Dynamics a Set of Notes on Theoretical Physical Chemistry - Jaclyn Steen

7/28/2019 Dynamics a Set of Notes on Theoretical Physical Chemistry - Jaclyn Steen

1/144

Dynamics: A Set of Notes on Theoretical Physical Chemistry

Jaclyn Steen, Kevin Range and Darrin M. York

December 5, 2003

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Contents

1 Vector Calculus 6

1.1 Properties of vectors and vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Fundamental operations involving vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Linear Algebra 11

2.1 Matrices, Vectors and Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Transpose of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Unit Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Trace of a (Square) Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5.1 Inverse of a (Square) Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 More on Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 More on [A, B] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.8 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.8.1 Laplacian expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.8.2 Applications of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.9 Generalized Greens Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.10 Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.11 Symmetric/Antisymmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.12 Similarity Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.13 Hermitian (self-adjoint) Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.14 Unitary Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.15 Comments about Hermitian Matrices and Unitary Tranformations . . . . . . . . . . . . . . . . 18

2.16 More on Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.17 Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.18 Anti-Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.19 Functions of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.20 Normal Marices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.21 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.21.1 Real Symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.21.2 Hermitian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.21.3 Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.21.4 Orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.21.5 Unitary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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CONTENTS CONTENTS

3 Calculus of Variations 22

3.1 Functions and Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Functional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Variational Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Functional Derivatives: Elaboration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.1 Algebraic Manipulations of Functional Derivatives . . . . . . . . . . . . . . . . . . . . 253.4.2 Generalization to Functionals of Higher Dimension . . . . . . . . . . . . . . . . . . . . 26

3.4.3 Higher Order Functional Variations and Derivatives . . . . . . . . . . . . . . . . . . . . 27

3.4.4 Integral Taylor series expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.5 The chain relations for functional derivatives . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.6 Functional inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5 Homogeneity and convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5.1 Homogeneity properties of functions and functionals . . . . . . . . . . . . . . . . . . . 31

3.5.2 Convexity properties of functions and functionals . . . . . . . . . . . . . . . . . . . . . 32

3.6 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7.1 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.7.2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7.3 Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7.3.1 Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7.3.2 Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7.3.3 Part C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7.3.4 Part D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7.3.5 Part E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7.4 Problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7.4.1 Part F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7.4.2 Part G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7.4.3 Part H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.7.4.4 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7.4.5 Part J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7.4.6 Part K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.7.4.7 Part L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.7.4.8 Part M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Classical Mechanics 40

4.1 Mechanics of a system of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.1 Newtons laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.2 Fundamental definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 DAlemberts principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4 Velocity-dependent potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5 Frictional forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Variational Principles 54

5.1 Hamiltons Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Comments about Hamiltons Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3 Conservation Theorems and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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6 Central Potential and More 61

6.1 Galilean Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.3 Motion in 1-Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.3.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.3.2 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.4 Classical Viral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.5 Central Force Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.6 Conditions for Closed Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.7 Bertrands Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.8 The Kepler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.9 The Laplace-Runge-Lenz Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7 Scattering 73

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.2 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.2.1 Rutherford Scattering Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.2.2 Rutherford Scattering in the Laboratory Frame . . . . . . . . . . . . . . . . . . . . . . 76

7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

8 Collisions 78

8.1 Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

9 Oscillations 82

9.1 Euler Angles of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

9.2 Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

9.3 General Solution of Harmonic Oscillator Equation . . . . . . . . . . . . . . . . . . . . . . . . . 85

9.3.1 1-Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

9.3.2 Many-Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869.4 Forced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

9.5 Damped Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10 Fourier Transforms 90

10.1 Fourier Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

10.2 Theorems of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

10.3 Derivative Theorem Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

10.4 Convolution Theorem Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

10.5 Parsevals Theorem Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

11 Ewald Sums 95

11.1 Rate of Change of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9511.2 Rigid Body Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

11.3 Principal Axis Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

11.4 Solving Rigid Body Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

11.5 Eulers equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

11.6 Torque-Free Motion of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

11.7 Precession in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

11.8 Derivation of the Ewald Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

11.9 Coulomb integrals between Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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11.10 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

11.11Linear-scaling Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

11.12 Greens Function Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

11.13 Discrete FT on a Regular Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

11.14FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

11.15 Fast Fourier Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

12 Dielectric 106

12.1 Continuum Dielectric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

12.2 Gauss Law I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

12.3 Gauss Law II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

12.4 Variational Principles of Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

12.5 Electrostatics - Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

12.6 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

13 Exapansions 115

13.1 Schwarz inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

13.2 Triangle inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11613.3 Schmidt Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

13.4 Expansions of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

13.5 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

13.6 Convergence Theorem for Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

13.7 Fourier series for different intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

13.8 Complex Form of the Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

13.9 Uniform Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

13.10Differentiation of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

13.11Integration of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

13.12Fourier Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

13.13M-Test for Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13613.14 Fourier Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

13.15Examples of the Fourier Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

13.16Parsevals Theorem for Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

13.17Convolution Theorem for Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

13.18Fourier Sine and Cosine Transforms and Representations . . . . . . . . . . . . . . . . . . . . . 143

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Chapter 1

Vector Calculus

These are summary notes on vector analysis and vector calculus. The purpose is to serve as a review. Although

the discussion here can be generalized to differential forms and the introduction to tensors, transformations and

linear algebra, an in depth discussion is deferred to later chapters, and to further reading.1,2,3,4,5

For the purposes of this review, it is assumed that vectors are real and represented in a 3-dimensional Carte-

sian basis (x, y, z), unless otherwise stated. Sometimes the generalized coordinate notation x1, x2, x3 will beused generically to refer to x,y ,z Cartesian components, respectively, in order to allow more concise formulasto be written using using i,j,k indexes and cyclic permutations.

If a sum appears without specification of the index bounds, assume summation is over the entire range of the

index.

1.1 Properties of vectors and vector space

A vector is an entity that exists in a vector space. In order to take for (in terms of numerical values for its

components) a vector must be associated with a basis that spans the vector space. In 3-D space, for example, a

Cartesian basis can be defined (x, y, z). This is an example of an orthonormal basis in that each componentbasis vector is normalized x x = y y = z z = 1 and orthogonal to the other basis vectors x y = y z =zx = 0. More generally, a basis (not necessarily the Cartesian basis, and not necessarily an orthonormal basis) isdenoted (e1, e2, e3. If the basis is normalized, this fact can be indicated by the hat symbol, and thus designated(e1, e2, e3.

Here the properties of vectors and the vector space in which they reside are summarized. Although the

present chapter focuses on vectors in a 3-dimensional (3-D) space, many of the properties outlined here are more

general, as will be seen later. Nonetheless, in chemistry and physics, the specific case of vectors in 3-D is so

prevalent that it warrants special attention, and also serves as an introduction to more general formulations.

A 3-D vector is defined as an entity that has both magnitude and direction, and can be characterized, provided

a basis is specified, by an ordered triple of numbers. The vector x, then, is represented as x = (x1, x2, x3).

Consider the following definitions for operations on the vectors x and y given by x = (x1, x2, x3) andy = (y1, y2, y3):

1. Vector equality: x = y ifxi = yi i = 1, 2, 3

2. Vector addition: x + y = z ifzi = xi + yi i = 1, 2, 3

3. Scalar multiplication: ax = (ax1, ax2, ax3)

4. Null vector: There exists a unique null vector 0 = (0, 0, 0)

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CHAPTER 1. VECTOR CALCULUS 1.2. FUNDAMENTAL OPERATIONS INVOLVING VECTORS

Furthermore, assume that the following properties hold for the above defined operations:

1. Vector addition is commutative and associative:

x + y = y + x(x + y) + z = x + (y + z)

2. Scalar multiplication is associative and distributive:(ab)x = a(bx)(a + b)(x + y) = ax + bx + ay + by

The collection of all 3-D vectors that satisfy the above properties are said to form a 3-D vector space.

1.2 Fundamental operations involving vectors

The following fundamental vector operations are defined.

Scalar Product:

a

b= axbx + ayby + azbz = i aibi = |a||b|cos()= b a (1.1)

where |a| = a a, and is the angle between the vectors a and b.Cross Product:

a b = x(aybz azby) + y(azbx axbz) + z(axby aybx) (1.2)or more compactly

c = a b (1.3)where (1.4)

ci = ajbk akbj (1.5)where i,j,k are x,y ,z and the cyclic permutations z ,x,y and y ,z ,x, respectively. The cross product can beexpressed as a determinant:

The norm of the cross product is

|a b| = |a||b|sin() (1.6)where, again, is the angle between the vectors a and b The cross product of two vectors a and b results in avector that is perpendicular to both a and b, with magnitude equal to the area of the parallelogram defined by a

and b.

