Classical Mechanicsfor the 19th century

T. Helliwell V. Sahakian

Contents

1 Newtonian particle mechanics 31.1 Inertial frames and the Galilean transformation . . . . . . . . 31.2 Newton’s laws of motion . . . . . . . . . . . . . . . . . . . . . 6Example 1-1: A bacterium with a viscous drag forceExample 1-2: A linearly damped oscillator1.3 Systems of particles . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . 16Example 1-3: A wrench in spaceExample 1-4: A particle moving in two dimensions with an attractive

spring forceExample 1-5: Particle in a magnetic fieldExample 1-6: A child on a swingExample 1-7: A particle attached to a spring revisitedExample 1-8: Newtonian central gravity and its potential energyExample 1-9: Dropping a particle in spherical gravityExample 1-10: Potential energies and turning points for positive power-

law forces1.5 Forces of Nature . . . . . . . . . . . . . . . . . . . . . . . . . 331.6 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 35Example 1-11: Find the rate at which molasses flows through a narrow

pipe1.7 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Relativity 492.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.1.1 The Postulates . . . . . . . . . . . . . . . . . . . . . . 492.1.2 The Lorentz transformation . . . . . . . . . . . . . . . 51

Example 2-1: Rotation and rapidity

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2.2 Relativistic kinematics . . . . . . . . . . . . . . . . . . . . . . 582.2.1 Proper time . . . . . . . . . . . . . . . . . . . . . . . . 582.2.2 Four-velocity . . . . . . . . . . . . . . . . . . . . . . . 62

Example 2-2: The transformation of ordinary velocityExample 2-3: Four-velocity invariant2.3 Relativistic dynamics . . . . . . . . . . . . . . . . . . . . . . . 67

2.3.1 Four-momentum . . . . . . . . . . . . . . . . . . . . . 67Example 2-4: Relativistic dispersion relationExample 2-5: Decay into two particles

2.3.2 Four-force . . . . . . . . . . . . . . . . . . . . . . . . . 732.3.3 Dynamics in practice . . . . . . . . . . . . . . . . . . . 75

Example 2-6: Uniformly accelerated motionExample 2-7: The Doppler effect

2.3.4 Minkowski diagrams . . . . . . . . . . . . . . . . . . . 80Example 2-8: Time dilationExample 2-9: Length contractionExample 2-10: The twin paradox

3 The Variational Principle 1013.1 Fermat’s principle . . . . . . . . . . . . . . . . . . . . . . . . . 1013.2 The calculus of variations . . . . . . . . . . . . . . . . . . . . 1033.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Example 3-1: Geodesics on a planeExample 3-2: Geodesics on a sphere3.4 Brachistochrone . . . . . . . . . . . . . . . . . . . . . . . . . . 114Example 3-3: Fermat again3.5 Several Dependent Variables . . . . . . . . . . . . . . . . . . . 120Example 3-4: Geodesics in three dimensions3.6 Mechanics from a variational principle . . . . . . . . . . . . . 1223.7 Motion in a uniform gravitational field . . . . . . . . . . . . . 1243.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4 Lagrangian mechanics 1394.1 The Lagrangian in Cartesian coordinates . . . . . . . . . . . . 1394.2 Hamilton’s principle . . . . . . . . . . . . . . . . . . . . . . . 141Example 4-1: A simple pendulumExample 4-2: A bead sliding on a vertical helixExample 4-3: Block on an inclined plane

CONTENTS iii

4.3 Generalized momenta and cyclic coordinates . . . . . . . . . . 149Example 4-4: Particle on a tabletop, with a central forceExample 4-5: The spherical pendulum4.4 Systems of particles . . . . . . . . . . . . . . . . . . . . . . . . 156Example 4-6: Two interacting particlesExample 4-7: Pulleys everywhereExample 4-8: A block on a movable inclined plane4.5 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 164Example 4-9: Bead on a rotating parabolic wire4.6 The moral of constraints . . . . . . . . . . . . . . . . . . . . . 1694.7 Small oscillations about equilibrium . . . . . . . . . . . . . . . 171Example 4-10: Particle on a tabletop with a central spring force4.8 Relativistic generalization . . . . . . . . . . . . . . . . . . . . 1744.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Appendices 187

