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Mathematics for Physics

A guided tour for graduate students

Michael Stone

and

Paul Goldbart

PIMANDER-CASAUBON

Alexandria • Florence • London

ii

Copyright ©c2002-2008 M. Stone, P. M. Goldbart

All rights reserved. No part of this material can be reproduced, stored or transmitted without the written permission of the authors. For information contact: Michael Stone or Paul Goldbart, Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, U.S.A.

iv DEDICATION

Acknowledgments

A great many people have encouraged us along the way: Our teachers at the University of Cambridge, the University of California-Los Angeles, and Imperial College London. Our students – your questions and enthusiasm have helped shape our under- standing and our exposition. Our colleagues—faculty and staff—at the University of Illinois at Urbana- Champaign – how fortunate we are to have a community so rich in both accomplishment and collegiality. Our friends and family: Kyre and Steve and Ginna; and Jenny, Ollie and Greta – we hope to be more attentive now that this book is done. Our editor Simon Capelin at Cambridge University Press – your patience is appreciated. The staff of the U.S. National Science Foundation and the U.S. Department of Energy, who have supported our research over the years. Our sincere thanks to you all.

v

Preface

This book is based on a two-semester sequence of courses taught to incoming graduate students at the University of Illinois at Urbana-Champaign, pri- marily physics students but also some from other branches of the physical sciences. The courses aim to introduce students to some of the mathematical methods and concepts that they will find useful in their research. We have sought to enliven the material by integrating the mathematics with its appli- cations. We therefore provide illustrative examples and problems drawn from physics. Some of these illustrations are classical but many are small parts of contemporary research papers. In the text and at the end of each chapter we provide a collection of exercises and problems suitable for homework assign- ments. The former are straightforward applications of material presented in the text; the latter are intended to be interesting, and take rather more thought and time.

We devote the first, and longest, part (Chapters 1 to 9, and the first semester in the classroom) to traditional mathematical methods. We explore the analogy between linear operators acting on function spaces and matrices acting on finite dimensional spaces, and use the operator language to pro- vide a unified framework for working with ordinary differential equations, partial differential equations, and integral equations. The mathematical pre- requisites are a sound grasp of undergraduate calculus (including the vector calculus needed for electricity and magnetism courses), elementary linear al- gebra, and competence at complex arithmetic. Fourier sums and integrals, as well as basic ordinary differential equation theory, receive a quick review, but it would help if the reader had some prior experience to build on. Contour integration is not required for this part of the book.

The second part (Chapters 10 to 14) focuses on modern differential ge- ometry and topology, with an eye to its application to physics. The tools of calculus on manifolds, especially the exterior calculus, are introduced, and

vii

viii PREFACE

used to investigate classical mechanics, electromagnetism, and non-abelian gauge fields. The language of homology and cohomology is introduced and is used to investigate the influence of the global topology of a manifold on the fields that live in it and on the solutions of differential equations that constrain these fields. Chapters 15 and 16 introduce the theory of group representations and their applications to quantum mechanics. Both finite groups and Lie groups are explored. The last part (Chapters 17 to 19) explores the theory of complex variables and its applications. Although much of the material is standard, we make use of the exterior calculus, and discuss rather more of the topological aspects of analytic functions than is customary. A cursory reading of the Contents of the book will show that there is more material here than can be comfortably covered in two semesters. When using the book as the basis for lectures in the classroom, we have found it useful to tailor the presented material to the interests of our students.

