic theory, string theory, and general relativity, it sub-sumes these subjects into classical mechanics. After these changes, the texthas become more nearly a Classical Theory of Fields for Pedestrians.

Geometric approaches have contributed importantly to the evolution ofmodern physical theories. The best example is general relativity; the mostmodern example is string theory. In fact general relativity and string theoryare the theories for which the adjective “geometric” is most unambiguouslyappropriate. There is now a chapter on each of these subjects in this text,along with material on (classical) field theory basic to these subjects. Also,because electromagnetic theory fits the same template, and is familiar to moststudents, that subject is here also formulated as a “branch” of classical me-chanics.

In grandiose terms, the plan of the text is to arrogate to classical mechanicsall of classical physics, where “classical” means nonquantum-mechanical and“all” means old-fashioned classical mechanics plus the three physical theoriesmentioned previously. Other classical theories, such as elasticity and hydro-dynamics, can be regarded as having already been subsumed into classicalmechanics, but they lie outside the scope of this text.

In more technical terms, the theme of the text is that all of classical physicsstarts from a Lagrangian, continues with Hamilton’s principle (also known

2 Introduction

as the principle of least action) and finishes with solving the resultant equa-tions and comparison with experiment. This program provides a unificationof classical physics. General principles, especially symmetry and special rela-tivity, limit the choices surprisingly when any new term is to be added to theLagrangian. Once a new term has been added the entire theory and its predic-tions are predetermined. These results can then be checked experimentally.The track record for success in this program has been astonishingly good. Asfar as classical physics is concerned the greatest triumphs are due to Maxwelland Einstein. The philosophic basis for this approach, apparently espousedby Einstein, is not that we live in the best of all possible worlds, but that welive in the only possible world. Even people who find this philosophy sillyfind that you don’t have to subscribe to this philosophy for the approach towork well.

There is an ambitious program in quantum field theory called “grand unifi-cation” of the four fundamental forces of physics. The present text can be re-garded as preparation for this program in that it describes classical physics inways consistent with this eventual approach. As far as I know, any imaginedgrand unification scheme will, when reduced to the classical level, resemblethe material presented here. (Of course most of the essence of the physics isquantum mechanical and cannot survive the reduction to classical physics.)

Converting the emphasis from applied mathematics to pure physics re-quired fewer changes to the text than might be supposed. Much of the ear-lier book emphasized specialized mathematics and computational descrip-tions that could be removed to make room for the “physics” chapters alreadymentioned. By no means does this mean that the text has been gutted of prac-tical worked examples of classical mechanics. For example, most of the longchapters on perturbation theory and on the application of adiabatic invariants(both of which are better thought of as physics than as mathematics) havebeen retained. All of the (admittedly unenthusiastic) discussion of canonicaltransformation methods has also been retained.

Regrettably, some material on the boundary between classical and quan-tum mechanics has had to be dropped. As well as helping to keep the booklength within bounds, this deletion was consistent with religiously restrictingthe subject matter to nothing but classical physics. There was a time when clas-sical Hamiltonian mechanics seemed like the best introduction to quantummechanics but, like the need to study Latin in school, that no longer seems tobe the case. Also, apart from its connections to the Hamilton–Jacobi theory(which every educated physicist has to understand) quantum mechanics isnot very geometric in character. It was relatively painless therefore, to removeunitary geometry, Bragg scattering (illustrating the use of covariant tensors),and other material on the margin between classical and quantum mechanics.

Introduction 3

In this book’s first manifestation the subject of mechanics was usefully, ifsomewhat artificially, segmented into Lagrangian, Hamiltonian, and Newto-nian formulations. Much was made of Poincaré’s extension to the Lagrangianapproach. Because this approach advances the math more than the physics, itnow has had to be de-emphasized (though most of the material remains). Onthe other hand, as mentioned already, the coverage of Lagrangian field theory,and especially its conservation laws, needed to be expanded. Reduced weightalso had to be assigned to Hamiltonian methods (not counting Hamilton’sprinciple.) Those methods provide the most direct connections to quantummechanics but, with quantum considerations now being ignored, they are lessessential to the program. Opposite comments apply to Newtonian methods,which stress fictitious forces (centrifugal and Coriolis), ideas that led naturallyto general relativity. Gauge invariant methods, which play such an importantrole in string theory, are also naturally introduced in the context of direct New-tonian methods. The comments in this paragraph, taken together, repudiatemuch of the preface to the first edition which has, therefore, been discarded.