Appendix A When is H 6= E? 187

5 Beyond The Basics I5.1 Classical waves . . . . . . . . . . . . . . . . . . . . . . . . . . 191Example 5-1: Two-slit interference of waves5.2 Two-slit experiments with light and atoms . . . . . . . . . . . 1995.3 Feynman sum-over-paths . . . . . . . . . . . . . . . . . . . . . 2035.4 Two slits and two paths . . . . . . . . . . . . . . . . . . . . . 2075.5 No barriers at all . . . . . . . . . . . . . . . . . . . . . . . . . 215Example 5-2: A class of paths near a straight-line pathExample 5-3: How classical is the path?5.6 Path shapes for light rays and particles . . . . . . . . . . . . . 220Example 5-4: Path shape for particles in uniform gravity5.7 Why Hamilton’s principle? . . . . . . . . . . . . . . . . . . . . 2235.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

6 Symmetries and Conservation Laws 2336.1 Cyclic coordinates and generalized momenta . . . . . . . . . . 234Example 6-1: A star orbiting a spheroidal galaxyExample 6-2: A charged particle moving outside a charged rod6.2 A less straightforward example . . . . . . . . . . . . . . . . . . 236

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6.3 Infinitesimal transformations . . . . . . . . . . . . . . . . . . . 238Example 6-3: TranslationsExample 6-4: RotationsExample 6-5: Lorentz transformations6.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2426.5 Noether’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 243Example 6-6: Space translations and momentumExample 6-7: Time translation and the HamiltonianExample 6-8: Rotations and angular momentumExample 6-9: Galilean BoostsExample 6-10: Lorentz invarianceExample 6-11: Sculpting Lagrangians from symmetry6.6 Some comments on symmetries . . . . . . . . . . . . . . . . . 256

7 Gravitation and Central-force motion 2657.1 Central forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 2657.2 The two-body problem . . . . . . . . . . . . . . . . . . . . . . 2687.3 The effective potential energy . . . . . . . . . . . . . . . . . . 271

7.3.1 Radial motion for the central-spring problem . . . . . . 2737.3.2 Radial motion in central gravity . . . . . . . . . . . . . 274

7.4 The shape of central-force orbits . . . . . . . . . . . . . . . . . 2767.4.1 Central spring-force orbits . . . . . . . . . . . . . . . . 2767.4.2 The shape of gravitational orbits . . . . . . . . . . . . 278

Example 7-1: Orbital geometry and orbital physics7.5 Bertrand’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 2847.6 Orbital dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 286

7.6.1 Kepler’s second law . . . . . . . . . . . . . . . . . . . . 2867.6.2 Kepler’s third law . . . . . . . . . . . . . . . . . . . . . 287

Example 7-2: Halley’s Comet7.6.3 Minimum-energy transfer orbits . . . . . . . . . . . . . 289

Example 7-3: A voyage to MarsExample 7-4: Gravitational assists

8 Electromagnetism 3058.1 The Lorentz force law . . . . . . . . . . . . . . . . . . . . . . . 305Example 8-1: Fixing a gauge8.2 The Lagrangian for electromagnetism . . . . . . . . . . . . . . 3108.3 The two-body problem, once again . . . . . . . . . . . . . . . 312

CONTENTS v

8.4 Coulomb scattering . . . . . . . . . . . . . . . . . . . . . . . . 315

Example 8-2: Snell scattering

8.5 Uniform magnetic field . . . . . . . . . . . . . . . . . . . . . . 319

Example 8-3: Bubble chamber

Example 8-4: Ion trapping

8.6 Contact forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

Example 8-5: A microscopic model

Example 8-6: Rolling down the plane

Example 8-7: Stacking barrels

Example 8-8: On the rope

9 Accelerating frames 349

9.1 Linearly accelerating frames . . . . . . . . . . . . . . . . . . . 349

Example 9-1: Pendulum in an accelerating spaceship

9.2 Rotating frames . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Example 9-2: Throwing a ball in a rotating space colony

Example 9-3: Polar orbits around the Earth

9.3 Pseudoforces in rotating frames . . . . . . . . . . . . . . . . . 356

9.4 Centrifugal and Coriolis pseudoforces . . . . . . . . . . . . . . 360

Example 9-4: Rotating space colonies revisited

9.5 Pseudoforces on Earth . . . . . . . . . . . . . . . . . . . . . . 363

Example 9-5: Coriolis pseudoforces in airflow

Example 9-6: Foucault’s pendulum

9.6 Spacecraft rendezvous and docking . . . . . . . . . . . . . . . 371

Example 9-7: Rendezvous with the space station?

