DOCSLIB.ORG

Lectures on Theoretical Mechanics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Lectures on Theoretical Mechanics Dietrich Belitz Department of Physics University of Oregon Contents Acknowledgments, and Disclaimer 0 1 Mathematical principles of mechanics 1 x 1 Philosophical comments . 1 x 1.1 The role of theoretical physics . 1 x 1.2 The structure of a physical theory . 2 x 2 Review of basic mathematical concepts and notation . 2 x 2.1 Number sets and functions . 2 x 2.2 Differentiation . 3 x 2.3 Stationary points . 5 x 2.4 Ordinary differential equations . 7 x 2.5 Integration . 8 x 2.6 Paths and path functionals . 9 x 2.7 Surfaces . 12 x 2.8 Line and surface integrals . 12 x 2.9 The implicit function theorem . 15 x 3 Calculus of variations . 16 x 3.1 Three classic problems . 16 x 3.2 Path neighborhoods . 17 x 3.3 A fundamental lemma . 18 x 3.4 The Euler-Lagrange equations . 19 x 4 General discussion of the Euler-Lagrange equations . 21 x 4.1 First integrals of the Euler-Lagrange equations . 21 x 4.2 Example: The brachistochrone problem . 22 x 4.3 Covariance of the Euler-Lagrange equations . 25 x 4.4 Gauge invariance . 26 x 4.5 Nöther’s theorem . 27 x 5 The principle of least action . 30 x 5.1 The axioms of classical mechanics . 30 x 5.2 Conservation laws . 31 x 6 Problems for Chapter 1 . 32 2 Mechanics of point masses 39 x 1 Point masses in potentials . 39 x 1.1 A single free particle . 39 i ii Contents x 1.2 Galileo’s principle of relativity . 41 x 1.3 Potentials . 42 x 1.4 The equations of motion . 44 x 1.5 Systems of point masses . 46 x 2 Simple examples for the motion of point masses . 49 x 2.1 Galileo’s law of falling bodies . 49 x 2.2 Particle on an inclined plane . 53 x 2.3 Particle on a rotating pole . 54 x 2.4 The harmonic oscillator . 55 x 3 One-dimensional conservative systems . 57 x 3.1 Definition of the model . 57 x 3.2 Solution to the equations of motion . 57 x 3.3 Unbounded motion . 58 x 3.4 Bounded motion . 62 x 3.5 Equilibrium positions . 63 x 4 Motion in a central field . 64 x 4.1 Reduction to a one-dimensional problem . 64 x 4.2 Kepler’s second law . 65 x 4.3 Radial motion . 67 x 4.4 Equation of the orbit . 68 x 4.5 Classification of orbits . 69 x 4.6 Kepler’s problem . 72 x 4.7 Kepler’s laws of planetary motion . 78 x 5 Introduction to perturbation theory . 79 x 5.1 General concept . 79 x 5.2 Digression - Fourier series . 80 x 5.3 Oscillations and Fourier series . 83 x 5.4 Summary thus far, and the anharmonic oscillator . 84 x 5.5 Perturbation theory for the central field problem . 92 x 5.6 Perturbations of Kepler’s problem . 94 x 5.7 Digression - introduction to potential theory . 96 x 5.8 Mercury’s perihelion advance due to an oblate sun . 102 x 6 Scattering theory . 104 x 6.1 Scattering experiments . 104 x 6.2 Classical theory for the scattering cross section . 106 x 6.3 Scattering by a central potential . 107 x 6.4 Rutherford scattering . 108 x 6.5 Scattering by a hard sphere . 110 x 7 N-particle systems . 111 x 7.1 Closed N-particle systems . 111 x 7.2 The two-body problem . 116 x 8 Problems for Chapter 2 . 119 Contents iii 3 The rigid body 128 x 1 Mathematical preliminaries - vector spaces and tensor spaces . 128 x 1.1 Metric vector spaces . 128 x 1.2 Coordinate transformations . 131 x 1.3 Proper coordinate systems . 132 x 1.4 Proper coordinate transformations . 133 x 1.5 Tensor spaces . 135 x 2 Orthogonal transformations . 138 x 2.1 Three-dimensional Euclidean space . 138 x 2.2 Special orthogonal transformations - the group SO (3) . 140 x 2.3 Rotations about a fixed axis . 142 x 2.4 Infinitesimal rotations - the generators of SO (3) . 144 x 2.5 Euler angles . 145 x 3 The rigid body model . 148 x 3.1 parameterization of a rigid body . 148 x 3.2 Angular velocity . 150 x 3.3 The inertia tensor . 155 x 3.4 Classification of rigid bodies . 157 x 3.5 Angular momentum of a rigid body . 157 x 3.6 The continuum model of rigid bodies . 159 x 3.7 The Euler equations . 160 x 4 Simple examples of rigid-body motion . 161 x 4.1 The physical pendulum . 161 x 4.2 Cylinder on an inclined plane . 162 x 4.3 The force-free symmetric top . 164 x 4.4 The force-free asymmetric top . 165 x 5 Problems for Chapter 3 . 170 Acknowledgments These lectures are based in part on notes taken in lectures delivered by Prof. Wolfgang Götze at the Technical University Munich between 1976 and 1985. In their current form they have been taught as a 15-week course for first-year graduate students at the University of Oregon. The original handwritten notes were typeset by Spencer Alexander during the academic year 2013/14. Julian Smith helped with the first round of proofreading. Thanks are due to everyone who has alerted me to typos since. All remaining mistakes and misconceptions are, of course, the author’s responsibility. Disclaimer These notes are still a work in progress. If you notice any mistakes, whether it’s trivial.
Recommended publications
  • Parametric Surfaces and 16.6 Their Areas Parametric Surfaces