• 1 Calculus of Variations Preface vii
  • 1.1 What is it good for?
  • 1.2 Functionals
  • 1.3 Lagrangian mechanics
  • 1.4 Variable endpoints
  • 1.5 Lagrange multipliers
  • 1.6 Maximum or minimum?
  • 1.7 Further exercises and problems
• 2 Function Spaces
  • 2.1 Motivation
  • 2.2 Norms and inner products
  • 2.3 Linear operators and distributions
  • 2.4 Further exercises and problems
• 3 Linear Ordinary Differential Equations
  • 3.1 Existence and uniqueness of solutions
  • 3.2 Normal form
  • 3.3 Inhomogeneous equations
  • 3.4 Singular points
  • 3.5 Further exercises and problems
• 4 Linear Differential Operators x CONTENTS
  • 4.1 Formal vs. concrete operators
  • 4.2 The adjoint operator
  • 4.3 Completeness of eigenfunctions
  • 4.4 Further exercises and problems
• 5 Green Functions
  • 5.1 Inhomogeneous linear equations
  • 5.2 Constructing Green functions
  • 5.3 Applications of Lagrange’s identity
  • 5.4 Eigenfunction expansions
  • 5.5 Analytic properties of Green functions
  • 5.6 Locality and the Gelfand-Dikii equation
  • 5.7 Further exercises and problems
• 6 Partial Differential Equations
  • 6.1 Classification of PDE’s
  • 6.2 Cauchy data
  • 6.3 Wave equation
  • 6.4 Heat equation
  • 6.5 Potential theory
  • 6.6 Further exercises and problems
• 7 The Mathematics of Real Waves
  • 7.1 Dispersive waves
  • 7.2 Making waves
  • 7.3 Non-linear waves
  • 7.4 Solitons
  • 7.5 Further exercises and problems
• 8 Special Functions
  • 8.1 Curvilinear co-ordinates
  • 8.2 Spherical harmonics
  • 8.3 Bessel functions
  • 8.4 Singular endpoints
  • 8.5 Further exercises and problems
• 9 Integral Equations CONTENTS xi
  • 9.1 Illustrations
  • 9.2 Classification of integral equations
  • 9.3 Integral transforms
  • 9.4 Separable kernels
  • 9.5 Singular integral equations
  • 9.6 Wiener-Hopf equations I
  • 9.7 Some functional analysis
  • 9.8 Series solutions
  • 9.9 Further exercises and problems
• 10 Vectors and Tensors
  • 10.1 Covariant and contravariant vectors
  • 10.2 Tensors
  • 10.3 Cartesian tensors
  • 10.4 Further exercises and problems
• 11 Differential Calculus on Manifolds
  • 11.1 Vector and covector fields
  • 11.2 Differentiating tensors
  • 11.3 Exterior calculus
  • 11.4 Physical applications
  • 11.5 Covariant derivatives
  • 11.6 Further exercises and problems
• 12 Integration on Manifolds
  • 12.1 Basic notions
  • 12.2 Integrating p-forms
  • 12.3 Stokes’ theorem
  • 12.4 Applications
  • 12.5 Further exercises and problems
• 13 An Introduction to Differential Topology
  • 13.1 Homeomorphism and diffeomorphism
  • 13.2 Cohomology
  • 13.3 Homology
  • 13.4 De Rham’s theorem
  • 13.5 Poincar´e duality
  • 13.6 Characteristic classes xii CONTENTS
  • 13.7 Hodge theory and the Morse index
  • 13.8 Further exercises and problems
• 14 Groups and Group Representations
  • 14.1 Basic ideas
  • 14.2 Representations
  • 14.3 Physics applications
  • 14.4 Further exercises and problems
• 15 Lie Groups
  • 15.1 Matrix groups
  • 15.2 Geometry of SU(2)
  • 15.3 Lie algebras
  • 15.4 Further exercises and problems
• 16 The Geometry of Fibre Bundles
  • 16.1 Fibre bundles
  • 16.2 Physics examples
  • 16.3 Working in the total space
• 17 Complex Analysis I
  • 17.1 Cauchy-Riemann equations
  • 17.2 Complex integration: Cauchy and Stokes
  • 17.3 Applications
  • 17.4 Applications of Cauchy’s theorem
  • 17.5 Meromorphic functions and the winding-number
  • 17.6 Analytic functions and topology
  • 17.7 Further exercises and problems
• 18 Applications of Complex Variables
  • 18.1 Contour integration technology
  • 18.2 The Schwarz reflection principle
  • 18.3 Partial-fraction and product expansions
  • 18.4 Wiener-Hopf equations II
  • 18.5 Further exercises and problems
• 19 Special Functions and Complex Variables CONTENTS xiii
  • 19.1 The Gamma function
  • 19.2 Linear differential equations
  • 19.3 Solving ODE’s via contour integrals
  • 19.4 Asymptotic expansions
  • 19.5 Elliptic functions
  • 19.6 Further exercises and problems
• A Linear Algebra Review
  • A.1 Vector space
  • A.2 Linear maps
  • A.3 Inner-product spaces
  • A.4 Sums and differences of vector spaces
  • A.5 Inhomogeneous linear equations
  • A.6 Determinants
  • A.7 Diagonalization and canonical forms
• B Fourier Series and Integrals.
  • B.1 Fourier series
  • B.2 Fourier integral transforms
  • B.3 Convolution
  • B.4 The Poisson summation formula
• C Bibliography

xiv CONTENTS

2 CHAPTER 1. CALCULUS OF VARIATIONS

1.2 Functionals

In variational problems we are provided with an expression J[y] that “eats” whole functions y(x) and returns a single number. Such objects are called functionals to distinguish them from ordinary functions. An ordinary func- tion is a map f : R → R. A functional J is a map J : C∞(R) → R where C∞(R) is the space of smooth (having derivatives of all orders) functions. To find the function y(x) that maximizes or minimizes a given functional J[y] we need to define, and evaluate, its functional derivative.

1.2.1 The functional derivative

We restrict ourselves to expressions of the form

J[y] =

∫ (^) x 2

x 1

f (x, y, y′, y′′, · · · y(n)) dx, (1.1)

where f depends on the value of y(x) and only finitely many of its derivatives. Such functionals are said to be local in x. Consider first a functional J =

f dx in which f depends only x, y and y′. Make a change y(x) → y(x) + εη(x), where ε is a (small) x-independent constant. The resultant change in J is

J[y + εη] − J[y] =

∫ (^) x 2

x 1

{f (x, y + εη, y′^ + εη′) − f (x, y, y′)} dx

∫ (^) x 2

x 1

εη

∂f ∂y

• ε

dη dx

∂f ∂y′^

• O(ε^2 )

dx

[

εη

∂f ∂y′

]x 2

x 1

∫ (^) x 2

x 1

(εη(x))

∂f ∂y

d dx

∂f ∂y′

dx + O(ε^2 ).