Example 9-8: Losing a wrench?

10 Beyond The Basics II

10.1 Beyond newtonian gravity . . . . . . . . . . . . . . . . . . . . 388

Example 10-1: The precession of Mercury’s perihelion

10.1.1 Magnetic gravity . . . . . . . . . . . . . . . . . . . . . 400

Example 10-2: Gravity inside the body of a star

Example 10-3: Cosmic string

10.2 Beyond the classical forces . . . . . . . . . . . . . . . . . . . . 403

10.3 Beyond deterministic forces . . . . . . . . . . . . . . . . . . . 403

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11 Hamiltonian formulation 409

11.1 Legendre transformations . . . . . . . . . . . . . . . . . . . . . 409Example 11-1: A simple Legendre transform

11.2 Hamilton’s equations . . . . . . . . . . . . . . . . . . . . . . . 414

11.3 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418Example 11-2: The simple harmonic oscillatorExample 11-3: A bead on a parabolic wireExample 11-4: A charged particle in a uniform magnetic field

11.4 Canonical transformations . . . . . . . . . . . . . . . . . . . . 423Example 11-5: Transforming the simple harmonic oscillatorExample 11-6: IdentitiesExample 11-7: Infinitesimal transformations and the HamiltonianExample 11-8: Point transformations

11.5 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . 432Example 11-9: Position and momentaExample 11-10: The simple harmonic oscillator once again

11.6 Liouville’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 437

12 Rigid Body Dynamics 447

12.1 Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

12.2 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

12.3 Infinitesimal Rotations . . . . . . . . . . . . . . . . . . . . . . 450Example 12-1: Rotations in higher dimensions

12.4 The Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . 452Example 12-2: Angular velocity transformation

12.5 Rotational kinetic energy . . . . . . . . . . . . . . . . . . . . . 455Example 12-3: A hoop

12.6 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . 460

12.7 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 461

12.8 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

12.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

12.10Principal Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . 464Example 12-4: Fixed Axis RotationExample 12-5: Principal Axis Shifts

12.11Torque Free Dynamics . . . . . . . . . . . . . . . . . . . . . . 468Example 12-6: Adding Angular Momenta

12.12Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

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13 Complex systems 47513.1 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

14 Small oscillations 479

15 Beyond The Basics III15.1 Beyond classical phase space . . . . . . . . . . . . . . . . . . . 480

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List of Figures

1.1 Various inertial frames in space. If one of these frames isinertial, any other frame moving at constant velocity relativeto it is also inertial. . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Two inertial frames, O and O′, moving relative to one anotheralong their mutual x or x′ axes. . . . . . . . . . . . . . . . . . 5

1.3 A bacterium in a fluid. What is its motion if it begins withvelocity v0 and then stops swimming? . . . . . . . . . . . . . . 10

1.4 Motion of an oscillator if it is (a) overdamped, (b) under-damped, or (c) critically damped, for the special case wherethe oscillator is released from rest (v0 = 0) at some position x0. 12

1.5 A system of particles, with each particle identified by a posi-tion vector r . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 A collection of particles, each with a position vector ri from afixed origin. The center of mass RCM is shown, and also theposition vector r′i of the ith particle measured from the centerof mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.7 The position vector for a particle. Angular momentum is al-ways defined with respect to a chosen point from where theposition vector originates. . . . . . . . . . . . . . . . . . . . . 18

1.8 A two-dimensional elliptical orbit of a ball subject to a Hooke’slaw spring force, with one end of the spring fixed at the origin. 20