    Parametric Surfaces and 16.6 Their Areas Parametric Surfaces

    Parametric Surfaces and 16.6 Their Areas Parametric Surfaces 2 Parametric Surfaces Similarly to describing a space curve by a vector function r(t) of a single parameter t, a surface can be expressed by a vector function r(u, v) of two parameters u and v. Suppose r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k is a vector-valued function defined on a region D in the uv-plane. So x, y, and, z, the component functions of r, are functions of the two variables u and v with domain D. The set of all points (x, y, z) in R3 s.t. x = x(u, v), y = y(u, v), z = z(u, v) and (u, v) varies throughout D, is called a parametric surface S. Typical surfaces: Cylinders, spheres, quadric surfaces, etc. 3 Example 3 – Important From Book The vector equation of a plane through (x0, y0, z0) and containing vectors is rather 4 Example – Point on Surface? Does the point (2, 3, 3) lie on the given surface? How about (1, 2, 1)? 5 Example – Identify the Surface Identify the surface with the given vector equation. 6 Example – Identify the Surface Identify the surface with the given vector equation. 7 Example – Find a Parametric Equation The part of the hyperboloid –x2 – y2 + z = 1 that lies below the rectangle [–1, 1] X [–3, 3]. 8 Example – Find a Parametric Equation The part of the cylinder x2 + z2 = 1 that lies between the planes y = 1 and y = 3. 9 Example – Find a Parametric Equation Part of the plane z = 5 that lies inside the cylinder x2 + y2 = 16.
  • Chapter 2 Introduction to Electrostatics

    Chapter 2 Introduction to Electrostatics

    Chapter 2 Introduction to electrostatics 2.1 Coulomb and Gauss’ Laws We will restrict our discussion to the case of static electric and magnetic fields in a homogeneous, isotropic medium. In this case the electric field satisfies the two equations, Eq. 1.59a with a time independent charge density and Eq. 1.77 with a time independent magnetic flux density, D (r)= ρ (r) , (1.59a) ∇ · 0 E (r)=0. (1.77) ∇ × Because we are working with static fields in a homogeneous, isotropic medium the constituent equation is D (r)=εE (r) . (1.78) Note : D is sometimes written : (1.78b) D = ²oE + P .... SI units D = E +4πP in Gaussian units in these cases ε = [1+4πP/E] Gaussian The solution of Eq. 1.59 is 1 ρ0 (r0)(r r0) 3 D (r)= − d r0 + D0 (r) , SI units (1.79) 4π r r 3 ZZZ | − 0| with D0 (r)=0 ∇ · If we are seeking the contribution of the charge density, ρ0 (r) , to the electric displacement vector then D0 (r)=0. The given charge density generates the electric field 1 ρ0 (r0)(r r0) 3 E (r)= − d r0 SI units (1.80) 4πε r r 3 ZZZ | − 0| 18 Section 2.2 The electric or scalar potential 2.2 TheelectricorscalarpotentialFaraday’s law with static fields, Eq. 1.77, is automatically satisfied by any electric field E(r) which is given by E (r)= φ (r) (1.81) −∇ The function φ (r) is the scalar potential for the electric field. It is also possible to obtain the difference in the values of the scalar potential at two points by integrating the tangent component of the electric field along any path connecting the two points E (r) d` = φ (r) d` (1.82) − path · path ∇ · ra rb ra rb Z → Z → ∂φ(r) ∂φ(r) ∂φ(r) = dx + dy + dz path ∂x ∂y ∂z ra rb Z → · ¸ = dφ (r)=φ (rb) φ (ra) path − ra rb Z → The result obtained in Eq.
  • The Celestial Mechanics of Newton