The Triple Scalar Product:

a

b

c = c

a

b = b

c

a (1.7)

and can also be expressed as a determinant

The triple scalar product is the volume of a parallelopiped defined by a, b, and c.

The Triple Vector Product:

a (b c) = b(a c) c(a b) (1.8)The above equation is sometimes referred to as the BAC CAB rule.

Note: the parenthases need to be retained, i.e. a (b c) = (a b) c in general.Lattices/Projection of a vector

a = (ax)x + (ay)y + (ay)y (1.9)

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1.2. FUNDAMENTAL OPERATIONS INVOLVING VECTORS CHAPTER 1. VECTOR CALCULUS

a x = ax (1.10)

r = r1a1 + r2a2 + r3a3 (1.11)

ai aj = ij (1.12)

ai =aj ak

ai (aj ak) (1.13)

Gradient, = x

x

+ y

y

+ z

z

(1.14)

f(|r|) = r

f

r

(1.15)

dr = xdx + ydy + zdz (1.16)

d = () dr (1.17)

(uv) = (u)v + u(v) (1.18)Divergence,

V =

Vxx

+

Vyy

+

Vzz

(1.19)

r = 3 (1.20)

(rf(r)) = 3f(r) + r dfdr

(1.21)

iff(r) = rn1 then rrn = (n + 2)rn1

(fv) = f v + f v (1.22)

Curl,x(

Vzy

Vyz

) + y(

Vxz

Vzx

) + z(

Vyx

Vxy

) (1.23)

(fv) = f v + (f) v (1.24)

r = 0 (1.25)

(rf(r)) = 0 (1.26)

(a b) = (b )a + (a )b + b ( a) + a ( b) (1.27)

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CHAPTER 1. VECTOR CALCULUS 1.2. FUNDAMENTAL OPERATIONS INVOLVING VECTORS

= = 2 =

2

2x

+

2

2y

+

2

2z

(1.28)

Vector Integration

Divergence theorem (Gausss Theorem)

V

f(r)d3r = S

f(r) d = S

f(r) nda (1.29)

let f(r) = uv then (uv) = u v + u2v (1.30)

V

u vd3r +

Vu2vd3r =

S

(uv) nda (1.31)

The above gives the second form of Greens theorem.

Let f(r) = uv vu then

V

u vd3r + V

u2vd3r V

u vd3r V

v2ud3r = S

(uv) nda S

(vu) nda (1.32)

Above gives the first form of Greens theorem.

Generalized Greens theoremV

uLu uLvd3r =

Sp(vu uv)) nda (1.33)

where L is a self-adjoint (Hermetian) Sturm-Lioville operator of the form:

L = [p] + q (1.34)

Stokes Theorem S

( v) nda =

CV d (1.35)

Generalized Stokes Theorem S

(d ) [] =

Cd [] (1.36)

where = ,,Vector Formulas

a (b c) = b (c a) = c (a b) (1.37)

a (b c) = (a c)b (a b)c (1.38)

(a b) (c d) = (a c)(b d) (a d)(b c) (1.39)

= 0 (1.40)

( a) = 0 (1.41)

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1.2. FUNDAMENTAL OPERATIONS INVOLVING VECTORS CHAPTER 1. VECTOR CALCULUS

( a) = ( a) 2a (1.42)

(a) = a + a (1.43)

(a) = a + a (1.44)

(a b) = (a )b + (b )a + a ( b) + b ( a) (1.45)

(a b) = b ( a) a ( b) (1.46)

(a b) = a( b) b( a) + (b )a (a )b (1.47)Ifx is the coordinate of a point with magnitude r = |x|, and n = x/r is a unit radial vector

x = 3 (1.48)

x = 0 (1.49)

n = 2/r (1.50)

n = 0 (1.51)

(a )n = (1/r)[a n(a n)] (1.52)

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Chapter 2

Linear Algebra

2.1 Matrices, Vectors and Scalars

Matrices - 2 indexes (2nd rank tensors) - Aij/A

Vectors - 1 index (1st

rank tensor) - ai/aScalar - 0 index (0 rank tensor) - a

Note: for the purpose of writing linear algebraic equations, a vector can be written as an N 1 Columnvector (a type of matrix), and a scalar as a 1 1 matrix.

2.2 Matrix Operations

Multiplication by a scalar .

C = A NN NN = A means Cij = Aij = Aij Addition/Subtraction

C = A B = B A means Cij = Aij BijMultiplication (inner product)

C = A B NN NMMN

means Cij =

k

AikBkj

AB = BA in general

A (B C) = (AB)C = ABC associative, not always communitiveMultiplication (outer product/direct product)

Cnmnm

= A B means C = Aij Bk

=n(i 1) + k =m(j 1) +

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2.3. TRANSPOSE OF A MATRIX CHAPTER 2. LINEAR ALGEBRA

A B =B A A (B C) =(A B)CNote, for vectors

C = a bT

N11Nmeans Cij = aibj

2.3 Transpose of a Matrix

A = BT NM (MN)T=NM

means Aij = (Bij)T = Bji

Note:

(AT)T = A

(Aij)TT

= [Aji ]T = Aij

(A B)T = BT AT

C =(A B)T = BT AT

Cij =

k

AikBkj

T=

k

Ajk Bki

=

k

BkiAjk =

k

(Bik)T(Akj )

T

2.4 Unit Matrix

(identity matrix) 1 0 00 . . . 00 0 1

1 (2.1)1ij ij

11 =1T = 1 A1 =1A = A

Commutator: a linear operation

[A, B] AB BA[A, B] = 0 ifA and B are diagonal matrices

Diagonal Matrices:

Aij = aiiij

Jacobi Identity:

[A, [B, C]] = [B, [A, C]] [C, [A, B]]

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CHAPTER 2. LINEAR ALGEBRA 2.5. TRACE OF A (SQUARE) MATRIX

2.5 Trace of a (Square) Matrix

(a linear operator)

Tr(A) =

iAii = Tr

AT

Tr(A B) = Tr(C) =

i

Cii =

i

k

AikBki

=

k

i

BkiAik = Tr(BA)

Note Tr([A, B]) = 0

Tr(A + B) = Tr(A) + Tr(B)

2.5.1 Inverse of a (Square) Matrix

A1 A = 1 = A A1

Note 1 1 = 1, thus 1 = 11

(A B)1 = B1 A1 prove

2.6 More on Trace

T r(ABC) = T r(CB A)

T r(a bT) = aT bT r(U+AU) = T r(A) U+U = 1 or T r(B1AB) = T r(A)

T r(S+S) 0 T r(BS B1BT B1 = T r(ST)T r(A) = T rA+ T r(ST)T r(AB) = T r(BA) = T r(B+A+)

T r(ABC) = T r(CAB) = T r(BC A)

T r([A, B]) = 0

T r(AB) = 0 ifA = AT and B = BT

2.7 More on [A,B]

[Sx, Sy] = iSz

Proof: (A B)1 = B1A1Ifx y = 1 then y = x1

associative (A B) (B1 A1) = A (B B1) A1 = (A A1) = 1 thus (B1 A1) = (A B)1

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2.8. DETERMINANTS CHAPTER 2. LINEAR ALGEBRA

2.8 Determinants

det(A) =

A11 A12 A1nA21 A22 A2n

.... . .

... Ann

=i,k...

ijk...A1iA2jA3k . . . (2.2)

ijk... : Levi-Civita symbol (1 even/odd permutation of1, 2, 3 . . ., otherwise 0. (has N! terms)A is a square matrix, (N N)

2.8.1 Laplacian expansion

D =Ni

(1)i+jMij Aij

=

Ni

cij Aij

Mij = minor ij

cij = cofactor = (1)i+jMij

D = kAkk (2.3)

A11 0A21 A22

(2.4)

Ni

= Aij cik = det(A)jk =

Aji cik

det(A) is an antisymmetrized product

Properties: for an N N matrix A

1. The value ofdet(A) = 0 if

any two rows (or columns) are equal

each element of a row (or column) is zero any row (or column) can be represented by a linear combination of the other rows (or columns). In

this case, A is called a singular matrix, and will have one or more of its eigenvalues equal to zero.