1.9 The work done by a force on a particle is its line integral alongthe path traced by the particle. . . . . . . . . . . . . . . . . . 23

1.10 Newtonian gravity pulling a probe mass m towards a sourcemass M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

ix

x LIST OF FIGURES

1.11 Potential energy functions for selected positive powers n. Apossible energy E is drawn as a horizontal line, since E isconstant. The difference between E and U(x) at any point isthe value of the kinetic energy T . The kinetic energy is zeroat the turning points, where the E line intersects U(x). Notethat for n = 1 there are two turning points for E > 0, but forn = 2 there is only a single turning point. . . . . . . . . . . . . 31

2.1 Inertial frames O and O′ . . . . . . . . . . . . . . . . . . . . . 51

2.2 Graph of the γ factor as a function of the relative velocity β.Note that γ ∼= 1 for nonrelativistic particles, and γ → ∞ asβ → 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3 The velocity vx as a function of v′x for fixed relative framevelocity V = 0.5c. . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4 A particle of mass m0 decays into two particles with massesm1 and m2. Both energy and momentum are conserved in thedecay, but mass is not conserved in relativistic physics. Thatis, m0 6= m1 +m2. . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.5 Plots of relativistic constant-acceleration motion. (a) showsvx(t), demonstrating that vx(t)→ c as t→∞, i.e., the speedof light is a speed limit in Nature. The dashed line showsthe incorrect Newtonian prediction. (b) shows the hyperbolictrajectory of the particle on a c t-x graph. Once again thedashed trajectory is the Newtonian prediction. . . . . . . . . . 78

2.6 Observer O′ shooting a laser towards observer O while movingtowards O. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.7 A point on a Minkowski diagram represents an event. A par-ticle’s trajectory appears as a curve with a slope that exceedsunity everywhere. . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.8 Three events on a Minkowski diagram. Events A and B aretimelike separated; A and C are lightlike separated; and B andC are spacelike separated. . . . . . . . . . . . . . . . . . . . . 81

2.9 The hyperbolic trajectory of a particle undergoing constantacceleration motion on a Minkowski diagram. . . . . . . . . . 82

2.10 The grid lines of two observers labeling the same event on aspacetime Minkowski diagram. . . . . . . . . . . . . . . . . . . 84

LIST OF FIGURES xi

2.11 The time dilation phenomenon. (a) Shows the scenario of aclock carried by observer O. (b) shows the case of a clockcarried by O′. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.12 The phenomenon of length contraction. (a) Shows the scenarioof a meter stick carried by observer O’. (b) shows the case ofa stick carried by O. . . . . . . . . . . . . . . . . . . . . . . . 86

2.13 Minkowski diagrams of the twin paradox. (a) shows simul-taneity lines according to John. During the first and thirdpart of the trip, a time 2 × T1 elapse on John’s clock; duringthe middle part, Jane is accelerating uniformly and the timeelapsed is denoted by T0(b) Shows simultaneity lines accord-ing to Jane, except for the two dotted lines sandwiching theaccelerating segment. Jane’s x′ axis is also shown for two in-stants in time. The segment labeled T0 is excised away andborrowed from John’s perspective since Jane is a not an in-ertial frame during this period. T ′1 and T ′2 on the other handcan be computed from Jane’s perspective. Notice how Jane’sx′ axis must smoothly flip around during the time interval T0,as she turns around. Her simultaneity lines during this periodwill hence be distorted and require general relativity to fullyunravel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.1 Light traveling by the least-time path between a and b, inwhich it moves partly through air and partly through a piece ofglass. At the interface the relationship between the angle θ1 inair, with index of refraction n1, and the angle θ2 in glass, withindex of refraction n2, is n1 sin θ1 = n2 sin θ2, known as Snell’slaw. This phenomenon is readily verified by experiment. . . . 102

3.2 A light ray from a star travels down through Earth’s atmo-sphere on its way to the ground. . . . . . . . . . . . . . . . . . 104

3.3 A function of two variables f(x1, x2) with a local minimum atpoint A, a local maximum at point B, and a saddle point at C. 105