Everything contained in this book is explained with more rigor, or moredepth, or more detail, or (especially) more sophistication, in at least one of thebooks listed at the end of this introduction. Were it not for the fact that mostof those books are fat, intimidating, abstract, formal, mathematical and (formany) unintelligible, the reader’s time would be better spent reading them(in the right order) than studying this book. But if this text renders books likethese both accessible and admirable, it will have achieved its main purpose. Ithas been said that bridge is a simple game; dealt thirteen cards, one has onlyto play them in the correct order. In the same sense mechanics is easy to learn;one simply has to study readily available books in a sensible order. I have triedto chart such a path, extracting material from various sources in an order that Ihave found appropriate. At each stage I indicate (at the end of the chapter) thereference my approach most closely resembles. In some cases what I provideis a kind of Reader’s Digest of a more general treatment and this may amountto my having systematically specialized and made concrete, descriptions thatthe original author may earlier have systematically labored to generalize andmake abstract. The texts to which these statements are most applicable arelisted at the end of each chapter, and keyed to the particular section to whichthey relate. It is not suggested that these texts should be systematically re-ferred to as they tend to be advanced and contain much unrelated material.But if particular material in this text is obscure, or seems to stop short of somedesirable goal, these texts should provide authoritative help.

Not very much is original in the text other than the selection and arrange-ment of the topics and the style of presentation. Equations (though not text)have been “borrowed,” in some cases verbatim, from various sources. Thisis especially true of Chapters 9, on special relativity, 11, on electromagnetic

4 Introduction

theory, and 13, on general relativity. These chapters follow Landau and Lif-schitz quite closely. Similarly, Chapter 12 follows Zwiebach closely. Thereare also substantial sections following Cartan, or Arnold, or others. As wellas occasional reminders in the text, of these sources, the bibliography at theend of each chapter lists the essential sources. Under “General References”are books, like the two just mentioned, that contain one or more chapters dis-cussing much the same material in at least as much, and usually far more,detail, than is included here. These references could be used instead of thematerial in the chapter they are attached to, and should be used to go deeperinto the subject. Under “References for Further Study” are sources that canbe used as well as the material of the chapter. In principle, none of these ref-erences should actually be necessary, as the present text is supposed to beself-sufficient. In practice, obscurities and the likelihood of errors or misun-derstandings, make it appropriate, or even necessary, to refer to other sourcesto obtain anything resembling a deep understanding of a topic.

The mathematical level strived for is only high enough to support a persua-sive (to a nonmathematician) trip through the physics. Still, “it can be shownthat” almost never appears, though the standards of what constitutes “proof”may be low, and the range of generality narrow. I believe that much mathe-matics is made difficult for the less-mathematically-inclined reader by the ab-sence of concrete instances of the abstract objects under discussion. This texttries to provide essentially correct instances of otherwise hard to grasp math-ematical abstractions. I hope and believe that this will provide a broad baseof general understanding from which deeper, more specialized, more math-ematical texts can be approached with a respectable general comprehension.This statement is most applicable to the excellent books by Arnold, who trieshard, but not necessarily successfully, to provide physical lines of reasoning.Much of this book was written with the goal of making one or another of hisdiscussions comprehensible.

In the early days of our weekly Laboratory of Nuclear Studies Journal Club,our founding leader, Robert Wilson, imposed a rule – though honored as muchin the breach as in the observance, it was not intended to be a joke – that theDirac γ-matrices never appear. The (largely unsuccessful) purpose of this rulewas to force the lectures to be intelligible to us theory-challenged experimen-talists. In this text there is a similar rule. It is that hieroglyphics such as

φ : {x ∈ R2 : |x| = 1} → R

not appear. The justification for this rule is that a “physicist” is likely to skipsuch a statement altogether or, once having understood it, regard it as obvi-ous. Like the jest that the French “don’t care what they say as long as theypronounce it properly” one can joke that mathematicians don’t care whattheir mappings do, as long as the spaces they connect are clear. Physicists,

Introduction 5

on the other hand, care primarily what their functions represent physicallyand are not fussy about what spaces they relate. Another “rule” has just beenfollowed; the word function will be used in preference to the (synonymous)word mapping. Other terrifying mathematical words such as flow, symplecto-morphism, and manifold will also be avoided except that, to avoid long-windedphrases such as “configuration space described by generalized coordinates,”the word manifold will occasionally be used. Of course one cannot alter theessence of a subject by denying the existence of mathematics that is manifestlyat its core. In spite of the loss of precision, I hope that sugar-coating the mate-rial in this way will make it more easily swallowed by nonmathematicians.