2. The value ofdet(A) is unchanged if

two rows (or columns) are swapped, sign changes a multiple of one row (or column) is added to another row (or column) A is transposed det(A) = det(A+) or det(A+) = det(A) = (det(A))

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CHAPTER 2. LINEAR ALGEBRA 2.8. DETERMINANTS

A undergoes unitary transformation det(A) = det(U+AU) (including the unitary tranformation thatdiagonalized A)

det(eA) = etr(A) (2.5)

3. If any row (or column) of A is multiplied by a scalar , the value of the determinat is det(A). If the

whole matrix is multiplied by , then

det(A) = Ndet(A) (2.6)

4. IfA = BC, det(A) = det(B) det(C), but ifA = B + C, det(A) = det(B) + det(C). (det(A) is nota linear operator)

det(AN) = [det(A)]N (2.7)

5. IfA is diagonalized, det(A) = iAii (also, det(1) = 1)

6. det(A)1) = (det(A))1

7. det(A) = (detA) = det(A+

)8. det(A) = det(U+AU) U+U = 1

2.8.2 Applications of Determinants

Wave function:

HFDFT

(x1, x2 xN) = 1N!

1(x1) 2(x1) N(x11(x2) 2(x2)

......

. . . N(xN)

(2.8)Evaluate:

(x1, x2, x3)(x1, x2, x3)d(write all the terms). How does this expression reduce if

i(x)j(x)d = iJ (orthonormal spin orbitals)

J=Jacobian |dxk >= J|dqj >=

i |dqi >< dqi|dxk >

dx1dx2dx3 = det(J)dq1dq2dq3, Jik =xiqk

J

x

q

: Jij =

xiqj

J

q

xij=

qixj

(2.9)

det

J q

x

=

q1x1

q1x2

...

. . .

(2.10)q1 =2x q2 =y q3 =z

dxdydz =1

2dq1 (2.11)

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2.9. GENERALIZED GREENS THEOREM CHAPTER 2. LINEAR ALGEBRA

x+2xx

x=

1

2

x

x=2 dx =

1

2dx

dx =dx

dq1dq1 =

1

2dq1 (2.12)

2.9 Generalized Greens Theorem

Use: v vd =

s

v d (2.13)v

(vLu uLv)d =

sp(xu uv) d (2.14)

L = [p] + q = p +p2 + q

v v(p u + 2u + q)u

u(

p

v +p

2v + q)v d

Note

vqud =

uqv dv

(v [p]u + v (pu) u [p]v v (pu)) d

=

v

( vpu upv) d (2.15)

v pu + v [p]u u pv u [p]v=

s(vpu upv) d

= sp(vu uv) ds

dFi =

k

Fixk

dxk +Fi

tdt =(dr )Fi + Fi

tdt

=

dr + dt

t

Fi (2.16)

L = r p = r mv, v = rA (B C) = B(AC) C(AB) r = rr

L =m(r ( r))=m[(r r) r(r )]=m[(r2r r) rr(rr )]=mr2[ r(r w)] (2.17)

I =mr2 L =I if r = 0

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CHAPTER 2. LINEAR ALGEBRA 2.10. ORTHOGONAL MATRICES

2.10 Orthogonal Matrices

(analogy to aiaj = ij = aTi aj )

AT A = 1 AT = A1AT = A1, therefore AT A = 1 = 1T = (AT A)T

Note det(A) = 1 ifA is orthogonal. Also: A and B orthogonal, then (AB) orthogonal.

Application: Rotation matrices

xi =

j

Aij xj or |xi >=

j

|xj xj |xi >

example:

r

=

sin cos sin sin cos cos cos cos sin

sin

sin cos 0 xyz (2.18) r

= Cxy

z

(2.19)xy

z

= C1 r

= CT r

(2.20)since C1 = CT (C is an orthogonal matrix)

Also Euler angles (we will use later. . .)

2.11 Symmetric/Antisymmetric Matrices

Symmetric means Aij = Aji , or A = AT

Antisymmetric means Aij = Aji , or A = AT

A =1

2

A + AT

+

1

2

A AT (2.21)

symmetric antisymmetric

Note also: AAT

T=

ATT

AT

=AAT

Note: Tr(AB) = 0 if A is symmetric and B is antisymmetric. Thus AAT ATA are symmetric, but

A AT = A AT

Quiz: IfA is an upper triangular matrix, use the Laplacian expansion to determine a formula for det(A) interms of the elements ofA.

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2.12. SIMILARITY TRANSFORMATION CHAPTER 2. LINEAR ALGEBRA

2.12 Similarity Transformation

A = BAB (2.22)

ifB1 = BT (B orthogonal) BABT = orthogonal similarity transformation.

2.13 Hermitian (self-adjoint) Matrices

Note:

(AB) = AB

(AB)+ = B+A+ also (A+)+ = A

H+ = H where H+ (H)T = (HT)A real symmetric matrix is Hermitian or a real Hermitian matrix is symmetric (if a matrix is real, Hermi-

tian=symmetric)

2.14 Unitary Matrix

U+ = U1

A real orthogonal matrix is unitary or a real unitary matrix is orthogonal (if a matrix is real, unitary=orthogonal)

2.15 Comments about Hermitian Matrices and Unitary Tranformations

1. Unitary tranformations are norm preserving

x = Ux (x)+x = x+U+Ux = x+ x

2. More generally, x = Ux, A = UAU+

Ax = UAU+Ux = U(Ax) and (y)+ A x = y+U+UAU+Uxoperation in transformation coordinates = transformation in uniform coordinates = y+Ax (invariant)

3. IfA+ = A, then (Ay)+ x = y+ x = y+ A+ x = y+ A x (Hermitian property)4. IfA+ = A, then (A)+ = (UAU+)+ = UAU+, or (A)+ = A

2.16 More on Hermitian Matrices

C =1

2(C + C+)

Hermitian+

1

2(C C+)

anti-Hermitian=

1

2(C + C+) +

1

2i i(C C+)

Hermitian!Note: C = i[A, B] is Hermitian even if A and B are not, (or iC = [A, B]). C = AB is Hermition if

A = A+, B = B+, and [A, B] = 0. A consequence of this is that Cn is Hermitian if C is Hermitian. Also,

C = eA is Hermitian ifA = A+, but C = eiA is not Hermitian (C is unitary). A unitary matrix in generalis not Hermitian.

f(A) =

k

CkAk is Hermitian ifCk are real

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CHAPTER 2. LINEAR ALGEBRA 2.17. EIGENVECTORS AND EIGENVALUES

2.17 Eigenvectors and Eigenvalues

Solve Ac = ac, (A a1)c = 0 det() = 0 secular equation.

Aci = aici Eigenvalue problem

Aci = iBci Generalized eigenvalue problem

Ac = Bc

Ifc+i B cj = ij then c+i A cj = iijrelation: A = B

12 AB

12 and ci = B

12 ci (Can always transform. . .Lowden) Example: Hartree-Fock/Kohn-

Shon Equations:

FC = SC, Hef fC = SC

For Aci = ici ifA = A+ ai = ai , c+i cj = ij can be chosenci form a complete set

IfA+A = 1, ai = 1, c+i cj = ij det(A) = 1

Hci = ici or Hc = c

c =

c1 c2 c3...

......

(2.23)

= 1 0. . .0 N

(2.24)since c+i cj = ij, c+ c = 1 and also c+ H c = c+c = c+c = hence c is a unitary matrix and is theunitary matrix that diagonalizes H. c+Hc = (eigenvalue spectrum).

2.18 Anti-Hermitian Matrices

Read about them. . .

2.19 Functions of Matrices

UAU+ = a A = U+aU or AU+ = U+a

f(A) = U+

f(a1) . . .f(an)

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2.19. FUNCTIONS OF MATRICES CHAPTER 2. LINEAR ALGEBRA

Power series e.g.

eA =

k=0

1

k!Ak

sin(A) =

k=0(1)k

(2k + 1)!A2k+1

cos(A) =

k=0

(1)k(2k)!

A2k

Note:

A2U+ =AUU+

=AU+Q

=U+QQ

=U+Q2

Q2 Q211 00

. . .

Ak U+ = U+QK

Note, iff(A) =

k CkAk and UAU+ = Q, then

F =f(A)U+

=

k

ckU+Qk

=U+k ckQk

=U+

f(a11) . . .f(ann)

= U+f (2.25)so ifUAU+ = Q, then UFU+ = f and fij = f(ai)ijAlso:

Trace Formula

det(eA) = eT r(A) (special case of det[f(A)] = if(aii))

Baker-Hausdorff Formula

eiGHeiG = H + [iGH] +1

2[iG, [iG, H]] +

Note, if

AU+ = U+QA() = A + 1

so has some eigenvectors, but eigenvalues are shifted.

