1. Getting Started
1.1 Introduction
1.1.1 Computers as a Tool
1.1.2 A Note About Notation and Style
1.1.3 Notation and Symbolize (For experts only)
1.2 Arithmetic and Algebra
1.2.1 Arithmetic and Notation
1.2.2 Algebraic Manipulations
1.2.3 PowerExpand
1.2.4 Simple Rules
1.2.5 A Home-made Complex Conjugate with SuperStar
1.2.6 Immediate and Delayed Substitutions
1.2.7 Selecting Parts of Expressions
1.2.8 Algebraic Equations
1.3 Functions and Procedures
1.3.1 Built-In Functions
1.3.2 User-Defined Functions
1.3.3 Pure Functions
1.3.4 Assigning Rules and Restrictions to Functions
1.3.5 Module
1.3.6 rootPlot
1.4 Packages
1.4.1 Loading Packages
1.4.2 Contexts
1.4.3 Shadowing
1.5 Calculus
1.5.1 Derivatives and Integrals
1.5.2 Differential Equations
1.5.3 Changing Variables and Pure Functions
1.5.4 Numerical Solutions of Differential Equations
1.6 Graphics
1.6.1 Using the Plot Command
1.6.2 Animated Plots
1.6.3 Vector Field Plots
1.6.4 Three-dimensional Graphics using Plot3D and ParametricPlot3D
1.7 Exercises
Exercise 1: Real Expressions
Exercise 2: Series
Exercise 3: Integration
Exercise 4: Superposition of Waves
Exercise 5: Tables of Zeros for Several Functions
Exercise 6: Plot of Roots
Exercise 7: Coupled Equations
Exercise 8: Statistical Data
Exercise 9: Differential Equation
Exercise 10: Coupled Differential Equations
Exercise 11: Trajectories and Vector Field Plots for Coupled Differential Equations
Exercise 12: Heat Equation
Exercise 13: Wave Equation
Exercise 14: Fit to Data
Exercise 15: Rules used to solve problems
Exercise 16: Procedures to select numbers
Exercise 17: Procedure to Select certain Pairs of Points
Exercise 18: Change
Exercise 19: Toss of a Die
Exercise 20: Three Dice
Exercise 21: Playing Darts and the Calculation of \[Pi]
Exercise 22: Dropping Needles
Exercise 23: Pacal's Wager
Exercise 24: The birthday problem
2. GENERAL PHYSICS
Introduction
2.1 Newtonian Mechanics in Inertial Frames
Overview of Newtonian Mechanics in Inertial Frames
Problem 1: Escape Velocity
Problem 2: Projectile in a Uniform Gravity Field
Problem 3: Reflecting Trajectories
Problem 4: Falling Projectile with Linear Drag
Problem 5: Projectile with Quadratic Drag
Problem 6: Rocket with varying Mass
Problem 7: Keplerian Orbits
2.2 Newtonian Mechanics in Rotating Frames
Overview of Newtonian Mechanics in Rotating Frames
Problem 1: Projectile Motion as Measured by an Observer on Earth
Problem 2: Foucault Pendulum
2.3 Electricity and Magnetism
Overview of Electricity and Magnetism
Problem 1: Charged Disk
Problem 2: Uniformly Charged Sphere
Problem 3: Electric Dipole
Problem 4: Magnetic Vector Potential for a Linear Current
Problem 5: Motion of a Charged Particle in a Uniform B Field
Problem 6: Motion of a Charged Particle in a Uniform B Field and Time Varying Electric Field
2.4 Modern Physics
Problem 1: Carbon Dating
Problem 2: Stable Isotopes
Problem 3: The Bohr Atom
Problem 4: Relativistic Collision
2.5 Exercises
Exercise 1: Exploding Projectiles
Exercise 2: Intersecting Trajectories
Exercise 3: Principle of Galilean Invariance