Notation: “Notation isn’t everything, it’s the only thing.” Grammaticallyspeaking, this statement, like the American football slogan it paraphrases,makes no sense. But its clearly intended meaning is only a mild exaggeration.After the need to evaluate some quantity has been expressed, a few straight-forward mathematical operations are typically all that is required to obtain thequantity. But specifying quantities is far from simple. The conceptual depthof the subject is substantial and ordinary language is scarcely capable of defin-ing the symbols, much less expressing the relations among them. This makesthe introduction of sophisticated symbols essential. Discussion of notationand the motivation behind its introduction is scattered throughout this text– probably to the point of irritation for some readers. Here we limit discus-sion to the few most important, most likely to be confusing, and most deviantfrom other sources: the qualified equality q= , the vector, the preferred referencesystem, the active/passive interpretation of transformations, and the terminologyof differential forms.

A fairly common occurrence in this subject is that two quantities A and Bare equal or equivalent from one point of view but not from another. Thiscircumstance will be indicated by “qualified equality” A q= B. This notationis intentionally vague (the “q” stands for qualified, or questionable, or query?as appropriate) and may have different meanings in different contexts; it onlywarns the reader to be wary of the risk of jumping to unjustified conclusions.Normally the qualification will be clarified in the subsequent text.

Next vectors. Consider the following three symbols or collections of sym-bols: −→, x, and (x, y, z)T. The first, −→, will be called an arrow (because itis one) and this word will be far more prevalent in this text than any other ofwhich I am aware. This particular arrow happens to be pointing in a horizon-tal direction (for convenience of typesetting) but in general an arrow can pointin any direction, including out of the page. The second, bold face, quantity,x, is an intrinsic or true vector; this means that it is a symbol that “stands for”an arrow. The word “intrinsic” means “it doesn’t depend on choice of coordi-nate system.” The third quantity, (x, y, z)T, is a column matrix (because the Tstands for transpose) containing the “components” of x relative to some pre-

6 Introduction

established coordinate system. From the point of view of elementary physicsthese three are equivalent quantities, differing only in the ways they are tobe manipulated; “addition” of arrows is by ruler and compass, addition ofintrinsic vectors is by vector algebra, and addition of coordinate vectors iscomponent wise. Because of this multiplicity of meanings, the word “vector”is ambiguous in some contexts. For this reason, we will often use the wordarrow in situations where independence of choice of coordinates is being em-phasized (even in dimensionality higher than 3.) According to its definitionabove, the phrase intrinsic vector could usually replace arrow, but some wouldcomplain of the redundancy, and the word arrow more succinctly conveys theintended geometric sense. Comments similar to these could be made concern-ing higher order tensors but they would be largely repetitive.

A virtue of arrows is that they can be plotted in figures. This goes a long waytoward making their meaning unambiguous but the conditions defining thefigure must still be made clear. In classical mechanics “inertial frames” havea fundamental significance and we will almost always suppose that there is a“preferred” reference system, its rectangular axes fixed in an inertial system.Unless otherwise stated, figures in this text are to be regarded as “snapshots”taken in that frame. In particular, a plotted arrow connects two points fixed inthe inertial frame at the instant illustrated. As mentioned previously, such anarrow is symbolized by a true vector such as x.

It is, of course, essential that these vectors satisfy the algebraic propertiesdefining a vector space. In such spaces “transformations” are important; a“linear” transformation can be represented by a matrix symbolized, for exam-ple, by M, with elements Mij. The result of applying this transformation tovector x can be represented symbolically as the “matrix product” y q= M xof “intrinsic” quantities, or spelled out explicitly in components yi = Mijx

j.Frequently both forms will be given. This leads to a notational difficulty in dis-tinguishing between the “active” and “passive” interpretations of the transfor-mation. The new components yi can belong to a new arrow in the old frame(active interpretation) or to the old arrow in a new frame (passive interpreta-tion). On the other hand, the intrinsic form y q= M x seems to support only anactive interpretation according to which M “operates” on vector x to yield adifferent vector y. To avoid this problem, when we wish to express a passiveinterpretation we will ordinarily use the form x q= M x and will insist that xand x stand for the same arrow. The significance of the overhead bar then is thatx is simply an abbreviation for an array of barred-frame coordinates xi. Whenthe active interpretation is intended the notation will usually be expanded toclarify the situation. For example, consider a moving point located initially atr(0) and at r(t) at later time t. These vectors can be related by r(t) = O(t) r(0)where O(t) is a time-dependent operator. This is an active transformation.