If η(x 1 ) = η(x 2 ) = 0, the variation δy(x) ≡ εη(x) in y(x) is said to have “fixed endpoints.” For such variations the integrated-out part [.. .]x x^21 van- ishes. Defining δJ to be the O(ε) part of J[y + εη] − J[y], we have

δJ =

∫ (^) x 2

x 1

(εη(x))

∂f ∂y

d dx

∂f ∂y′

dx

∫ (^) x 2

x 1

δy(x)

δJ δy(x)

dx. (1.2)

1.2. FUNCTIONALS 3

The function δJ δy(x)

∂f ∂y

d dx

∂f ∂y′

is called the functional (or Fr´echet) derivative of J with respect to y(x). We can think of it as a generalization of the partial derivative ∂J/∂yi, where the discrete subscript “i” on y is replaced by a continuous label “x,” and sums over i are replaced by integrals over x:

δJ =

i

∂J

∂yi

δyi →

∫ (^) x 2

x 1

dx

δJ δy(x)

δy(x). (1.4)

1.2.2 The Euler-Lagrange equation

Suppose that we have a differentiable function J(y 1 , y 2 ,... , yn) of n variables and seek its stationary points — these being the locations at which J has its maxima, minima and saddlepoints. At a stationary point (y 1 , y 2 ,... , yn) the variation

δJ =

∑^ n

i=

∂J

∂yi

δyi (1.5)

must be zero for all possible δyi. The necessary and sufficient condition for this is that all partial derivatives ∂J/∂yi, i = 1,... , n be zero. By analogy, we expect that a functional J[y] will be stationary under fixed-endpoint vari- ations y(x) → y(x)+δy(x), when the functional derivative δJ/δy(x) vanishes for all x. In other words, when

∂f ∂y(x)

d dx

∂f ∂y′(x)

= 0, x 1 < x < x 2. (1.6)

The condition (1.6) for y(x) to be a stationary point is usually called the Euler-Lagrange equation. That δJ/δy(x) ≡ 0 is a sufficient condition for δJ to be zero is clear from its definition in (1.2). To see that it is a necessary condition we must appeal to the assumed smoothness of y(x). Consider a function y(x) at which J[y] is stationary but where δJ/δy(x) is non-zero at some x 0 ∈ [x 1 , x 2 ]. Because f (y, y′, x) is smooth, the functional derivative δJ/δy(x) is also a smooth function of x. Therefore, by continuity, it will have the same sign throughout some open interval containing x 0. By taking δy(x) = εη(x) to be

1.2. FUNCTIONALS 5

minimal surface will be a surface of revolution about the x axis. We therefore seek the profile y(x) that makes the area

J[y] = 2π

∫ (^) x 2

x 1

y

1 + y′^2 dx (1.9)

of the surface of revolution the least among all such surfaces bounded by the circles of radii y(x 1 ) = y 1 and y(x 2 ) = y 2. Because a minimum is a stationary point, we seek candidates for the minimizing profile y(x) by setting the functional derivative δJ/δy(x) to zero. We begin by forming the partial derivatives ∂f ∂y

= 4πσ

1 + y′^2 ,

∂f ∂y′^

4 πσyy′ √ 1 + y′^2

and use them to write down the Euler-Lagrange equation

√ 1 + y′^2 −

d dx

yy′ √ 1 + y′^2

Performing the indicated derivative with respect to x gives √ 1 + y′^2 − √(y′)^2 1 + y′^2

√yy′′ 1 + y′^2

y(y′)^2 y′′ (1 + y′^2 )^3 /^2

After collecting terms, this simplifies to

1 √ 1 + y′^2

yy′′ (1 + y′^2 )^3 /^2

The differential equation (1.13) still looks a trifle intimidating. To simplify further, we multiply by y′^ to get

0 = √ y′ 1 + y′^2

yy′y′′ (1 + y′^2 )^3 /^2

=

d dx

y √ 1 + y′^2

The solution to the minimization problem therefore reduces to solving

y √ 1 + y′^2

= κ, (1.15)

6 CHAPTER 1. CALCULUS OF VARIATIONS

where κ is an as yet undetermined integration constant. Fortunately this non-linear, first order, differential equation is elementary. We recast it as

dy dx

y^2 κ^2

and separate variables (^) ∫

dx =

dy √ y^2 κ^2 −^1

We now make the natural substitution y = κ cosh t, whence ∫ dx = κ

dt. (1.18)

Thus we find that x + a = κt, leading to

y = κ cosh

x + a κ

We select the constants κ and a to fit the endpoints y(x 1 ) = y 1 and y(x 2 ) = y 2.

x

y

h

−L +L

Figure 1.2: Hanging chain

Example: Heavy Chain over Pulleys. We cannot yet consider the form of the catenary, a hanging chain of fixed length, but we can solve a simpler problem of a heavy flexible cable draped over a pair of pulleys located at x = ±L, y = h, and with the excess cable resting on a horizontal surface as illustrated in figure 1.2.