A()U+ = (A + 1)U+ = U+Q + U+1 = U+(Q + 1) = U+Q()

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CHAPTER 2. LINEAR ALGEBRA 2.20. NORMAL MARICES

2.20 Normal Marices

[A, A+] = 0 (Hermitian and real symmetry are specific cases)

Aci = aici, A+ci = a

i ci, c

+i cj = 0

2.21 Matrix

2.21.1 Real Symmetric

A = AT = A+ ai real cTi cj = ij Hermitian normal

2.21.2 Hermitian

A = A+ ai real c+i cj = ij normal

2.21.3 Normal

[A, A+] = 0 ifAci = aici c+i cj = ij

2.21.4 Orthogonal

UT U = 1 (UT = U1) ai (1) cTi cj = ij unitary, normal

2.21.5 Unitary

U+U = 1 (U+ = U1) ai real (1) c+i cj = ij normalIfUAU+ = a, and U+U = 1 then [A, A+] = 0 and conversely.

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Chapter 3

Calculus of Variations

3.1 Functions and Functionals

Here we consider functions and functionals of a single argument (a variable and function, respectively) in

order to introduce the extension of the conventional function calculus to that of functional calculus.

A function f(x) is a prescription for transforming a numerical argument x into a number; e.g. f(x) =1 + x2 + ex.

A functional F[y] is a prescription for transforming a function argument y(x) into a number; e.g. F[y] =x2x1

y2(x) esx dx.Hence, a functional requires knowledge of its function argument (say y(x)) not at a single numerical point x, ingeneral, but rather over the entire domain of the functions numerical argument x (i.e., over all x in the case ofy(x)). Alternately stated, a functional is often written as some sort of integral (see below) where the argument ofy (we have been referring to it as x) is a dummy integration index that gets integrated out.

In general, a functional F[y] of the 1-dimensional function y(x) may be written

F[y] =x2

x1

f

x, y(x), y(x), y(n)(x),

dx (3.1)

where f is a (multidimensional) function, and y dy/dx , yn dny/dxn. For the purposes here, we willconsider the bounday conditions of the function argument y are such that y, y y(n1) have fixed values atthe endpoints; i.e.,

y(j)(x1) = y(j)1 , y(j)(x2) = y

(j)2 for j = 0, (n 1) (3.2)

where y(j)1 and y

(j)2 are constants, and y

(0) y.In standard function calculus, the derivative of a function f(x) with respect to x is defined as the limiting

process dfdx

x

lim0

f(x + ) f(x)

(3.3)

(read the derivative off with respect to x, evaluated at x). The function derivative indicates how f(x) changeswhen x changes by an infintesimal amount from x to x + .

Analogously, we define the functional derivative of F[y] with respect to the function y at a particular pointx0 by

F

y(x)

y(x0)

lim0

F[y + (x x0)] F[y]

(3.4)

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CHAPTER 3. CALCULUS OF VARIATIONS 3.2. FUNCTIONAL DERIVATIVES

(read, the functional derivative ofF with respect to y, evaluated at the point y(x0)). This functional derivativeindicates how F[y] changes when the function y(x) is changed by an infintesimal amount at the point x = x0from y(x) to y(x) + (x x0).

We now procede formally to derive these relations.

3.2 Functional Derivatives: Formal Development in 1-Dimension

Consider the problem of finding a function y(x) that corresponds to a stationary condition (an extremumvalue) of the functional F[y] of Eq. 3.1, subject to the boundary conditions of Eq. 3.2; i.e., that the function yand a sufficient number of its derivatives are fixed at the boundary. For the purposes of describing variations, we

define the function

y(x, ) = y(x) + (x) = y(x, 0) + (x) (3.5)

where (x) is an arbitrary differentiable function that satisfies the end conditions (j)(x1) = (j)(x2) = 0

for j = 1, (n 1) such that in any variation, the boundary conditions of Eq. 3.2 are preserved; i.e., thaty(j)(x1) = y

(j)

1

and y(j)(x2) = y(j)

2

for j = 0,

(n

1). It follows the derivative relations

y(j)(x, ) = y(j)(x) + (j)(x) (3.6)

d

dy(j)(x, ) = (j)(x) (3.7)

where superscript j (j = 0, n) in parentheses indicates the order of the derivative with respect to x. Toremind, here n is the highest order derivative of y that enters the functional F[y]. For many examples in physicsn = 1 (such as we will see in classical mechanics), and only the fixed end points ofy itself are required; however,in electrostatics and quantum mechanics often higher order derivatives are involved, so we consider the more

general case.

If y(x) is the function that corresponds to an extremum of F[y], we expect that any infintesimal variation(x) away from y that is sufficiently smooth and subject to the fixed-endpoint boundary conditions of Eq. 3.2)

will have zero effect (to first order) on the extremal value. The arbitrary function (x) of course has beendefined to satisfy the differentiability and boundary conditions, and the scale factor allows a mechanism foreffecting infintesimal variations through a limiting procedure approaching = 0. At = 0, the varied functiony(x, ) is the extremal value y(x). Mathematically, this implies

d

d[F[y + ]]=0 = 0

=d

d

x2x1

f(x, y(x, ), y(x, ), ) dx

=0

(3.8)

=

x2x1

f

y

y(x, )

+

f

y

y (x, )

+

dx

= x2x1

fy (x) + f

y (x) + dx

where we have used y(x, )/ = (x), y (x, )/ = (x), etc... If we integrate by parts the term in Eq. 3.8involving (x) we obtainx2

x1

f

y

(x) dx =

f

y

(x)

x2x1

x2

x1

d

dx

f

y

(x) dx

= x2

x1

d

dx

f

y

(x) dx (3.9)

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3.3. VARIATIONAL NOTATION CHAPTER 3. CALCULUS OF VARIATIONS

where we have used the fact that (x1) = (x2) = 0 to cause the boundary term to vanish. More generally, forall the derivative terms in Eq. 3.8 we obtainx2

x1

f

y(j)

(j)(x) dx = (1)(j)

x2x1

dj

dxj

f

y(j)

(x) dx (3.10)

for j = 1, n where we have used the fact that (j)

(x1) = (j)

(x2) = 0 for j = 0, (n 1) to cause theboundary terms to vanish. Substituting Eq. 3.10 in Eq. 3.8 gives

d

d[F[y + ]]=0 =

x2x1

f

y

d

dx

f

y

+

+(1)n dn

dxn

f

y (n)

(x) dx

= 0 (3.11)

Since (x) is an arbitrary differential function subject to the boundary conditions of Eq. 3.2, the terms in bracketsmust vanish. This leads to a generalized form of the Euler equation in one dimension for the extremal value of

F[y] of Eq. 3.4 fy d

dx f

y + + (1)n dn

dxn f

y(n) = 0 (3.12)

In other words, for the function y(x) to correspond to an extremal value of F[y], Eq. 3.12 must be satisfied forall y(x); i.e., over the entire domain of x of y(x). Consequently, solution of Eq. 3.12 requires solving for anentire function - not just a particular value of the function argument x as in function calculus. Eq. 3.12 is referredto as the Euler equation (1-dimensional, in this case), and typically results in a differential equation, the solution

of which (subject to the boundary conditions already discussed) provides the function y(x) that produces theextremal value of the functional F[y].

We next define a more condensed notation, and derive several useful techniques such as algebraic manipu-

lation of functionals, functional derivatives, chain relations and Taylor expansions. We also explicitly link the

functional calculus back to the traditional function calculus in certain limits.

3.3 Variational Notation

We define F[y], the variation of the functional F[y], as

F[y] dd

[F[y + ]]=0

=

x2x1

f

y

d

dx

f

y

+ + (1)n d

n

dxn

f

y (n)

(x) dx

=

x2x1

F

y(x)y(x) dx (3.13)

where (analagously) y(x), the variation of the function y(x), is defined by

y(x) dd

[y(x) + (x)]=0

= (x)

= y(x, ) y(x, 0) (3.14)where we have again used the identity y(x, 0) = y(x). Relating the notation of the preceding section, we have

y(x, ) = y(x) + (x) = y(x) + y(x) (3.15)

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3.4. FUNCTIONAL DERIVATIVES: ELABORATION CHAPTER 3. CALCULUS OF VARIATIONS

3.4.2 Generalization to Functionals of Higher Dimension

The expression for the functional derivative in Eq. 3.18 can be generalized to multidimensional functions in a

straight forward manner.