    The Celestial Mechanics of Newton

    GENERAL I ARTICLE The Celestial Mechanics of Newton Dipankar Bhattacharya Newton's law of universal gravitation laid the physical foundation of celestial mechanics. This article reviews the steps towards the law of gravi­ tation, and highlights some applications to celes­ tial mechanics found in Newton's Principia. 1. Introduction Newton's Principia consists of three books; the third Dipankar Bhattacharya is at the Astrophysics Group dealing with the The System of the World puts forth of the Raman Research Newton's views on celestial mechanics. This third book Institute. His research is indeed the heart of Newton's "natural philosophy" interests cover all types of which draws heavily on the mathematical results derived cosmic explosions and in the first two books. Here he systematises his math­ their remnants. ematical findings and confronts them against a variety of observed phenomena culminating in a powerful and compelling development of the universal law of gravita­ tion. Newton lived in an era of exciting developments in Nat­ ural Philosophy. Some three decades before his birth J 0- hannes Kepler had announced his first two laws of plan­ etary motion (AD 1609), to be followed by the third law after a decade (AD 1619). These were empirical laws derived from accurate astronomical observations, and stirred the imagination of philosophers regarding their underlying cause. Mechanics of terrestrial bodies was also being developed around this time. Galileo's experiments were conducted in the early 17th century leading to the discovery of the Keywords laws of free fall and projectile motion. Galileo's Dialogue Celestial mechanics, astronomy, about the system of the world was published in 1632.
  • Multivariable and Vector Calculus

    Multivariable and Vector Calculus

    Multivariable and Vector Calculus Lecture Notes for MATH 0200 (Spring 2015) Frederick Tsz-Ho Fong Department of Mathematics Brown University Contents 1 Three-Dimensional Space ....................................5 1.1 Rectangular Coordinates in R3 5 1.2 Dot Product7 1.3 Cross Product9 1.4 Lines and Planes 11 1.5 Parametric Curves 13 2 Partial Differentiations ....................................... 19 2.1 Functions of Several Variables 19 2.2 Partial Derivatives 22 2.3 Chain Rule 26 2.4 Directional Derivatives 30 2.5 Tangent Planes 34 2.6 Local Extrema 36 2.7 Lagrange’s Multiplier 41 2.8 Optimizations 46 3 Multiple Integrations ........................................ 49 3.1 Double Integrals in Rectangular Coordinates 49 3.2 Fubini’s Theorem for General Regions 53 3.3 Double Integrals in Polar Coordinates 57 3.4 Triple Integrals in Rectangular Coordinates 62 3.5 Triple Integrals in Cylindrical Coordinates 67 3.6 Triple Integrals in Spherical Coordinates 70 4 Vector Calculus ............................................ 75 4.1 Vector Fields on R2 and R3 75 4.2 Line Integrals of Vector Fields 83 4.3 Conservative Vector Fields 88 4.4 Green’s Theorem 98 4.5 Parametric Surfaces 105 4.6 Stokes’ Theorem 120 4.7 Divergence Theorem 127 5 Topics in Physics and Engineering .......................... 133 5.1 Coulomb’s Law 133 5.2 Introduction to Maxwell’s Equations 137 5.3 Heat Diffusion 141 5.4 Dirac Delta Functions 144 1 — Three-Dimensional Space 1.1 Rectangular Coordinates in R3 Throughout the course, we will use an ordered triple (x, y, z) to represent a point in the three dimensional space.
  • Math 1300 Section 4.8: Parametric Equations A