3.4 Various paths y(x) that can be used as input to the functionalI[f(x)]. We look for that special path from which an arbitrarysmall displacement δy(x) leaves the functional unchanged tolinear order in δy(x). Note that δy(a) = δy(b) = 0. . . . . . . 107

3.5 A discretization of a path. . . . . . . . . . . . . . . . . . . . . 108

3.6 The coordinates θ and ϕ on a sphere. . . . . . . . . . . . . . 112

xii LIST OF FIGURES

3.7 (a) Great circles on a sphere are geodesics; (b) Two pathsnearby the longer of the two great-circle routes of a path. . . 114

3.8 Possible least-time paths for a sliding block. . . . . . . . . . . 1153.9 A cycloid. If in darkness you watch a wheel rolling along a

level surface, with a lighted bulb attached to a point on theouter rim of the wheel, the bulb will trace out the shape of acycloid. In the diagram the wheel is rolling along horizontallybeneath the surface. For xb < (π/2)yb, the rail may look likethe segment from a to b1; for xb > (π/2)yb, the segment froma to b2 would be needed. . . . . . . . . . . . . . . . . . . . . . 117

3.10 (a) A light ray passing through a stack of atmospheric layers;(b) The same problem visualized as a sequence of adjacentslabs of air of different index of refraction. . . . . . . . . . . . 119

3.11 Two spaceships, one accelerating in gravity-free space (a), andthe other at rest on the ground (b). Neither observers in theaccelerating ship nor those in the ship at rest on the groundcan find out which ship they are in on the basis of any exper-iments carried out solely within their ship. . . . . . . . . . . . 125

3.12 A laser beam travels from the bow to the stern of the acceler-ating ship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.1 Cartesian, cylindrical, and spherical coordinates . . . . . . . . 1424.2 A bead sliding on a vertically-oriented helical wire . . . . . . . 1484.3 Block sliding down an inclined plane . . . . . . . . . . . . . . 1484.4 Particle moving on a tabletop . . . . . . . . . . . . . . . . . . 1504.5 The effective radial potential energy for a mass m moving with

an effective potential energy Ueff = (pϕ)2/2mr2 + (1/2)kr2 forvarious values of pϕ, m, and k. . . . . . . . . . . . . . . . . . . 153

4.6 Coordinates of a ball hanging on an unstretchable string . . . 1544.7 A sketch of the effective potential energy Ueff for a spherical

pendulum. A ball at the minimum of Ueff is circling the verticalaxis passing through the point of suspension, at constant θ.The fact that there is a potential energy minimum at someangle θ0 means that if disturbed from this value the ball willoscillate back and forth about θ0 as it orbits the vertical axis. 155

4.8 Two interacting beads on a one-dimensional frictionless rail.The interaction between the particles depends only on thedistance between them. . . . . . . . . . . . . . . . . . . . . . . 157

LIST OF FIGURES xiii

4.9 A contraption of pulleys. We want to find the accelerations ofall three weights. We assume that the pulleys have negligiblemass so they have negligible kinetic and potential energies. . 159

4.10 A block slides along an inclined plane. Both block and inclinedplane are free to move along frictionless surfaces. . . . . . . . . 161

4.11 A bead slides without friction on a vertically-oriented parabolicwire that is forced to spin about its axis of symmetry. . . . . . 166

4.12 The effective potential Ueff for the Hamiltonian of a bead on arotating parabolic wire with z = αr2, depending upon whetherthe angular velocity ω is less than, greater than, or equal toωcrit =

√2g α. . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.13 An effective potential energy Ueff with a focus near a mini-mum. Such a point is a stable equilibrium point. The dot-ted parabola shows the leading approximation to the poten-tial near its minimum. As the energy drains out, the systemsettles into its minimum with the final moments being wellapproximated with harmonic oscillatory dynamics. . . . . . . . 172

4.14 The shape of the two-dimensional orbit of a particle subjectto a central spring force, for small oscillations about the equi-librium radius. . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.1 A transverse small displacement of a string. . . . . . . . . . . 192

5.2 (a) A small slice of string; (b) Tension forces on the slice. . . . 193

5.3 Two paths for waves from slit system to detectors. . . . . . . . 198

5.4 (a) The relationship between s2−s1, d, and θ; (b) The two-slitinterference pattern. . . . . . . . . . . . . . . . . . . . . . . . 199