F

(x1, xN)= f+

n

j=1(1)jN

i=1j

xji f(j)xi (3.24)The functional derivative in the 3-dimensional case is

F

(r)=

f

f

+ 2

f

2

(3.25)

Example: Variational principle for a 1-particle quantum system.

Consider the energy of a 1-particle system in quantum mechanics subject to an external potential v(r). Thesystem is described by the 1-particle Hamiltonian operator

H =

h2

2m2

+ v(r) (3.26)

The expectation value of the energy given a trial wave function (r) is given by

E[] =

(r)

h22m2 + v(r)

(r)d3r

(r)(r)d3r(3.27)

=

H

|Let us see what equation results when we require that the energy is an extremal value with respect to the wave

function; i.e.,

E[](r)

= 0 (3.28)

Where we denote the wave function that produces this extremal value . Since this is a 1-particle system inthe absence of magnetic fields, there is no need to consider spin explicitly or to enforce antisymmetry of the

wave function as we would in a many-particle system. Moreover, there is no loss of generality if we restrict

the wave function to be real (one can double the effort in this example by considering the complex case, but

the manipulations are redundant, and it does not add to the instructive value of the variational technique - and

a students time is valuable!). Finally, note that we have constructed the energy functional E[] to take on anun-normalized wave function and return a correct energy (that is to say, the normalization is built into the energy

expression), alleviating the need to explicitly constrain the wave function to be normalized in the variational

process. We return to this point in a later example using the method of Lagrange multipliers.

Recall from Eq. 3.23 we have

E[]

(r)=

H

(r)| |

(r)

H

/|2 (3.29)Let us consider in more detail the first functional derivative term,

H

(r)=

(r)

(r)

h2

2m2 + v(r)

(r)d3r (3.30)

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CHAPTER 3. CALCULUS OF VARIATIONS 3.4. FUNCTIONAL DERIVATIVES: ELABORATION

where we have dropped the complex conjugation since we consider the case where is real. It is clear that theintegrand, as written above, depends explicitly only on and 2. Using Eq. 3.25 (the specific 3-dimensionalcase of Eq. 3.24), we have that

f

(r)=

h22m

2(r) + 2v(r)(r) (3.31)

f2(r)

=

h22m

(r)

and thus

H

(r)= 2

h22m

2 + v(r)

(r) = 2H(r) = 2H| (3.32)where the last equality simply reverts back to Bra-Ket notation. Similarly, we have that

|(r)

= 2(r) = 2| (3.33)

This gives the Euler equation

E[]

(r)=

2H|| 2|

H /|2 (3.34)

= 2

H||

|

H

|2

= 0

Multiplying through by |, dividing by 2 and substituting in the expression for E[] above we obtain

H| =

H

|

| = E[]| (3.35)

which is, of course, just the stationary-state Schrodinger equation.

3.4.3 Higher Order Functional Variations and Derivatives

Higher order functional variations and functional derivatives follow in a straight forward manner from the cor-

responding first order definitions. The second order variation is 2F = (F), and similarly for higher ordervariations. Second and higher order functional derivatives are defined in an analogous way.

The solutions of the Euler equations are extremals - i.e., stationary points that correspond to maxima, minima

or saddle points (of some order). The nature of the stationary point can be discerned (perhaps not completely) by

consideration of the second functional variation. For a functional F[f], suppose f0 is the function that solves theEuler equation; i.e., that satisfies F[f] = 0 or equivalently [F[f]/f(x)]f=f0 = 0, then

2F =1

2

f(x)

2Ff(x)f(x)

f=f0

f(x) dxdx (3.36)

The stationary point ofF[f0] at f0 can be characterized by

2F 0 : minimum2F = 0 : saddle point (order undetermined)

2F 0 : maximum(3.37)

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3.4. FUNCTIONAL DERIVATIVES: ELABORATION CHAPTER 3. CALCULUS OF VARIATIONS

Example: Second functional derivatives.

Consider the functional for the classical electrostatic energy

J[] =1

2 (r)(r)

|r

r

|d3rd3r (3.38)

what is the second functional derivative

2J[](r)(r)

?

The first functional derivative with respect to (r) is

J[]

(r)=

1

2

(r)

|r r|d3r +

1

2

(r)

|r r|d3r (3.39)

=

(r)

|r r|d3r

Note, r is merely a dummy integration index - it could have been called x, y, r, etc... The important featureis that after integration what results is a function of r - the same r that is indicated by the functional derivativeJ[](r) .

The second functional derivative with respect to (r) is

2J[]

(r)(r)

=

(r)J[]

(r)(3.40)

=

(r)

(r)

|r r|d3r

=

1

|r r|

3.4.4 Integral Taylor series expansions

An integral Taylor expansion for F[f0 + f] is defined as

F[f0 + f] = F[f0]

+

n=11

n! (n)F

f(x1)f(x2)

f(xn)f0f(x1)f(x2) f(xn) dx1dx2 dxn (3.41)For functionals of more than one function, e.g. F[f, g], mixed derivatives can be defined. Typically, for

sufficiently well behaved functionals, the order of the functional derivative operations is not important (i.e., they

commute),

2F

f(x)g(x)=

2F

g(x)f(x)(3.42)

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CHAPTER 3. CALCULUS OF VARIATIONS 3.4. FUNCTIONAL DERIVATIVES: ELABORATION

An integral Taylor expansion for F[f0 + f, g0 + g] is defined as

F[f0 + f, g0 + g] = F[f0, g0]

+

F

f(x)

f0,g0

f(x) dx

+ Fg(x)f0,g0 g(x) dx+

1

2

f(x)

2F

f(x)f(x)

f0,g0

f(x) dxdx

+

f(x)

2F

f(x)g(x)

f0,g0

g(x) dxdx

+1

2

g(x)

2F

g(x)g(x)

f0,g0

g(x) dxdx

+ (3.43)

3.4.5 The chain relations for functional derivatives

From Eq. 3.13, the variation of a functional F[f] is given by

F =

F

f(x)f(x) dx (3.44)

(where it is understood we have dropped the definite integral notation with endpoinds x1 and x2 - it is also validthat the boundaries be at plus or minus infinity). If at each point x, f(x) itself is a functional of another functiong, we write f = f[g(x), x] (an example is the electrostatic potential (r) which at every point r is a functionalof the charge density at all points), we have

f(x) = f(x)g(x) g(x) dx (3.45)which gives the integral chain relation

F =

F

f(x)

f(x)

g(x)g(x) dxdx

=

F

g(x)g(x) dx (3.46)

hence,F

g(x)=

F

f(x)

f(x)

g(x)dx (3.47)

Suppose F[f] is really an ordinary function (a special case of a functional); i.e. F = F(f), then written as afunctional

F(f(x)) =

F(f(x))(x x)dx (3.48)

it follows thatF(f(x))

f(x)=

dF

df(x x) (3.49)

If we take F(f) = f itself, we see thatf(x)

f(x)= (x x) (3.50)

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3.5. HOMOGENEITY AND CONVEXITY CHAPTER 3. CALCULUS OF VARIATIONS

If instead we have a function that takes a functional argument, e.g. g = g(F[f(x)]), then we have

g

f(x)=

g

F[f, x]F[f, x]

f(x)dx

=

dg

dF[f, x](x x) F[f, x

]f(x)

dx

=dg

dF

F

f(x)(3.51)

If the argument f of the functional F[f] contains a parameter ; i.e., f = f(x; ), then the derivative ofF[f]with respect to the parameter is given by

F[f(x; )]

=

F

f(x; )

f(x; )

dx (3.52)

3.4.6 Functional inverses

For ordinary function derivatives, the inverse is defined as (dF/df)1 = df/dF such that (dF/df)1

(df/dF) =

1, and hence is unique. In the case of functional derivatives, the relation between F[f] and f(x) is in general areduced dimensional mapping; i.e., the scalar F[f] is determined from many (often an infinite number) values ofthe function argument f(x).