Introduction 7

The beauty and power of vector analysis as it is applied to physics is thata bold face symbol such as V indicates that the quantity is intrinsic and alsoabbreviates its multiple components Vi into one symbol. Though these areboth valuable purposes, they are not the same. The abbreviation works invector analysis only because vectors are the only multiple component objectsoccurring. That this will no longer be the case in this book will cause con-siderable notational difficulty because the reader, based on experience withvector analysis, is likely to jump to unjustified conclusions concerning boldface quantities.1 We will not be able to avoid this problem however since wewish to retain familiar notation. Sometimes we will be using bold face sym-bols to indicate intrinsically, sometimes as abbreviation, and sometimes both.Sometimes the (redundant) notation �v will be used to emphasize the intrinsicaspect. Though it may not be obvious at this point, notational insufficiencywas the source of the above-mentioned need to differentiate verbally betweenactive and passive transformations. In stressing this distinction the text differsfrom a text such as Goldstein that, perhaps wisely, de-emphasizes the issue.

According to Arnold “it is impossible to understand mechanics without theuse of differential forms.” Accepting the validity of this statement only grudg-ingly (and trying to corroborate it) but knowing from experience that typicalphysics students are innocent of any such knowledge, a considerable portionof the text is devoted to this subject. Briefly, the symbol dx will stand for anold-fashioned differential displacement of the sort familiar to every student ofphysics. But a new quantity d̃x to be known as a differential form, will alsobe used. This symbol is distinguished from dx both by being bold face andhaving an overhead tilde. Displacements dx1, dx2, . . . in spaces of higher di-

mension will have matching forms d̃x1, d̃x

2, . . . . This notation is mentioned

at this point only because it is unconventional. In most treatments one or theother form of differential is used, but not both at the same time. I have found itimpossible to cause classical formulations to morph into modern formulationswithout this distinction (and others to be faced when the time comes.)

It is hard to avoid using terms whose meanings are vague. (See the previousparagraph, for example.) I have attempted to acknowledge such vagueness, atleast in extreme cases, by placing such terms in quotation marks when they arefirst used. Since quotation marks are also used when the term is actually beingdefined, a certain amount of hunting through the surrounding sentences maybe necessary to find if a definition is actually present. (If it is not clear whetheror not there is a definition then the term is without any doubt vague.) Italicsare used to emphasize key phrases, or pairs of phrases in opposition, that are

1) Any computer programmer knows that, when two logically distinctquantities have initially been given the same symbol, because theyare expected to remain equal, it is hard to unscramble the code whenlater on it becomes necessary to distinguish between the two usages.

8 Introduction

central to the discussion. Parenthesized sentences or sentence fragments aresupposedly clear only if they are included right there but they should not beallowed to interrupt the logical flow of the surrounding sentences. Footnotes,though sometimes similar in intent, are likely to be real digressions, or techni-cal qualifications or clarifications.

The text contains at least enough material for a full year course and far morethan can be covered in any single term course. At Cornell the material hasbeen the basis for several distinct courses: (a) Junior/senior level classical me-chanics (as that subject is traditionally, and narrowly, defined.) (b) First yeargraduate classical mechanics with geometric emphasis. (c) Perturbative andadiabatic methods of solution and, most recently, (d) “Geometric Concepts inPhysics.” Course (d) was responsible for the math/physics reformulation ofthis edition. The text is best matched, therefore, to filling a curricular slot thatallows variation term-by-term or year-by-year.

Organization of the book: Chapter 1, containing review/examples, providesappropriate preparation for any of the above courses; it contains a briefoverview of elementary methods that the reader may wish (or need) to re-view. Since the formalism (primarily Lagrangian) is assumed to be familiar,this review consists primarily of examples, many worked out partially or com-pletely. Chapter 2 and the first half of Chapter 3 contain the geometric con-cepts likely to be both “new” and needed. The rest of Chapter 3 as well asChapter 4 contain geometry that can be skipped until needed. Chapters 5, 6,7, and 8 contain, respectively, the Lagrangian, Newtonian, Hamiltonian andHamilton–Jacobi, backbone of course labeled (a) above. The first half of Chap-ter 10, on conservation laws, is also appropriate for such a course, and meth-ods of solution should be drawn from Chapters 14, 15, and 16.