Exercise 4: Elastic Collision
Exercise 5: Projectile and Earth's Rotation
Exercise 6: Projectile Thrown up an Incline Plane
Exercise 7: Falling Object with Air Resistance as Observed from the Rotating Earth
Exercise 8: Projectile Thrown up with Air Resistance
Exercise 9: Quadrupole
Exercise 10: Motion of a Charged Particle
Exercise 11: B Field of a Spinning Disk
3. Oscillating Systems
Introduction
3.1 Linear Oscillations
Overview of Linear Oscillations
1. Phase plot for one-dimensional system
2. Time behavior of phase plot for a one-dimensional system
3. Combination of the phase plot and the time behavior of the phase plot
Problem 1: Linear Oscillator
Problem 2: Series expansion Solution
Problem 3: Potential and Phase Diagrams for the Linear Oscillator
Problem 4: Damped Linear Oscillator
Problem 5: Damped Harmonic Oscillator and Driving Forces
3.2 Small Oscillations
Overview of Small Oscillations and Normal Modes
Eigenvalues and eigenvectors for small oscillating systems
Problem 1: Two Coupled Oscillators along a Line
Problem 2: Three Coupled Oscillators along a Line
Problem 3: Three Coupled Oscillators along a Circle
Problem 4: Double Pendulum
Problem 5: Understanding the User-defined procedure smallOsc[]
3.3 Oscillating Circuits
Overview of Circuits
Problem 1: Series RC Circuit
Problem 2: Series RL Loop
Problem 3 : RLC Loop
3.4 Exercises
Exercise 1: Blocks connect with a Spring on a Table
Exercise 2: Falling Blocks connect with a Spring
Exercise 3: Parabolic coordinates and Harmonic Oscillator
Exercise 4: Forced Oscillator
Exercise 5: Pendulums and Spring
Exercise 6: Five Particles connected with Springs
Exercise 7: Vibrating Atoms on an Equilateral Triangle
Exercise 8: Vibrating Charged Spheres
Exercise 9: Quadratic Damping
Exercise 10: Motion of a Charged Particle in a Harmonic Potential
4. NonLinear Oscillating Systems
Introduction
4.1 Nonlinear Pendulum
Overview of the Nonlinear Pendulum
Problem 1: Analytic Solution for the Planar Pendulum
Problem 2: Damped Pendulum
Problem 3: Periodic Solutions for the Driven Pendulum
Problem 4: Looping Solutions for the Driven Pendulum
Problem 5: Chaotic motion for the Driven Pendulum
4.2 Duffing Equation
Overview of the Duffing Equation
1. User-defined procedure to plot the Duffing displacement motion
2. User-defined procedure to plot the Duffing phase
3. User-defined procedure to plot the Duffing Poincaré map
Problem 1: Potential and Phase Diagrams for the Duffing Oscillator
Problem 2: Phase Diagram and Orbits for the Damped, Duffing Equation
Problem 3: Driven Duffing Orbits with No Damping
Problem 4: Two Well Driven Duffing Oscillator with Damping
4.3 Exercises
Exercise 1: van der Pol Oscillator and Limiting Cycles
Exercise 2: Springs
Exercise 3: Buckling column system
Exercise 4: Nonlinear Equation
Exercise 5: Inverted Pendulum
Exercise 6: Driven Nonlinear Equation
5. Discrete Dynamical Systems
Introduction
5.1 Logistic Map
Overview of the Logistic Map
Problem 1: Logistic Map
Problem 2: Logistic Fixed Points
Problem 3: Logistic Cobwebs.
