Classical Mechanics

John R. Taylor University of Colorado

RED material is the focus of our 2011 JHU Course

Contents

Part I: THE ESSENTIALS

1. Newton's Laws of Motion
1.1 Classical Mechanics
1.2 Space and Time
1.3 Mass and Force
1.4 Newton's First and Second Laws; Inertial Frames
1.5 The Third Law and Conservation of the Momentum
1.6 Newton's Second Law in Cartesian Coordinates
1.7 Two-Dimensional Polar Coordinates
1.8 Problems for Chapter 1

2. Projectiles and Charged Particles
2.1 Air Resistance
2.2 Linear Air Resistance
2.3 Trajectory and Range in a Linear Motion
2.5 Motion of a Charge in a Uniform Magnetic Field
2.6 Complex Exponentials
2.7 Solution for the Charge in a B Field
2.8 Problems for Chapter 2

3. Momentum and Angular Momentum
3.1 Conservation of Momentum
3.2 Rockets
3.3 The Center of Mass
3.4 Angular Momentum for a Single Particle
3.5 Angular Momentum for Several Particles
3.6 Problems for Chapter 3

4. Energy
4.1 Kinetic Energy and Work
4.2 Potential Energy and Conservative Forces
4.3 Force as the Gradient of Potential Energy
4.4 The Second Condition that F be Conservative
4.5 Time-Dependent Potential Energy
4.6 Energy for Linear One-Dimensional Systems
4.7 Curvilinear One-Dimensional Systems
4.8 Central Forces
4.9 Energy of Interaction of Two Particles
4.10 The Energy of a Multiparticle System
4.11 Problems for Chapter 4

5. Oscillations
5.1 Hooke's Law
5.2 Simple Harmonic Motion
5.3 Two-Dimensional Oscillators
5.4 Damped Oscillators
5.5 Driven Damped Oscillations
5.6 Resonance
5.7 Fourier Series
5.8 Fourier Series Solution for the Driven Oscillator
5.9 The RMS Displacement; Parseval's Theorem
5.10 Problems for Chapter 5

6. Calculus of Variations
6.1 Two Examples
6.2 The Euler-Lagrange Equation
6.3 Applications of the Euler-Lagrange Equation
6.4 More than Two Variables
6.5 Problems for Chapter 6

7. Lagrange's Equations
7.1 Lagrange's Equations for Unconstrained Motion
7.2 Constrained Systems; an Example
7.3 Constrained Systems in General
7.4 Proof of Lagrange's Equations with Constraints
7.5 Examples of Lagrange's Equations
7.6 Conclusion
7.7 Conservation Laws in Lagrangian Mechanics
7.8 Lagrange's Equations for Magnetic Forces
7.9 Lagrange Multipliers and Constraint Forces
7.10 Problems for Chapter 7

8. Two-Body Central Force Problems
8.1 The Problem
8.2 CM and Relative Coordinates; Reduced Mass
8.3 The Equations of Motion
8.4 The Equivalent One-Dimensional Problems
8.5 The Equation of the Orbit
8.6 The Kepler Orbits
8.7 The Unbonded Kepler Orbits
8.8 Changes of Orbit
8.9 Problems for Chapter 8

9. Mechanics in Noninertial Frames
9.1 Acceleration without Rotation
9.2 The Tides
9.3 The Angular Velocity Vector
9.4 Time Derivatives in a Rotating Frame
9.5 Newton's Second Law in a Rotating Frame
9.6 The Centrifugal Force
9.7 The Coriolis Force
9.8 Free Fall and The Coriolis Force
9.9 The Foucault Pendulum
9.10 Coriolis Force and Coriolis Acceleration
9.11 Problems for Chapter 9

10. Motion of Rigid Bodies
10.1 Properties of the Center of Mass
10.2 Rotation about a Fixed Axis
10.3 Rotation about Any Axis; the Inertia Tensor
10.4 Principal Axes of Inertia
10.5 Finding the Principal Axes; Eigenvalue Equations
10.6 Precession of a Top Due to a Weak Torque
10.7 Euler's Equations
10.8 Euler's Equations with Zero Torque
1
0.9 Euler Angles
10.10 Motion of a Spinning Top

10.11 Problems for Chapter 10

11. Coupled Oscillators and Normal Modes
11.1 Two Masses and Three Springs
11.2 Identical Springs and Equal Masses
11.3 Two Weakly Coupled Oscillators
11.4 Lagrangian Approach; the Double Pendulum
11.5 The General Case
11.6 Three Coupled Pendulums
11.7 Normal Coordinates
11.8 Problems for Chapter 11

Part II: FURTHER TOPICS

12. Nonlinear Mechanics and Chaos
12.1 Linearity and Nonlinearity
12.2 The Driven Damped Pendulum or DDP
12.3 Some Expected Features of the DDP
12.4 The DDP; Approach to Chaos
12.5 Chaos and Sensitivity to Initial Conditions
12.6 Bifurcation Diagrams
12.7 State-Space Orbits
12.8 Poincare Sections
12.9 The Logistic Map
12.10 Problems for Chapter 12

13. Hamiltonian Mechanics
13.1 The Basic Variables
13.2 Hamilton's Equations for One-Dimensional Systems
13.3 Hamilton's Equations in Several Dimensions
13.4 Ignorable Coordinates
13.5 Lagrange's Equations vs. Hamilton's Equations
13.6 Phase-Space Orbits
13.7 Liouville's Theorem
13.8 Problems for Chapter 13

14. Collision Theory
14.1 The Scattering Angle and Impact Parameter
14.2 The Collision Cross Section
14.3 Generalizations of the Cross Section
14.4 The Differential Scattering Cross Section
14.5 Calculating the Differential Cross Section
14.6 Rutherford Scattering
14.7 Cross Sections in Various Frames
14.8 Relation of the CM and Lab Scattering Angles
14.9 Problems for Chapter 14

15. Special Relativity
15.1 Relativity
15.2 Galilean Relativity
15.3 The Postulates of Special Relativity
15.4 The Relativity of Time; Time Dilation
15.5 Length Contraction
15.6 The Lorentz Transformation
15.8 Four-Dimensional Space-Time; Four-Vectors
15.9 The Invariant Scalar Product
15.10 The Light Cone
15.11 The Quotient Rule and Doppler Effect
15.12 Mass, Four-Velocity, and Four-Momentum
15.13 Energy, the Fourth Component of Momentum
15.14 Collisions
15.15 Force in Relativity
15.16 Massless Particles; the Photon
15.17 Tensors
15.18 Electrodynamics and Relativity
15.19 Problems for Chapters 15

16. Continuum Mechanics
16.1 Transverse Motion of a Taut String
16.2 The Wave Equation
16.3  Boundary Conditions; Waves on a Finite String
16.4 The Three-Dimensional Wave Equation
16.5 Volume and Surface Forces
16.6 Stress and Strain: the Elastic Moduli
16.7 The Stress Tensor
16.8 The Strain Tensor for a Solid
16.9 Relation between Stress and Strain: Hooke's Law
16.10 The Equation of Motion for an Elastic Solid
16.11 Longitudinal and Transverse Waves in a Solid
16.12 Fluids: Description of the Motion
16.13 Waves in a Fluid
16.14 Problems for Chapter 16

Appendix: Diagonalizing Real Symmetric Matrices