Suppose we have the case where we have a function f(x), each value of which is itself a functional of anotherfunction g(x). Moreover, assume that this relation is invertable; i.e., that each value of g(x) can be written asa functional of f(x). A simple example would be if f(x) were a smooth function, and g(x) was the Fouriertransform off(x) (usually the x would be called k or or something...). For this case we can write

f(x) =

f(x)

g(x)g(x) dx (3.53)

g(x) = g(x)f(x

)

f(x) dx (3.54)

which leads to

f(x) =

f(x)

g(x)g(x)f(x)

f(x) dx dx (3.55)

providing the reciprocal relation f(x)

g(x)g(x)f(x)

dx =f(x)

f(x)= (x x) (3.56)

We now define the inverse as

f(x)

g(x)1

=g(x)

f(x)

(3.57)

from which we obtain f(x)

g(x)

f(x)

g(x)

1dx = (x x) (3.58)

3.5 Homogeneity and convexity properties of functionals

In this section we define two important properties, homogeneity and convexity, and discuss some of the powerful

consequences and inferences that can be ascribed to functions and functionals that have these properties.

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CHAPTER 3. CALCULUS OF VARIATIONS 3.5. HOMOGENEITY AND CONVEXITY

3.5.1 Homogeneity properties of functions and functionals

A function f(x1, x2 ) is said to be homogeneous of degree k(in all of its degrees of freedom) iff(x1, x2, ) = kf(x1, x2 ) (3.59)

and similarly, a functional F[f] is said to be homogeneous of degree kif

F[f] = kF[f] (3.60)

Thus homogeneity is a type ofscaling relationship between the value of the function or functional with unscaled

arguments and the corresponding values with scaled arguments. If we differentiate Eq. 3.59 with respect to weobtain for the term on the left-hand side

df(x1, x2, )d

=f(x1, x2, )

(xi)

d(xi)

d(3.61)

=

i

xif(x1, x2, )

(xi)(3.62)

(where we note that d(xi)/d = xi), and for the term on the right-hand side

d

d

kf(x1, x2, )

= kk1f(x1, x2, ) (3.63)

Setting = 1 and equating the left and right-hand sides we obtain the important relationi

xif(x1, x2, )

xi= kf(x1, x2, ) (3.64)

Similarly, for homogeneous functionals we can derive an analogous relation

F

f(x)

f(x) dx = kF[f] (3.65)

Sometimes these formulas are referred to as Eulers theorem for homogeneous functions/functionals. These

relations have the important conseuqnce that, for homogeneous functions (functionals), the value of the function

(functional) can be derived from knowledge only of the function (functional) derivative.

For example, in thermodynamics, extensive quantities (such as the Energy, Enthalpy, Gibbs free energy,

etc...) are homogeneous functionals of degree 1 in their extensive variables (like entropy, volume, the number of

particles, etc...). and intensive quantities (such as the pressure, etc...) are homogeneous functionals of degree 0.

Consider the energy as a function of entropy S, volume V, and particle number ni for each type of particle (irepresents a type of particle). Then we have that E = E(S ,V,n1, n2 ) and

E = E(S ,V,n1, n2 ) (3.66)

dE = ESV,ni dS+EVS,ni dV +i EniS,V,nj=i dni (3.67)= T dSpdV +

i

idni

where we have used the identities

ES

V,ni

= T,

EV

S,ni

= p, and

Eni

S,V,nj=i

= i. From the first-order

homogeneity of the extensive quantity E(S ,V,n1, n2 ) we have thatE(S,V,n1, n2, ) = E(S ,V,n1, n2 ) (3.68)

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3.5. HOMOGENEITY AND CONVEXITY CHAPTER 3. CALCULUS OF VARIATIONS

The Euler theorem for first order homogeneous functionals then gives

E =

E

S

V,ni

S+

E

V

S,ni

V +

i

E

ni

S,V,nj=i

ni

= T SpV +i ini (3.69)Taking the total differential of the above equation yields

dE = T dS+ SdT pdV V dp +

i

idni + nidi

Comparison of Eqs. 3.68 and 3.70 gives the well-known Gibbs-Duhem equation

SdT V dp +

i

nidi = 0 (3.70)

Another example is the classical and quantum mechanical virial theorem that uses homogeneity to relate the

kinetic and potential energy. The virial theorem (for 1 particle) can be stated asx

V

x+ y

V

y+ z

V

z

= 2 K (3.71)

Note that the factor 2 arises from the fact that the kinetic energy is a homogeneous functional of degree 2 in theparticle coordinates. If the potential energy is a central potential; i.e., V(r) = C rn (a homogeneous functionalof degree n) we obtain

n V = 2 K (3.72)

In the case of atoms (or molecules - if the above is generalized slightly) V(r) is the Coulomb potential 1/r,

n = 1 and we have < V >= 2 < K > or E = (1/2) < V > since E =< V > + < K >.3.5.2 Convexity properties of functions and functionals

Powerful relations can be derived for functions and functionals that possess certain convexity properties. We

define convexity in three cases, starting with the most general: 1) general functions (functionals), 2) at least

once-differentiable functions (functionals), and 3) at least twice-differentiable functions (functionals).

For a general function (functional) to be convex on the interval I (for functions) or the domain D (for func-tionals) if, for 0 1

f(x1 + (1 )x2) f(x1) + (1 )f(x2) (3.73)F[f1 + (1

)f2]

F[f1] + (1

)F[f2] (3.74)

for x1, x2 I and f1, f2 D. f(x) (F[f]) is said to be strictly convex on the interval if the equality holds onlyfor x1 = x2 (f1 = f2). f(x) (F[f]) is said to be concave (strictly concave) iff(x) (-F[f]) is convex (strictlyconvex).

A once-differentiable function f(x) (or functional F[f]) is convex if and only if

f(x1) f(x2) f(x2) (x1 x2) 0 (3.75)F[f1] F[f2]

F[f]

f(x)

f=f2

(f1(x) f2(x)) dx 0 (3.76)

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CHAPTER 3. CALCULUS OF VARIATIONS 3.5. HOMOGENEITY AND CONVEXITY

for x1, x2 I and f1, f2 D.A twice-differentiable function f(x) (or functional F[f]) is convex if and only if

f(x) 0 (3.77)

2F[f]

f(x)f(x) 0 (3.78)

for x I and f D.An important property of convex functions (functionals) is known as Jensens inequality: For a convext

function f(x) (functional F[f])

f(x) f(x) (3.79)F[f] F[f] (3.80)

where denotes an average (either discreet of continuous) over a positive semi-definite set of weights. Infact, Jensens inequality can be extended to a convex function of a Hermitian operator

f

O

f(O)

(3.81)

where in this case

denotes the quantum mechanical expectation value

||

. Hence, Jensens inequal-

ity is valid for averages of the form

f =

i

Pifi where Pi 0,

i

Pi = 1 (3.82)

f =

P(x)f(x) dx; where P(x) 0,

P(x) dx = 1 (3.83)

f =

(x)f(x) dx; where

(x)(x) dx = 1 (3.84)

The proof is elementary but I dont feel like typing it at 3:00 in the morning. Instead, here is an example of

an application - maybe in a later version...

Example: Convex functionals in statistical mechanics.

Consider the convect function ex (it is an infinitely differentiable function, and has d2(ex)/dx2 = ex, a positive definite secondderivative). Hence e exp(x).

e exp(x) (3.85)

This is a useful relation in statistical mechanics.

Similarly the function xln(x) is convex for x 0. If we consider two sets of probabilities Pi and Pi such that Pi, P

i 0 and

i Pi =

i Pi = 1, then we obtain from Jensens inequality

x ln(x) xln(x) (3.86)Ni

Pi PiPi

ln

Ni

Pi PiPi

Ni

Pi PiPi

ln

PiPi

(3.87)

Note thatN

i Pi Pi/P

i =N

i Pi = 1 and hence the left hand side of the above inequality is zero since 1 ln(1) = 0. The right handside can be reduced by canceling out the Pi factors in the numerator and denominator, which results in

Ni

Pi lnPiPi

0 (3.88)

which is a famous inequality derived by Gibbs, and is useful in providing a lower bound on the entropy: let Pi = 1/Nthen, taking minusthe above equation, we get

Ni

Pi ln (Pi) ln(N) (3.89)

If the entropy is defined as kBN

iPi ln (Pi) where kB is the Boltzmann constant (a positive quantity), then the largest value

the entropy can have (for an ensemble of N discreet states) is kB ln(N), which occurs when all probabilities are equal (the infinitetemperature limit).

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3.6. LAGRANGE MULTIPLIERS CHAPTER 3. CALCULUS OF VARIATIONS

3.6 Lagrange Multipliers

In this section, we outline an elegant method to introduce constraints into the variational procedure. We begin

with the case of a discreet constraint condition, and then outline the generalization to a continuous (pointwise)

set of constraints.