5.5 (a) At very low intensity light, individual photons appear toland on the screen randomly; (b) as the intensity is crankedup, the interference pattern emerges. . . . . . . . . . . . . . . 200

xiv LIST OF FIGURES

5.6 Helium atoms with speeds between 2.1 and 2.2 km/s reachingthe rear detectors, with both slits open. The detectors observethe arrival of individual atoms, but the distribution shows aclear interference pattern as we would expect for waves!. Wesee how the interference pattern builds up one atom at a time.The first data set is taken after 5 minutes of counting, whilethe last is taken after 42 hours of counting. The experimentswere carried out by Ch. Kurtsiefer, T. Pfau, and J. Mlynek;see their article in Nature 386, 150 (1997). (The “hotspot”in the data arises from an enhanced dark count due to animpurity in the microchannel plate detector.) . . . . . . . . . . 202

5.7 A phasor z(t0)eiφ ≡ |z(t0)|ei(φ+φ0) drawn in the complex plane.The real axis is horizontal and the imaginary axis is vertical.The absolute length of the phasor is |z(t0)| and the angle be-tween the phasor and the real axis is the phase (φ+φ0), whereφ0 is the phase of z(t0) alone. . . . . . . . . . . . . . . . . . . 206

5.8 The sum of two individual phasors with the same magnitudes|z(t0)| but different phases. The result is a phasor that extendsfrom the tail of the first to the tip of the second, as in vectoraddition. The difference in their angles in the complex plane isthe difference in their phase angles. Shown are examples withphase differences equal to (a) zero (b) 45◦ (c) 90◦ (d) 135◦ (e)180◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

5.9 Two paths from a source to a detector. . . . . . . . . . . . . . 208

5.10 High-velocity helium atoms, with speeds above 30 km/s, reach-ing the rear detectors, with both slits open. The detectorsobserve the arrival of individual atoms, and the distributionis what we would expect for classical particles. Experimentscarried out by Ch. Kurtsiever, T. Pfau, and J. Mlynek, Nature386, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

5.11 (a) Path length as a function of position y within the slit. (b)The single-slit diffraction pattern. . . . . . . . . . . . . . . . . 211

5.12 The double slit, with a screen at distance D. We can view theintensity on the screen as a function of the transverse distancex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

LIST OF FIGURES xv

5.13 Interference/diffraction patterns for a double slit with a = d/4and D = 1000d. The diffraction curves, shown in dashed lines,serve as envelopes for the more rapidly oscillating interferencepattern. (a) The pattern in the case d = 0.1x1/2, where x1/2

is the distance on the detecting plane between the center andthe first minimum of the diffraction envelope. The diffractioncurves of the two slits strongly overlap in this case, giving ineffect a single diffraction envelope. (b) The pattern in thecase d = 2x1/2, showing that the two diffraction patterns havebecome separated, with the first minimum due to each slit atthe same location in the center. This case corresponds to awavelength smaller by a factor of 20 than the pattern shownin (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

5.14 The sum of a large number of phasors (a) that are about thesame (b) that differ by constant amounts. . . . . . . . . . . . 215

5.15 A class of kinked paths between a source and detector. Thestraight line is the shortest path, and the midpoint of theothers is a distance D = |n|D0 from the straight line, where(n = ±1,±2, ...). . . . . . . . . . . . . . . . . . . . . . . . . . 217

5.16 Phasors up to n = ±25. The more distant paths wind up inspirals, contributing very little to the overall phasor sum. . . . 218

6.1 An elliptical galaxy (NGC 1132) pulling on a star at the outerfringes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

6.2 The two types of transformations considered: direct on theleft, indirect on the right. . . . . . . . . . . . . . . . . . . . . . 239

6.3 Sliding pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . 261

7.1 Newtonian gravity pulling a probe mass m2 towards a sourcemass m1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

7.2 Angular momentum conservation and the planar nature of cen-tral force orbits. . . . . . . . . . . . . . . . . . . . . . . . . . . 267