The need for relativistic mechanics is what characterizes Chapters 9, 11, 12,and 13. These chapters can provide the “physics” content for a course suchas (d) above. The rest of the book does not depend on the material in thesechapters. A course should therefore include either none of this material or allof it, though perhaps emphasizing either, but not both, of general relativityand string theory.

Methods of solution are studied in Chapters 14, 15, and 16. These chapterswould form an appreciable fraction of a course such as (c) above.

Chapter 17 is concerned mainly with the formal structure of mechanics inHamiltonian form. As such it is most likely to be of interest to students plan-ning to take a subsequent courses in dynamical systems, chaos, plasma or ac-celerator physics. Somehow the most important result of classical mechanics– Liouville’s theorem – has found its way to the last section of the book.

The total number of problems has been almost doubled compared to thefirst edition. However, in the chapters covering areas of physics not tradition-ally regarded as classical mechanics, the problems are intended to require nospecial knowledge of those subjects beyond what is covered in this text.

Bibliography 9

Some Abbreviations Used in This Text

E.D. exterior derivativeB.C. bilinear covariantO.P.L. optical path lengthI.I. integral invariantH.I. Hamiltonian variational line integralL.I.I. Lagrange invariant integralR.I.I. relative integral invariant

Bibliography

General Mechanics Texts

1 V.I. Arnold, Mathematical Methods of Classi-cal Mechanics, Springer, New York, 1978.

2 N.G. Chetaev, Theoretical Mechanics,Springer, Berlin, 1989.

3 H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1980.

4 L.D. Landau and E.M. Lifshitz, Mechanics,Pergamon, Oxford, 1976.

5 L.A. Pars, Analytical Dynamics, Ox BowPress, Woodbridge, CT, 1979.

6 K.R. Symon, Mechanics, Addison-Wesley,Reading, MA, 1971.

7 D. Ter Haar, Elements of Hamiltonian Me-chanics, 2nd ed., Pergamon, Oxford, 1971.

8 E.T. Whittaker, Treatise on the AnalyticalDynamics of Particles and Rigid Bodies, Cam-bridge University Press, Cambridge, UK,1989

Specialized Mathematical Books onMechanics

9 V.I. Arnold, V.V. Kozlov, and A.I. Neish-tadt, Dynamical Systems III, Springer, Berlin,1980.

10 J.E. Marsden, Lectures on Mechanics, Cam-bridge University Press, Cambridge, UK,1992.

11 K.R. Meyer and R. Hall, Introduction toHamiltonian Dynamical Systems and the N-Body Problem, Springer, New York, 1992.

Relevant Mathematics

12 E. Cartan, The Theory of Spinors, Dover,New York, 1981.

13 E. Cartan, Leçons sur la géometrie des espacesde Riemann, Gauthiers-Villars, Paris, 1951.(English translation available.)

14 B.A. Dubrovin, A.T. Fomenko, andS.P. Novikov, Modern Geometry I, Springer,Berlin, 1985

15 H. Flanders, Differential Forms With Appli-cations to the Physical Sciences, Dover, NewYork, 1989

16 D.H. Sattinger and O.L. Weaver, Lie Groupsand Algebras, Applications to Physics, Geom-etry, and Mechanics, Springer, New York,1986

17 B.F. Schutz, Geometrical Methods of Math-ematical Physics, Cambridge UniversityPress, Cambridge, UK, 1980

18 V.A. Yakubovitch and V.M. Starzhinskii,Linear Differential Equations With PeriodicCoefficients, Wiley, New York, 1975

Physics

19 L.D. Landau and E.M. Lifshitz, The ClassicalTheory of Fields, Pergamon, Oxford, 1975.

20 S. Weinberg, Gravitation and Cosmology,Wiley, New York, 1972.

21 B. Zwiebach, A First Course in String Theory,Cambridge University Press, Cambridge,UK, 2004.

11

1Review of Classical Mechanics and String Field Theory

1.1Preview and Rationale

This introductory chapter has two main purposes. The first is to review La-grangian mechanics. Some of this material takes the form of worked exam-ples, chosen both to be appropriate as examples and to serve as bases for top-ics in later chapters.