Consider the problem to extremize the functional F[f] subject to a functional constraint condition

G[f] = 0 (3.90)

In this case, the method of Lagrange multipliers can be used. We define the auxiliary function

[f] F[f] G[f] (3.91)

where is a parameter that is yet to be determined. We then solve the variational condition

f(x)=

F

f(x) G

f(x)= 0 (3.92)

Solution of the above equation results, in general, in a infinite set of solutions depending on the continuousparameter lambda. We then have the freedom to choose the particular value of that satisfies the constraintrequirement. Hopefully there exists such a value of, and that value is unique - but sometimes this is not thecase. Note that, iff0(x) is a solution of the constrained variational equation (Eq. 3.92), then

=

F

f(x)

f=f0

/

G

f(x)

f=f0

(3.93)

for any and all values of f0(x)! Often in chemistry and physics the Lagrange multiplier itself has a physicalmeaning (interpretation), and is sometimes referred to as a sensitivity coefficient.

Constraints can be discreet, such as the above example, or continuous. A continuous (or pointwise) set of

constraints can be written

g[f, x] = 0 (3.94)

where the notation g[f, x] is used to represent a simultaneous functional of f and function of x - alternatelystated, g[f, x] is a functional off at every point x. We desire to impose a continuous set of constraints at everypoint x, and for this purpose, we require a Lagrange multiplier (x) that is itself a continuous function ofx. Wethen define (similar to the discreet constraint case above) the auxiliary function

[f] F[f]

(x)g[f, x] dx (3.95)

and then solve

f(x)=

F

f(x)

(x)g[f, x]

f(x)dx = 0 (3.96)

As before, the Lagrange multiplier (x) is determined to satisy the constraint condition of Eq. 3.94. One canconsider (x) to be a continuous (infinite) set of Lagrange multipliers associated with a constraint condition ateach point x.

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CHAPTER 3. CALCULUS OF VARIATIONS 3.7. PROBLEMS

3.7 Problems

3.7.1 Problem 1

Consider the functional

TW

[] =1

8 (r) (r)(r) dddr (3.97)a. Evaluate the functional derivative TW/(r).

b. Let (r) = |(r)|2 (assume is real), and rewrite T[] = TW[|(r)|2]. What does this functionalrepresent in quantum mechanics?

c. Evaluate the functional derivative T[]/(r) directly and verify that it is identical to the functionalderivative obtained using the chain relation

T[]

(r)=

TW[]

(r)=

TW[]

(r) (r

)(r)

d3r

3.7.2 Problem 2


Ex[] =

4/3(r)

c b2x3/2(r)

d3r (3.98)

where c and b are constants, and

x(r) =|(r)|4/3(r)

(3.99)

evaluate the functional derivativeEx[](r)

.

3.7.3 Problem 3

This problem is an illustrative example of variational calculus, vector calculus and linear algebra surrounding an

important area of physics: classical electrostatics.

The classical electrostatic energy of a charge distribution (r) is given by

J[] =1

2

(r)(r)|r r| d

3rd3r (3.100)

This is an example of a homogeneous functional of degree 2, that is to say J[] = 2J[].

3.7.3.1 Part A

Show that

J[] =1

k

J

(r)

(r)d3r (3.101)

where k = 2. Show that the quantity

J(r)

= (r) where (r) is the electrostatic potential due to the charge

distribution (r).

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3.7. PROBLEMS CHAPTER 3. CALCULUS OF VARIATIONS

3.7.3.2 Part B

The charge density (r) and the the electrostatic potential (r) are related by formula derived above. Withappropriate boundary conditions, an equivalent condition relating (r) and (r) is given by the Poisson equation

2(r) = 4(r)

Using the Poisson equation and Eq. 3.101 above, write a new functional W1[] for the electrostatic energy with as the argument instead of and calculate the functional derivative W1(r) .

3.7.3.3 Part C

Rewrite W1[] above in terms of the electrostatic (time independant) field E = assuming that the quantity(r)(r) vanishes at the boundary (e.g., at |r| = ). Denote this new functional W2[]. W2[] should haveno explicit dependence on itself, only through terms involving .

3.7.3.4 Part D

Use the results of the Part C to show that

J[] = W1[] = W2[] 0for any and connected by the Poisson equation and subject to the boundary conditions described in Part C.Note: (r) can be either positive OR negative or zero at different r.

3.7.3.5 Part E

Show explicitlyW1

(r)=

W2(r)

3.7.4 Problem 4

This problem is a continuation of problem 3.

3.7.4.1 Part F

Perform an integral Taylor expansion of J[] about the reference charge density 0(r). Let (r) = (r) 0(r). Similarly, let (r), 0(r) and (r) be the electrostatic potentials associated with (r), 0(r) and (r),respectively.

Write out the Taylor expansion to infinite order. This is not an infinite problem! At what order does the

Taylor expansion become exact for any 0(r) that is sufficiently smooth?

3.7.4.2 Part G

Suppose you have a density (r), but you do not know the associated electrostatic potential (r). In other words,for some reason it is not convenient to calculate (r) via

(r) =

(r)

|r r|d3r

However, suppose you know a density 0(r) that closely resembles (r), and for which you do know the asso-ciated electrostatic potential 0(r). So the knowns are (r), 0(r) and 0(r) (but NOT (r)!) and the goal isto approximate J[] in the best way possible from the knowns. Use the results of Part F to come up with a newfunctional W3[, 0, 0] for the approximate electrostatic energy in terms of the known quantities.

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3.7.4.3 Part H


U[] =

(r)(r)d3r +

1

8

(r)2(r)d3r (3.102)

where (r) is notnecessarily the electrostatic potential corresponding to (r), but rather a trial function indepen-

dent of(r). Show thatU[]

(r)= 0

leads to the Poisson equation; i.e., the (r) that produces an extremum of U[] is, in fact, the electrostaticpotential (r) corresponding to (r).

Note: we could also have written the functional U[] in terms of the trial density (r) as

U[] =

(r)(r)|r r| d

3rd3r 12

(r)(r)|r r| d

3rd3r (3.103)

withe variational conditionU[]

(r) = 0

3.7.4.4 Part I

Show that U[0] and U[0] of Part H are equivalent to the expression for W3[, 0, 0] in Part G. In other words,the functional W3[, 0, 0] shows you how to obtain the best electrostatic energy approximation for J[]given a reference density 0(r) for which the electrostatic potential 0(r) is known. The variational conditionU[]

(r)= 0 or

U[](r) = 0 of Part H provides a prescription for obtaining the best possible model density and

model potential (r). This is really useful!!

3.7.4.5 Part JWe now turn toward casting the variational principle in Part H into linear-algebraic form. We first expand the

trial density (r) as

(r) =

Nfk

ckk(r)

where the k(r) are just a set ofNf analytic functions (say Gaussians, for example) for which it assumed we cansolve or in some other way conveniently obtain the matrix elements

Ai,j =

i(r)j (r

)|r r| d

3rd3r

and

bi = (r)i(r)

|r r| d3rd3r

so A is an Nf Nf square, symmetric matrix and b is an Nf 1 column vector. Rewrite U[] of Part I as amatrix equation U[c] involving the A matrix and b vector defined above. Solve the equation

U[c]

c= 0

for the coefficient vector c to give the best model density (r) (in terms of the electrostatic energy).

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3.7. PROBLEMS CHAPTER 3. CALCULUS OF VARIATIONS

3.7.4.6 Part K

Repeat the excercise in Part J with an additional constraint that the model density (r) have the same normaliza-tion as the real density (r); i.e., that

(r)d3r = (r)d3r = N (3.104)or in vector form

cT d = N (3.105)where di =

i(r)d3r. In other words, solve

U[c] (cT d N)

= 0

for c() in terms of the parameter , and determine what value of the Lagrange multiplier satisfies theconstraint condition of Eq. 3.105.

3.7.4.7 Part L

In Part J you were asked to solve an unconstrained variational equation, and in Part K you were asked to solve

for the more general case of a variation with a single constraint. 1) Show that the general solution of c()(the superscript indicates that c() variational solution and not just an arbitrary vector) of Part K reduces tothe unconstrained solution of Part J for a particular value of (which value?). 2) Express c() as c() =c(0) + c() where c() is the unconstrained variational solution c(0) when the constraint condition isturned on. Show that indeed, U[c

(0)] U[c()]. Note that this implies the extremum condition correspondsto a maximum. 3) Suppose that the density (r) you wish to model by (r) can be represented by

(r) =

k

xkk(r) (3.106)

where the functions k(r) are the same functions that were used to expand (r). Explicitly solve for c(0), andc() for this particular (r).