7.3 The classical two-body problem in physics. . . . . . . . . . . . 268

7.4 The effective potential for the central-spring potential. . . . . 273

7.5 The effective gravitational potential. . . . . . . . . . . . . . . 275

7.6 Elliptical orbits due to a central spring force F = −kr. . . . . 278

7.7 Conic sections: circles, ellipses, parabolas, and hyperbolas. . . 281

xvi LIST OF FIGURES

7.8 An elliptical gravitational orbit, showing the foci, the semi-major axis a, semiminor axis b, the eccentricity ε, and theperiapse and apoapse. . . . . . . . . . . . . . . . . . . . . . . 282

7.9 Parabolic and hyperbolic orbits . . . . . . . . . . . . . . . . . 283

7.10 The four types of gravitational orbits . . . . . . . . . . . . . . 285

7.11 The area of a thin pie slice . . . . . . . . . . . . . . . . . . . . 287

7.12 The orbit of Halley’s comet . . . . . . . . . . . . . . . . . . . 289

7.13 A minimum-energy transfer orbit to an outer planet. . . . . . 290

7.14 Insertion from a parking orbit into the transfer orbit. . . . . . 292

7.15 A spacecraft flies by Jupiter, in the reference frames of(a) Jupiter (b) the Sun . . . . . . . . . . . . . . . . . . . . . . 295

8.1 The electrostatic Coulomb force between two charged particles. 307

8.2 Hyperbolic trajectory of a probe scattering off a charged target.315

8.3 Definition of the scattering cross section in terms of change inimpact area 2πbdb and scattering area 2π sin ΘdΘ on the unitsphere centered at the target. . . . . . . . . . . . . . . . . . . 317

8.4 The Rutherford scattering cross section. The graph showslog σ(Θ) as a function of log Θ superimposed on actual datain scattering of protons off gold atoms. . . . . . . . . . . . . . 318

8.5 Scattering of light off a reflecting bead. . . . . . . . . . . . . . 318

8.6 (a) Top view of a charged particle in a uniform magnetic field;(b) The helical trajectory of the charged particle. . . . . . . . 321

8.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

8.8 The effective Penning potential. At the minimum, we have astable circular trajectory. In general however, the radial extentwill oscillate with frequency ω0. . . . . . . . . . . . . . . . . . 325

8.9 The full trajectory of an ion in a Penning trap. A vertical oscil-lation along the z axis with frequency ωz is superimposed ontoan fast oscillation of frequency ω0, while the particle traces alarge circle with characteristic frequency ωm. . . . . . . . . . . 327

8.10 The electric field from a neutral atom leaks out in a dipolepattern due to small asymmetries in the charge distribution ofthe atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

8.11 A layer of perfectly aligned dipole at the surface of a floor onwhich a block is to rest. . . . . . . . . . . . . . . . . . . . . . 329

8.12 A hoop rolling down an inclined plane without slipping. . . . . 335

LIST OF FIGURES xvii

8.13 Two barrels stacked on top of each other. The lower barrel isstationary, while the upper one rolls down without slipping. . 337

8.14 A pendulum with a single constraint given by the fixed lengthof the rope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

9.1 A ball is thrown sideways in an accelerating spaceship (a) asseen by observers within the ship (b) as seen by a hypotheticalinertial observer outside the ship . . . . . . . . . . . . . . . . 351

9.2 A simple pendulum in an accelerating spaceship . . . . . . . . 352

9.3 spacecolony living on the inside rim of a rotating cylindricalspace colony . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

9.4 Throwing a ball in a rotating space colony (a) From the pointof view of an external inertial observer (b) From the point ofview of a colonist . . . . . . . . . . . . . . . . . . . . . . . . . 356

9.5 Path of a satellite orbiting Earth, in Earth’s rest frame. Dashedlines represent the equator and longitude lines. . . . . . . . . 357

9.6 A vector that is constant in a rotating frame changes in aninertial frame: (a) simple two dimensional case; (b) three di-mensional general case. . . . . . . . . . . . . . . . . . . . . . . 358

9.7 (a) The angular velocity vector for a rotating frame (b) thetriple cross product ω × ω × r . . . . . . . . . . . . . . . . . . 361