    Math 1300 Section 4.8: Parametric Equations A

    MATH 1300 SECTION 4.8: PARAMETRIC EQUATIONS A parametric equation is a collection of equations x = x(t) y = y(t) that gives the variables x and y as functions of a parameter t. Any real number t then corresponds to a point in the xy-plane given by the coordi- nates (x(t); y(t)). If we think about the parameter t as time, then we can interpret t = 0 as the point where we start, and as t increases, the parametric equation traces out a curve in the plane, which we will call a parametric curve. (x(t); y(t)) • The result is that we can \draw" curves just like an Etch-a-sketch! animated parametric circle from wolfram Date: 11/06. 1 4.8 2 Let's consider an example. Suppose we have the following parametric equation: x = cos(t) y = sin(t) We know from the pythagorean theorem that this parametric equation satisfies the relation x2 + y2 = 1 , so we see that as t varies over the real numbers we will trace out the unit circle! Lets look at the curve that is drawn for 0 ≤ t ≤ π. Just picking a few values we can observe that this parametric equation parametrizes the upper semi-circle in a counter clockwise direction. (0; 1) • t x(t) y(t) 0 1 0 p p π 2 2 •(1; 0) 4 2 2 π 2 0 1 π -1 0 Looking at the curve traced out over any interval of time longer that 2π will indeed trace out the entire circle.
  • Parametric Curves You Should Know

    Parametric Curves You Should Know

    Parametric Curves You Should Know Straight Lines Let a and c be constants which are not both zero. Then the parametric equations determining the straight line passing through (b; d) with slope c=a (i.e., the line y − d = c=a(x − b)) are: x(t) = at + b ; −∞ < t < 1: y(t) = ct + d Note that when c = 0, the line is horizontal of the form y = d, and when a = 0, the line is vertical of the form x = b. 15 10 x(t)= 2t+ 3, y(t)=t+4 5 x(t)=3-2t, y(t)= 2t+1 �������� x(t)=t+ 2, y(t)=3- 3t x(t)= 3, y(t)=2+ 3t -10 -5 5 10 15 x(t)=2+3t, y(t)=3 -5 -10 Figure 1 A collection of lines whose parametric equations are given as above. Circles Let r > 0 and let x0 and y0 be real numbers. Then the parametric equations determining the circle of radius r centered at (x0; y0) are: x(t) = r cos t + x 0 ; 0 < t < 2π: (1) y(t) = r sin t + y0 To get a circular arc instead of the full circle, restrict the t-values in (1) to t1 < t < t2. 1 1 1 2 3 4 5 2 3 4 5 (3,-1) -1 -1 (3,-1) -2 -2 -3 -3 (a) 0 ≤ t ≤ 2π (b) π=4 ≤ t ≤ 7π=6 Figure 2 The circle x(t) = 2 cos t + 3, y(t) = 2 sin t − 1 and a circular arc thereof. Note that the center is at (3; 1).
  • Polynomial Curves and Surfaces

    Polynomial Curves and Surfaces

    Polynomial Curves and Surfaces Chandrajit Bajaj and Andrew Gillette September 8, 2010 Contents 1 What is an Algebraic Curve or Surface? 2 1.1 Algebraic Curves . .3 1.2 Algebraic Surfaces . .3 2 Singularities and Extreme Points 4 2.1 Singularities and Genus . .4 2.2 Parameterizing with a Pencil of Lines . .6 2.3 Parameterizing with a Pencil of Curves . .7 2.4 Algebraic Space Curves . .8 2.5 Faithful Parameterizations . .9 3 Triangulation and Display 10 4 Polynomial and Power Basis 10 5 Power Series and Puiseux Expansions 11 5.1 Weierstrass Factorization . 11 5.2 Hensel Lifting . 11 6 Derivatives, Tangents, Curvatures 12 6.1 Curvature Computations . 12 6.1.1 Curvature Formulas . 12 6.1.2 Derivation . 13 7 Converting Between Implicit and Parametric Forms 20 7.1 Parameterization of Curves . 21 7.1.1 Parameterizing with lines . 24 7.1.2 Parameterizing with Higher Degree Curves . 26 7.1.3 Parameterization of conic, cubic plane curves . 30 7.2 Parameterization of Algebraic Space Curves . 30 7.3 Automatic Parametrization of Degree 2 Curves and Surfaces . 33 7.3.1 Conics . 34 7.3.2 Rational Fields . 36 7.4 Automatic Parametrization of Degree 3 Curves and Surfaces . 37 7.4.1 Cubics . 38 7.4.2 Cubicoids . 40 7.5 Parameterizations of Real Cubic Surfaces . 42 7.5.1 Real and Rational Points on Cubic Surfaces . 44 7.5.2 Algebraic Reduction . 45 1 7.5.3 Parameterizations without Real Skew Lines . 49 7.5.4 Classification and Straight Lines from Parametric Equations . 52 7.5.5 Parameterization of general algebraic plane curves by A-splines .
  • TOPICS in CELESTIAL MECHANICS 1. the Newtonian N-Body Problem