The second purpose is to introduce the mechanics of classical strings. Thistopic is timely, being introductory to the modern subject of (quantum fieldtheoretical) string theory. But, also, the Lagrangian theory of strings is an ap-propriate area in which to practice using supposedly well-known conceptsand methods in a context that is encountered (if at all) toward the end ofa traditional course in intermediate mechanics. This introduces the topic ofLagrangian field theory in a well-motivated and elementary way. Classicalstrings have the happy properties of being the simplest system for which La-grangian field theory is appropriate.

The motivation for emphasizing strings from the start comes from the dar-ing, and apparently successful, introduction by Barton Zwiebach, of stringtheory into the M.I.T. undergraduate curriculum. This program is fleshed outin his book A First Course in String Theory. The present chapter, and especiallyChapter 12 on relativistic strings, borrows extensively from that text. UnlikeZwiebach though, the present text stops well short of quantum field theory.

An eventual aim of this text is to unify “all” of classical physics within suit-ably generalized Lagrangian mechanics. Here “all” will be taken to be ade-quately represented by the following topics: mechanics of particles, specialrelativity, electromagnetic theory, classical (and, eventually, relativistic) stringtheory, and general relativity. This list, which is to be regarded as definingby example what constitutes “classical physics,” is indeed ambitious, thoughit leaves out many other important fields of classical physics, such as elastic-ity and fluid dynamics.1 The list also includes enough varieties of geometry

1) By referring to a text such as Theoretical Mechanics of Particles andContinua, by Fetter and Walecka, which covers fluids and elasticsolids in very much the same spirit as in the present text, it shouldbe clear that these two topics can also be included in the list of fieldsunified by Lagrangian mechanics.

12 1 Review of Classical Mechanics and String Field Theory

to support another aim of the text, which is to illuminate the important roleplayed by geometry in physics.

An introductory textbook on Lagrangian mechanics (which this is not)might be expected to begin by announcing that the reader is assumed to befamiliar with Newtonian mechanics – kinematics, force, momentum and en-ergy and their conservation, simple harmonic motion, moments of inertia, andso on. In all likelihood such a text would then proceed to review these verysame topics before advancing to its main topic of Lagrangian mechanics. Thiswould not, of course, contradict the original assumption since, apart from thesimple pedagogical value of review, it makes no sense to study Lagrangianmechanics without anchoring it firmly in a Newtonian foundation. The stu-dent who had not learned this material previously would be well advised tostart by studying a less advanced, purely Newtonian mechanics textbook. Somany of the most important problems of physics can be solved cleanly with-out the power of Lagrangian mechanics; it is uneconomical to begin with anabstract formulation of mechanics before developing intuition better acquiredfrom a concrete treatment. One might say that Newtonian methods give better“value” than Lagrangian mechanics because, though ultimately less powerful,Newtonian methods can solve the most important problems and are easier tolearn. Of course this would only be true in the sort of foolish system of ac-counting that might attempt to rate the relative contributions of Newton andEinstein. One (but not the only) purpose of this textbook, is to go beyond La-grange’s equations. By the same foolish system of accounting just mentioned,these methods could be rated less valuable than Lagrangian methods since,though more powerful, they are more abstract and harder to learn.

It is assumed the reader has had some (not necessarily much) experiencewith Lagrangian mechanics.2 Naturally this presupposes familiarity with theabove-mentioned elementary concepts of Newtonian mechanics. Neverthe-less, for the same reasons as were described in the previous paragraph, westart by reviewing material that is, in principle, already known. It is assumedthe reader can define a Lagrangian, can write it down for a simple mechanicalsystem, can write down (or copy knowledgeably) the Euler–Lagrange equa-tions and from them derive the equations of motion of the system, and finally(and most important of all) trust these equations to the same extent that sheor he trusts Newton’s law itself. A certain (even if grudging) acknowledge-ment of the method’s power to make complicated systems appear simple isalso helpful. Any reader unfamiliar with these ideas would be well advised

2) Though “prerequisites” have been men-tioned, this text still attempts to be “nottoo advanced.” Though the subject matterdeviates greatly from the traditional cur-riculum at this level (as represented, say,

by Goldstein, Classical Mechanics) it is myintention that the level of difficulty and theanticipated level of preparation be muchthe same as is appropriate for Goldstein.