9.8 Stroboscopic pictures of a ball thrown from the center of arotating space colony (a) as seen in an inertial frame (b) asseen in the colony . . . . . . . . . . . . . . . . . . . . . . . . . 363

9.9 The length of the day relatively to the stars (sidereal time) isslightly longer than the length of the day relative to the Sun. . 364

9.10 The Earth bulges at the equator due to its rotation, whichproduces a centrifugal pseudoforce in the rotating frame. Aplumb bob hanging near the surface experiences both gravita-tion and the centrifugal pseudoforce. . . . . . . . . . . . . . . 365

9.11 (a) A set of three Cartesian coordinates placed on the Earth(b) The horizontal coordinates x and y. . . . . . . . . . . . . 366

9.12 Inflowing air develops a counterclockwise rotation in the north-ern hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . 368

9.13 Foucault’s pendulum at the North Pole. . . . . . . . . . . . . 369

9.14 Foucault’s pendulum . . . . . . . . . . . . . . . . . . . . . . . 370

xviii LIST OF FIGURES

9.15 (a) A spacecraft trying to rendezvous and dock with a spacestation in circular orbit around the Earth. (b) A strandedastronaut trying to return to the space station by throwing awrench. (c) An astronaut accidentally lets a wrench escapefrom the ISS. What is its subsequent trajectory? . . . . . . . . 372

9.16 Coordinates of the space station and object . . . . . . . . . . 373

9.17 The spacecraft trajectory in the nonrotating frame. . . . . . . 377

9.18 Rendezvous with the ISS? The bizarre trajectory, after start-ing off in the desired direction. . . . . . . . . . . . . . . . . . . 378

9.19 Rendezvous with the ISS? The initial boost. . . . . . . . . . . 378

9.20 Trajectory of a wrench in the rotating frame in which the ISSis at rest. The wrench is thrown from the ISS vertically, awayfrom the Earth. It returns like a boomerang. . . . . . . . . . . 379

9.21 Trajectory of the wrench in the nonrotating frame where theISS is in circular orbit around the Earth. . . . . . . . . . . . . 380

9.22 (a) a balloon in a car (b) a cork in a fishtank . . . . . . . . . . 381

9.23 Tilt of the northward-flowing gulf stream surface, looking north386

10.1 An ant colony measures the radius and circumference of aturntable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

10.2 Non-Euclidean geometry: circumferences on a sphere. . . . . . 391

10.3 Successive light rays sent to a clock at altitude h from a clockon the ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

10.4 Effective potential for the Schwarzschild geometry. . . . . . . . 397

10.5 A gaussian surface probing the gravity inside a star of uniformvolume mass density. . . . . . . . . . . . . . . . . . . . . . . . 402

10.6 An infinite linear mass distribution moves upward with speedV while a probe of mass M ventures nearby. . . . . . . . . . . 404

11.1 (a) Two functions A(x, y), differing by a shift, whose naivetransformation through y → z lead to the same transformedfunction B(x, z); (b) The envelope of A(x, y) consisting ofslopes and intercepts completely describe the shape of A(x, y). 411

11.2 The Legendre transformation of A(x, y) as B(x, z). . . . . . . 412

11.3 The two dimensional cross section of a phase space for a sys-tem. The flow lines depict Hamiltonian time evolution. . . . . 418

11.4 The phase space of the one dimensional simple harmonic os-cillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

LIST OF FIGURES xix

11.5 The phase space of the one dimensional particle on a parabolaproblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

11.6 The flow lines in the x-px cross section of phase space for acharged particle in a uniform magnetic field. . . . . . . . . . . 424

11.7 (a) The flow lines in a given phase space; (b) The same flowlines as described by transformation coordinates and momenta. 424

11.8 The transformation of phase space under a canonical transfor-mation. Volume elements may get distorted in shape, but thevolume of each element must remain unchanged. . . . . . . . . 432

11.9 A depiction of Liouville’s theorem: the density of states of asystem evolves in phase space in such as way that its totaltime derivative is zero. . . . . . . . . . . . . . . . . . . . . . . 438

xx LIST OF FIGURES

LIST OF FIGURES

1