    TOPICS in CELESTIAL MECHANICS 1. the Newtonian N-Body Problem

    TOPICS IN CELESTIAL MECHANICS RICHARD MOECKEL 1. The Newtonian n-body Problem Celestial mechanics can be defined as the study of the solution of Newton's differ- ential equations formulated by Isaac Newton in 1686 in his Philosophiae Naturalis Principia Mathematica. The setting for celestial mechanics is three-dimensional space: 3 R = fq = (x; y; z): x; y; z 2 Rg with the Euclidean norm: p jqj = x2 + y2 + z2: A point particle is characterized by a position q 2 R3 and a mass m 2 R+. A motion of such a particle is described by a curve q(t) where t runs over some interval in R; the mass is assumed to be constant. Some remarks will be made below about why it is reasonable to model a celestial body by a point particle. For every motion of a point particle one can define: velocity: v(t) =q _(t) momentum: p(t) = mv(t): Newton formulated the following laws of motion: Lex.I. Corpus omne perservare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare 1 Lex.II. Mutationem motus proportionem esse vi motrici impressae et fieri secundem lineam qua vis illa imprimitur. 2 Lex.III Actioni contrarium semper et aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi. 3 The first law is statement of the principle of inertia. The second law asserts the existence of a force function F : R4 ! R3 such that: p_ = F (q; t) or mq¨ = F (q; t): In celestial mechanics, the dependence of F (q; t) on t is usually indirect; the force on one body depends on the positions of the other massive bodies which in turn depend on t.
  • Perturbation Theory in Celestial Mechanics

    Perturbation Theory in Celestial Mechanics

    Perturbation Theory in Celestial Mechanics Alessandra Celletti Dipartimento di Matematica Universit`adi Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) ([email protected]) December 8, 2007 Contents 1 Glossary 2 2 Definition 2 3 Introduction 2 4 Classical perturbation theory 4 4.1 The classical theory . 4 4.2 The precession of the perihelion of Mercury . 6 4.2.1 Delaunay action–angle variables . 6 4.2.2 The restricted, planar, circular, three–body problem . 7 4.2.3 Expansion of the perturbing function . 7 4.2.4 Computation of the precession of the perihelion . 8 5 Resonant perturbation theory 9 5.1 The resonant theory . 9 5.2 Three–body resonance . 10 5.3 Degenerate perturbation theory . 11 5.4 The precession of the equinoxes . 12 6 Invariant tori 14 6.1 Invariant KAM surfaces . 14 6.2 Rotational tori for the spin–orbit problem . 15 6.3 Librational tori for the spin–orbit problem . 16 6.4 Rotational tori for the restricted three–body problem . 17 6.5 Planetary problem . 18 7 Periodic orbits 18 7.1 Construction of periodic orbits . 18 7.2 The libration in longitude of the Moon . 20 1 8 Future directions 20 9 Bibliography 21 9.1 Books and Reviews . 21 9.2 Primary Literature . 22 1 Glossary KAM theory: it provides the persistence of quasi–periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i.e. a non– degeneracy of the integrable system and a diophantine condition of the frequency of motion.
  • Lecture 1: Explicit, Implicit and Parametric Equations

    Lecture 1: Explicit, Implicit and Parametric Equations

    4.0 3.5 f(x+h) 3.0 Q 2.5 2.0 Lecture 1: 1.5 Explicit, Implicit and 1.0 Parametric Equations 0.5 f(x) P -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x x+h -0.5 -1.0 Chapter 1 – Functions and Equations Chapter 1: Functions and Equations LECTURE TOPIC 0 GEOMETRY EXPRESSIONS™ WARM-UP 1 EXPLICIT, IMPLICIT AND PARAMETRIC EQUATIONS 2 A SHORT ATLAS OF CURVES 3 SYSTEMS OF EQUATIONS 4 INVERTIBILITY, UNIQUENESS AND CLOSURE Lecture 1 – Explicit, Implicit and Parametric Equations 2 Learning Calculus with Geometry Expressions™ Calculus Inspiration Louis Eric Wasserman used a novel approach for attacking the Clay Math Prize of P vs. NP. He examined problem complexity. Louis calculated the least number of gates needed to compute explicit functions using only AND and OR, the basic atoms of computation. He produced a characterization of P, a class of problems that can be solved in by computer in polynomial time. He also likes ultimate Frisbee™. 3 Chapter 1 – Functions and Equations EXPLICIT FUNCTIONS Mathematicians like Louis use the term “explicit function” to express the idea that we have one dependent variable on the left-hand side of an equation, and all the independent variables and constants on the right-hand side of the equation. For example, the equation of a line is: Where m is the slope and b is the y-intercept. Explicit functions GENERATE y values from x values.
  • Plotting Circles and Ellipses Parametrically Example 1: the Unit Circle

