Dynamics and Relativity

Dr Jeff Forshaw, Department of Physics & Astronomy, University of Manchester, Oxford Road, Manchester, UK.Dr Gavin Smith, Department of Physics & Astronomy, University of Manchester, Oxford Road, Manchester, UK.

'Dynamics and Relativity by Forshaw and Smith was the ideal book for the upper level course Physics from Newton to Einstein at Yale since it was thoroughly modern and extremely lucid. Its most admirable features are the countless solved examples that made it ideal for self-study and a very detailed and lucid introduction to relativity.' - Ramamurti Shankar, John Randolph Huffman Professor of Physics, Professor of Applied Physics, Yale University

• Editors’ Preface to the Manchester Physics Series xiAuthor’s Preface xiiiI Introductory Dynamics 11 Space, Time and Motion 31.1 Defining Space and Time 31.1.1 Space and the classical particle 41.1.2 Unit vectors 61.1.3 Addition and subtraction of vectors 61.1.4 Multiplication of vectors 71.1.5 Time 81.1.6 Absolute space and space-time 101.2 Vectors and Co-ordinate Systems 111.3 Velocity and Acceleration 141.3.1 Frames of reference 161.3.2 Relative motion 161.3.3 Uniform acceleration 181.3.4 Velocity and acceleration in plane-polar co-ordinates: uniform circular motion 201.4 Standards and Units 212 Force, Momentum and Newton’s Laws 252.1 Force and Static Equilibrium 252.2 Force and Motion 312.2.1 Newton’s Third Law 352.2.2 Newton’s bucket and Mach’s principle 392.3 Applications of Newton’s Laws 412.3.1 Free body diagrams 412.3.2 Three worked examples 422.3.3 Normal forces and friction 462.3.4 Momentum conservation 492.3.5 Impulse 512.3.6 Motion in fluids 513 Energy 553.1 Work, Power and Kinetic Energy 563.2 Potential Energy 613.2.1 The stability of mechanical systems 643.2.2 The harmonic oscillator 653.2.3 Motion about a point of stable equilibrium 673.3 Collisions 683.3.1 Zero-momentum frames 683.3.2 Elastic and inelastic collisions 713.4 Energy Conservation in Complex Systems 754 Angular Momentum 814.1 Angular Momentum of a Particle 814.2 Conservation of Angular Momentum in Systems of Particles 834.3 Angular Momentum and Rotation About a Fixed Axis 864.3.1 The parallel-axis theorem 944.4 Sliding and Rolling 954.5 Angular Impulse and the Centre of Percussion 974.6 Kinetic Energy of Rotation 99II Introductory Special Relativity 1035 The Need for a New Theory of Space and Time 1055.1 Space and Time Revisited 1055.2 Experimental Evidence 1085.2.1 The Michelson-Morley experiment 1085.2.2 Stellar aberration 1105.3 Einstein’s Postulates 1136 Relativistic Kinematics 1156.1 Time Dilation, Length Contraction and Simultaneity 1156.1.1 Time dilation and the Doppler effect 1166.1.2 Length contraction 1216.1.3 Simultaneity 1236.2 Lorentz Transformations 1246.3 Velocity Transformations 1296.3.1 Addition of velocities 1296.3.2 Stellar aberration revisited 1307 Relativistic Energy and Momentum 1357.1 Momentum and Energy 1357.1.1 The equivalence of mass and energy 1427.1.2 The hint of an underlying symmetry 1447.2 Applications in Particle Physics 1457.2.1 When is relativity important? 1467.2.2 Two useful relations and massless particles 1497.2.3 Compton scattering 152III Advanced Dynamics 1578 Non-inertial Frames 1598.1 Linearly Accelerating Frames 1598.2 Rotating Frames 1618.2.1 Motion on the earth 1659 Gravitation 1739.1 Newton’s Law of Gravity 1749.2 The Gravitational Potential 1779.3 Reduced Mass 1829.4 Motion in a Central Force 1849.5 Orbits 18610 Rigid Body Motion 19710.1 The Angular Momentum of a Rigid Body 19810.2 The Moment of Inertia Tensor 20010.2.1 Calculating the moment of inertia tensor 20310.3 Principal Axes 20710.4 Fixed-axis Rotation in the Lab Frame 21210.5 Euler’s Equations 21410.6 The Free Rotation of a Symmetric Top 21610.6.1 The body-fixed frame 21610.6.2 The lab frame 21810.6.3 The wobbling earth 22310.7 The Stability of Free Rotation 22410.8 Gyroscopes 22610.8.1 Gyroscopic precession 22610.8.2 Nutation of a gyroscope 232IV Advanced Special Relativity 23711 The Symmetries of Space and Time 23911.1 Symmetry in Physics 23911.1.1 Rotations and translations 24011.1.2 Translational symmetry 24511.1.3 Galilean symmetry 24611.2 Lorentz Symmetry 24712 Four-vectors and Lorentz Invariants 25312.1 The Velocity Four-vector 25412.2 The Wave Four-vector 25512.3 The Energy-momentum Four-vector 25812.3.1 Further examples in relativistic kinematics 25912.4 Electric and Magnetic Fields 26213 Space-time Diagrams and Causality 26713.1 Relativity Preserves Causality 27013.2 An Alternative Approach 27214 Acceleration and General Relativity 27914.1 Acceleration in Special Relativity 27914.1.1 Twins paradox 28014.1.2 Accelerating frames of reference 28214.2 A Glimpse of General Relativity 28814.2.1 Gravitational fields 290A Deriving the Geodesic Equation 295B Solutions to Problems 297