3.7.4.8 Part M

In Part J you were asked to solve an unconstrained variational equation, and in Part K you were asked to solve

for the more general case of a variation with a single constraint. You guessed it - now we generalize the solution

to an arbitrary number of constraints (so long as the number of constraints Nc does not exceed the number ofvariational degrees of freedom Nf - which we henceforth will assume). For example, we initially considered asingle normalization constraint that the model density (r) integrate to the same number as the reference density(r), in other words

(r)d3r = (r)d3r NThis constraint is a specific case of more general form oflinear constraint

(r)fn(r)d3r =

(r)fn(r)d

3r yn

For example, iff1(r) = 1 we recover the original constraint condition with y1 = N, which was that the monopolemoment of equal that of. As further example, if f2(r) = x, f3(r) = y, and f4(r) = z, then we would alsorequire that each component of the dipole moment of equal those of , and in general, if fn(r) = r

lYlm(r)

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where the functions Ylm are spherical harmonics, we could constrain an arbitrary number of multipole momentsto be identical.

This set ofNc constraint conditions can be written in matrix form as

DT c = y

where the matrix D is an Nf Nc matrix defined as

Di,j =

i(r)fj(r)d

3r

and the y is an Nc 1 column vector defined by

yj =

(r)fj (r)d

3r

Solve the general constrained variation

U[c] T (DT c y) = 0for the coefficients c() and the Nc 1 vector of Lagrange multipliers . Verify that 1) if = 0 one recoversthe unconstrained solution of Part J, and 2) = 1 (where 1 is just a vector of 1s) recovers the solution for thesingle constraint condition of Part K.

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Chapter 4

Elementary Principles of Classical Mechanics

4.1 Mechanics of a system of particles

4.1.1 Newtons laws

1. Every object in a state of uniform motion tends to remain in that state unless acted on by an external force.

vi = ri =d

dtri (4.1)

pi = miri = mivi (4.2)

2. The force is equal to the change in momentum per change in time.

Fi =d

dtpi pi (4.3)

3. For every action there is an equal and opposite reaction.

Fij = Fij (weak law), not always true

e.g. Fij = iV(rij), e.g. Biot-Savart law moving e

Fij = fij rij = Fij = fji rji (strong law)

e.g. Fij = iV(|rij |), e.g. Central force problem

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CHAPTER 4. CLASSICAL MECHANICS 4.1. MECHANICS OF A SYSTEM OF PARTICLES

4.1.2 Fundamental definitions

vi =d

dtri ri (4.4)

ai =d

dt

vi =d2

dt2

ri = ri (4.5)

pi =mivi = miri (4.6)

Li =ri pi (4.7)Ni =ri Fi = Li (4.8)Fi =pi =

d

dt(miri) (4.9)

Ni = ri pi (4.10)= ri d

dt(miri)

= ddt

(ri pi)

=d

dtLi

= Li

Proof:

d

dt(ri pi) =ri pi + ri pi (4.11)

=vi mvi + ri pi=0 + ri

pi

If ddt A(t) = 0, At =constant, and A is conserved.

A conservative force field (or system) is one for which the work required to move a particle in a closed

loop vanishes.F ds = 0 note: F = V(r), then

cV(r) ds =

s(V(r)

) nda (4.12)

c

A ds = s

( A) nda (Stokes Theorem) (4.13)

System of particles mi = mi(t)

Fixed mass, strong law ofa and r on internal forces.

Suppose Fi =

j Fji + F(e)i and fij = fji

Fij = 0

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4.1. MECHANICS OF A SYSTEM OF PARTICLES CHAPTER 4. CLASSICAL MECHANICS

rij Fij = rji Fji = 0 (4.14)Strong law on Fij

Fi = Pi =d

dt mid

dtri (4.15)

=d2

dt2miri, for mi = mi(t)

let R =i miriM and M =

i mi

i

Fi =d2

dt2

i

miri (4.16)

=Md2

dt2

i

miriM

=Md2

Rdt2

=

i

F(e)i +

i

j

Fij

Note:

i

j Fij =

i

j>i(Fij + Fji ) = 0

P Md2R

dt2=

i

F(e)i F(e) (4.17)

P =i M

dri

dt (4.18)

=d

dtP

Ni =ri Fi (4.19)=ri Pi=Li

Ni =i Ni = i Li = L (4.20)=

i

ri Fi =

i

ri

j

Fji + F(e)i

=

ij

ri Fji +

i

ri F(e)i

=

i

ri F(e)i = N(e) = Li

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i

ji

ri Fji = 0 (4.21)

i

ji

ri Fji

Note, although

P =

i

Pi (4.22)

=

i

mid

dtri

=md

dt(4.23)

V dRdt

= i vi = idridt

L =

i

Li (4.24)

=

i

ri Pi

ri =(ri R) + R = ri + R (4.25)ri =r

i + R (4.26)

pi =pi + miV (4.27)

(Vi =Vi + V) (4.28)

L =

i

ri Pi +

i

Ri Pi +

i

ri miVi +

i

Ri miVi (4.29)

Note i

miri =

i

miri

i

miR (4.30)

=MR MR = 0hence

i

ri miV = i

miri V = 0 (4.31)

i

R Pi =

i

R miVi (4.32)

=R ddt

i

miri

=0

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4.1. MECHANICS OF A SYSTEM OF PARTICLES CHAPTER 4. CLASSICAL MECHANICS

L =

i

ri Pi L about C.O.M.

+ R MV L of C.O.M.

(4.33)

For a system of particles obeying Newtons equations of motion, the work done by the system equals the

difference in kinetic energy.

Fi = miVi, dri =dridt

dt (4.34)

W12 =

i

21

Fi dri (4.35)

=

i

21

miVi Vidt

=

i21

1

2mi

d

dt(Vi Vi)dt

=i

12

miVi Vi21

= 12

miV2i21

= T2 T1

The kinetic energy can be expressed as a kinetic energy of the center of mass and a T of the particles relative to

the center of mass.

T =1

2

i

mi(V + Vi) (V + Vi) (4.36)

=1

2

i

miV2

12

MV2

+

i

miVi V

(i miV

i)V

+1

2

i

mi(Vi)2

internalNote i

miVi =

i

miVi MV = 0

Proof i

miVi =

d

dt

i

miri (4.37)

=Md

dt

i

miriM

=MdR

dt=MV

W12 = T2 T1 =

i

21

Fi dri (4.38)

In the special case that

Fi = F(e)i +

j

Fij where F(e)i = iVi and Fij = iVij (|ri rj |)

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Note

iVij (rij ) =jVi, Vij(rij) (4.39)=Vji(rij)

=1rij

dV

drij (ri rj )= Fji

W12 =T2 T1 (4.40)

=

i

21

iVi dri +

i

j

21

iVij (rij ) fri

Note

2

1

iVi dri = 2

1

(dri i)Vi (4.41)

= 21

dVi = Vi21

where

dri i = dx ddx

+ dyd

dy+ dz

d

dz(4.42)

i

j

Aij =

i

ji

Aji +

i

Aii (4.43)

i j 2

1 iV

ijr

ij dr

i=

i j


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4.2. CONSTRAINTS CHAPTER 4. CLASSICAL MECHANICS

W12 =T2 T1 (4.46)= (V2 V1)=V1 V2

4.2 Constraints

Constraints are scleronomous if they do not depend explicitly on time, and are rheonomous if they do.

(fr1, r2, . . ., rN, t)= 0 (holonomic)

e.g. rigid body (ri rj)2 = C2ijNonholonomic: r2a2 0, e.g. container boundary

Nonholonomic constaints cannot be used to eliminate dependent variables.

r1 . . . rN q1 . . . qNK; qNK+1 . . . qNFor holonomic systems with applied forces derivable from a scalar potential with workless constraints, a La-grangian can always be defined.

Constraints are artificial...name one that is not...?

ri(q1, q2 . . . q NK, t; qNK+1, qN)

4.3 DAlemberts principle

Fi = F(a)i + fi F

(a)i = applied force

fi = constraint force

ri(t) = virtual (infintesimal) displacement, so small that Fi does not change, consistent with the forces andconstraints at the time t.

We consider only constraint forces fi that do no net virtual work on the system (

i fi vi = 0) since:

W12 =

21

fi dri (4.47)

Dynamics a Set of Notes on Theoretical Physical Chemistry - Jaclyn Steen

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Transcript of Dynamics a Set of Notes on Theoretical Physical Chemistry - Jaclyn Steen