Problem 4: Logistic Bifurcations
Problem 5: Logistic Lyapunov Exponent and Entropy
5.2 Other Maps
Overview of Other Maps
Problem 1: Salmon Map
Problem 2: Sine-Circle Map
Problem 3: Taylor-Greene-Chirikov map
Problem 4: Henon Map
5.3 Fractals
Overview of Fractals
Problem 1: Mandelbrot Set
Problem 2: Julia Set
5.4 Exercises
Exercise 1: Gaussian Map
Exercise 2: Return Maps for the Henon Map
Exercise 3: Cubic Maps
Exercise 4: Two Dimensional Map
Exercise 5: Tent map
6. Lagrangians and Hamiltonians
Introduction
6.1 Lagrangian Problems without Lagrange Multipliers
Overview of Lagrangian Problems without Lagrange Multipliers
Problem 1: Particle Sliding on a Movable Incline
Problem 2: Bead Sliding on a Rotating Wire
Problem 3: Bead on a Rotating Hoop
Problem 4: Springs Mounted on Top of a Carriage
Problem 5: Mass Falling Through a Hole in a Table
Problem 6: Spring Pendulum
6.2 Lagrangian Problems with Lagrange Multipliers
Overview of Nonholonomic Constraints and Lagrange Multipliers
Problem 1: Atwood Machine
Problem 2: Hoop Rolling on an Incline
Problem 3: Sphere Rolling on a Fixed Sphere
6.3 Hamiltonian Problems
Overview of Hamilton's Equations
Problem 1: Harmonic Oscillator
Problem 2: Nonlinear Oscillator
Problem 3: Cylindrical Coordinates
Problem 4: Swinging Atwood Machine
Problem 5: Spherical Pendulum
6.4 Hamilton-Jacobi Problems
Overview
Problem 1: Harmonic Oscillator
Problem 2: Particle in a Constant Gravity Field
Problem 3: Kepler's Problem and Hamilton-Jacobi Equations
6.5 Exercises
Exercise 1: Atwood Machine
Exercise 2: Double Atwood machine
Exercise 3: Spherical Pendulum
Exercise 4: Double Pendulum
Exercise 5: Spring Pendulum
Exercise 6: Hamilton-Jacobi in parabolic coordinates
Exercise 7: How Hamilton Works
Exercise 8: How HamiltonJacobi Works
Exercise 9: How firstDiffSeries Works
Exercise 10: How firstOrderPert Works
7. Orbiting Bodies
Introduction
7.1 Two Body Problem
Overview of the Two-Body Problem
Problem 1: Equivalent One-body Problem
Problem 2: Kepler Orbits
Problem 3: Precessing Ellipse
Problem 4: Numerical Solution
7.2 Restricted Three Body Problem
Overview of the Three Body Problem
Problems on the Equal Mass Primaries (\[Mu]=1/2 )
Problem 1: Lagrangian Points for Equal Mass Binaries (\[Mu]=1/2 )
Problem 2: Looping Motion in an Equal Mass Binary System (\[Mu]=1/2)
Problem 3: Symmetric orbits about the y-axis for \[Mu]=1/2
Problem 4: Mass exchange Between Equal Mass Binaries
Problems on the Sun-Jupiter System (\[Mu]=.000954)
Problem 5: Lagrangian Points for the Sun-Jupiter System
Problem 6: Numerical Solution for the Trojan Asteroids
Problem 7: Perturbative Solution for the Trojan Asteroids
Problems on the Earth-Moon System (\[Mu]= .01215)
Problem 8: Lagrangian Points for the Earth-Moon System
Problem 9: Motion about L[4] in the Earth-Moon System
Problem 10: Orbit around the Earth and Moon
7.3 Exercises
Exercise 1: Central force problems
Exercise 2: Central force procedure
Exercise 3: Central forces and elliptical solutions
Exercise 4: Attractive Inverse fifth power force
Exercise 5: Eccentric Anomaly
Exercise 6: Eccentric Anomaly
Exercise 7: Kepler problem with drag
Exercise 8: Lagrange points
Exercise 9: Orbit around the Sun and Jupiter
Exercise 10: Exact Solution for the Three-Body Problem
8. Electrostatics
Introduction
Mathematica Commands for All Sections
Example: Equipotential surface and electric field of two-point charges
8.1 Point Charges, Multipoles, and Image Charges
Overview of Point Charges, Multipoles, and Image Charges
Mathematica Commands for Section 8.1
Monopole
TrigToY
TrigToP
Problem 1: Superposition of point charges
Problem 2: Point charges and grounded plane
Problem 3: Point charges and grounded sphere
Problem 4: Line charge and grounded plane
