Lent Term, 2013

Dynamics and Relativity

University of Cambridge Part IA Mathematical Tripos

David Tong

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/relativity.html d.tong@damtp.cam.ac.uk

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Recommended Books and Resources

• Tom Kibble and Frank Berkshire, “Classical Mechanics” • Douglas Gregory, “Classical Mechanics” Both of these books are well written and do an excellent job of explaining the fundamentals of classical mechanics. If you’re struggling to understand some of the basic concepts, these are both good places to turn. • S. Chandrasekhar, “Newton’s Principia (for the common reader)” Want to hear about Newtonian mechanics straight from the horse’s mouth? This is an annotated version of the Principia with commentary by the Nobel prize winning astrophysicist Chandrasekhar who walks you through Newton’s geometrical proofs. Although, in fairness, Newton is sometimes easier to understand than Chandra. • A.P. French, “Special Relativity” A clear introduction, covering the theory in some detail. • Wolfgang Pauli, “Theory of Relativity” Pauli was one of the founders of quantum mechanics and one of the great physicists of the last century. Much of this book was written when he was just 21. It remains one of the most authoritative and scholarly accounts of special relativity. It’s not for the faint of heart. (But it is cheap). A number of excellent lecture notes are available on the web. Links can be found on the course webpage: http://www.damtp.cam.ac.uk/user/tong/relativity.html

Contents

1. Newtonian Mechanics 1.1 Newton’s Laws of Motion 1.1.1 Newton’s Laws 1.2 Inertial Frames and Newton’s First Law 1.2.1 Galilean Relativity 1.3 Newton’s Second Law 1.4 Looking Forwards: The Validity of Newtonian Mechanics 1 2 3 4 5 7 9

2. Forces 10 2.1 Potentials in One Dimension 10 2.1.1 Moving in a Potential 12 2.1.2 Equilibrium: Why (Almost) Everything is a Harmonic Oscillator 15 2.2 Potentials in Three Dimensions 17 2.2.1 Central Forces 19 2.2.2 Angular Momentum 20 2.3 Gravity 21 2.3.1 The Gravitational Field 21 2.3.2 Escape Velocity 23 2.3.3 Inertial vs Gravitational Mass 24 2.4 Electromagnetism 24 2.4.1 The Electric Field of a Point Charge 26 2.4.2 Circles in a Constant Magnetic Field 26 2.4.3 An Aside: Maxwell’s Equations 29 2.5 Friction 29 2.5.1 Dry Friction 30 2.5.2 Fluid Drag 30 2.5.3 An Example: The Damped Harmonic Oscillator 31 2.5.4 Terminal Velocity with Quadratic Friction 33 3. Interlude: Dimensional Analysis 38

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4. Central Forces 4.1 Polar Coordinates in the Plane 4.2 Back to Central Forces 4.2.1 The Eﬀective Potential: Getting a Feel for Orbits 4.2.2 The Stability of Circular Orbits 4.3 The Orbit Equation 4.3.1 The Kepler Problem 4.3.2 Kepler’s Laws of Planetary Motion 4.3.3 Orbital Precession 4.4 Scattering: Throwing Stuﬀ at Other Stuﬀ 4.4.1 Rutherford Scattering 5. Systems of Particles 5.1 Centre of Mass Motion 5.1.1 Conservation of Momentum 5.1.2 Angular Momentum 5.1.3 Energy 5.1.4 In Praise of Conservation Laws 5.1.5 Why the Two Body Problem is Really a One Body Problem 5.2 Collisions 5.2.1 Bouncing Balls 5.2.2 More Bouncing Balls and the Digits of π 5.3 Variable Mass Problems 5.3.1 Rockets: Things Fall Apart 5.3.2 Avalanches: Stuﬀ Gathering Other Stuﬀ 5.4 Rigid Bodies 5.4.1 Angular Velocity 5.4.2 The Moment of Inertia 5.4.3 Parallel Axis Theorem 5.4.4 The Inertia Tensor 5.4.5 Motion of Rigid Bodies 6. Non-Inertial Frames 6.1 Rotating Frames 6.1.1 Velocity and Acceleration in a Rotating Frame 6.2 Newton’s Equation of Motion in a Rotating Frame 6.3 Centrifugal Force 6.3.1 An Example: Apparent Gravity

46 46 48 50 51 53 54 58 60 61 62 65 65 66 66 67 68 69 70 71 72 74 75 78 79 80 80 83 84 85 90 90 91 92 94 94

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6.4

Coriolis Force 6.4.1 Particles, Baths and Hurricanes 6.4.2 Balls and Towers 6.4.3...