    Plotting Circles and Ellipses Parametrically Example 1: the Unit Circle

    Lesson 1 Plotting Circles and Ellipses Parametrically Example 1: The Unit Circle Let's compare traditional and parametric equations for the unit circle : Traditional : Parametric : x22 y 1 x(t) cos(t) y(t) sin(t) P(t) cos(t),sin(t) t is called a parameter 0 t 2 You can see that the parametric equation satisfies the traditional equation by substituting one into the other: x22 y 1 (cos(t))22 (sin(t)) 1 11 Created by Christopher Grattoni. All rights reserved. Example 2: Circle of Radius r Let's compare traditional and parametric equations for a circle of radius r centeredTraditional at the : origin : Parametric : 2 2 2 x y r x(t) r cos(t) y(t) r sin(t) P(t) r cos(t),sin(t) 0 t 2 Orientation: Counterclockwise Think of this as dilating the unit circle by a factor of r. Created by Christopher Grattoni. All rights reserved. Example 3: Recentering the Circle Let's compare traditional and parametric equations for a circle of radius r centeredTraditional at (h,k) : : Parametric : 2 2 2 (x h) (y k) r x(t) r cos(t) h y(t) r sin(t) k P(t) r cos(t),sin(t) (h,k) 0 t 2 Think of this as dilating the unit circle by a factor of r and translating by the point (h,k). Let's add the orientation: Created by Christopher Grattoni. All rights reserved. Example 4: Ellipses Let's compare traditional and parametric equations for an ellipse centeredTraditional at (h,k) : : Parametric : 2 2 xh yk x(t) acos(t) h 1 ab y(t) bsin(t) k P(t) acos(t),bsin(t) (h,k) 0 t 2 Think of this as dilating the unit circle by a factor of "a" in the x-direction, a factor of "b" in the y-direction, and translated by the Created by Christopherpoint Grattoni.
  • 2 Classical Field Theory

    2 Classical Field Theory

    2 Classical Field Theory In what follows we will consider rather general field theories. The only guiding principles that we will use in constructing these theories are (a) symmetries and (b) a generalized Least Action Principle. 2.1 Relativistic Invariance Before we saw three examples of relativistic wave equations. They are Maxwell’s equations for classical electromagnetism, the Klein-Gordon and Dirac equations. Maxwell’s equations govern the dynamics of a vector field, the vector potentials Aµ(x) = (A0, A~), whereas the Klein-Gordon equation describes excitations of a scalar field φ(x) and the Dirac equation governs the behavior of the four- component spinor field ψα(x)(α =0, 1, 2, 3). Each one of these fields transforms in a very definite way under the group of Lorentz transformations, the Lorentz group. The Lorentz group is defined as a group of linear transformations Λ of Minkowski space-time onto itself Λ : such that M M→M ′µ µ ν x =Λν x (1) The space-time components of Λ are the Lorentz boosts which relate inertial reference frames moving at relative velocity ~v. Thus, Lorentz boosts along the x1-axis have the familiar form x0 + vx1/c x0′ = 1 v2/c2 − x1 + vx0/c x1′ = p 1 v2/c2 − 2′ 2 x = xp x3′ = x3 (2) where x0 = ct, x1 = x, x2 = y and x3 = z (note: these are components, not powers!). If we use the notation γ = (1 v2/c2)−1/2 cosh α, we can write the Lorentz boost as a matrix: − ≡ x0′ cosh α sinh α 0 0 x0 x1′ sinh α cosh α 0 0 x1 = (3) x2′ 0 0 10 x2 x3′ 0 0 01 x3 The space components of Λ are conventional three-dimensional rotations R.