Problem 5: Multipole expansion of a charge distribution
8.2 Laplace's Equation in Cartesian and Cylindrical Coordinates
Overview of Cartesian and Cylindrical Coordinates
Problem 1: Separation of variables in Cartesian and cylindrical coordinates
Problem 2: Potential in a rectangular groove
Problem 3: Rectangular conduit
Problem 4: Potential inside a rectangular box with five sides at zero potential
Problem 5: Conducting cylinder with a potential on the surface
8.3 Laplace's Equation in Spherical Coordinates
Overview of Spherical Coordinates
Problem 1: A charged ring
Problem 2: Grounded sphere in an electric field
Problem 3: Sphere with an axially symmetric charge distribution
Problem 4: Sphere with a given axially symmetric potential
Problem 5: Sphere with upper hemisphere V0 and lower hemisphere -V0
8.4 Exercises
Exercise 1: Parallel plates and strips
Exercise 2: Parallel plates and a point charge
Exercise 3: Cylinder divided into segments
Exercise 4: Elliptical cylinder in a uniform electric field
Exercise 5: Cylindrical box and n charges
Exercise 6: Distorted sphere with a potential
Exercise 7: Concentric spheres with different potentials on the two hemispheres
Exercise 8: Ring charge
Exercise 9: Laplace's equations in other coordinates
9. Quantum Mechanics
Introduction
9.1 One-Dimensional Schroedinger's Equation
Problem 1: Particle bound in an Infinite Potential Well
Problem 2: Particle bound in a Finite Potential Well
Problem 3: Particle Hitting a Finite Step Potential
Problem 4: Particle Propagating Towards a Rectangular Potential
Problem 5: The One-Dimensional Harmonic Oscillator
9.2 Three-Dimensional Schroedinger's Equation
Problem 1: Three-Dimensional Harmonic Oscillator in Cartesian Coordinates
Problem 2: Schroedinger's Equation for Spherically Symmetric Potentials
Problem 3: Particle in an Infinite, Spherical Well
Problem 4: Particle in a Finite, Spherical Well
Problem 5: The Hydrogen Atom in Spherical Coordinates
9.3 Exercises
Exercise 1: Infinite Potential Well with Rectangular Perturbation
Exercise 2: Tilted Square Well
Exercise 3: The Wentzel-Kramers-Brillouin Approximation
Exercise 4: Plot of the Electron Probability Density
Exercise 5: Perturbed Harmonic Oscillator
Exercise 6: Perturbation Theory
Exercise 7: Separation of variables in Cylindrical Coordinates
10. Relativity and Cosmology
Introduction
10.1 Special Relativity
Overview of Special Relativity
Problem 1: Decay of a particle
Problem 2: Two-particle collision
Problem 3: Compton scattering
Problem 4: Moving mirror and generalized Snell's law
Problem 5: One-dimensional motion of a relativistic particle with constant acceleration
Problem 6: Two-dimensional motion of a relativistic particle in a uniform electric field
10.2 General Relativity
Overview of General Relativity
Problem 1: Killing vectors and Spherical Symmetry
Problem 2: Schwarzschild solution
Problem 3: Geodesics for the Schwarzschild metric
Problem 4: Time it takes to fall into a black hole
Problem 5: Circular geodesics for the Schwarzschild metric
Problem 6: Precession of the Perihelion
10.3 Cosmology
Overview of Friedmann, Robertson, and Walker Cosmology
Problem 1: Field equations for Friedmann-Robertson-Walker cosmology
Problem 2: Zero-pressure (Dust) Cosmological Models
Problem 3: The Expansion and Age for the Friedmann-Robertson-Walker models
10.4 Exercises
Exercise 1: Lorentz boosts
Exercise 2: Radioactive decay of moving Nucleus
Exercise 3: Schwarzschild solution
Exercise 4: Potential analysis for timelike geodesics
Exercise 5: Schwarzschild solution in null coordinates
Exercise 6: Null geodesics for the Schwarzschild metric
Exercise 7: Mercury's Anomalous Precession and Modification of Newton's Gravity
Exercise 8: Cosmological Force and Mercury