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Preface

About the Book

1: Introductory Statics - the Catenary and the Arch

• 1.1: The Catenary
• 1.2: A Tight String
• 1.3: Not So Tight
• 1.4: Ideal Arches

2: The Calculus of Variations

• 2.1: The Catenary and the Soap Film
• 2.2: A Soap Film Between Two Horizontal Rings- the Euler-Lagrange Equation
• 2.3: General Method for the Minimization Problem
• 2.4: An Important First Integral of the Euler-Lagrange Equation
• 2.5: Fastest Curve for Given Horizontal Distance
• 2.6: The Perfect Pendulum
• 2.7: Calculus of Variations with Many Variables
• 2.8: Multivariable First Integral
• 2.9: The Soap Film and the Chain
• 2.10: Lagrange Multipliers
• 2.11: Lagrange Multiplier for the Chain
• 2.12: The Brachistochrone

3: Fermat's Principle of Least Time

• 3.1: Another Minimization Problem…
• 3.2: Huygens' Picture of Wave Propagation
• 3.3: Fermat’s Principle
• 3.4: Reflection, Too
• 3.5: The Bottom Line- Geometric Optics and Wave Optics

4: Hamilton's Principle and Noether's Theorem

• 4.1: Introduction- Galileo and Newton
• 4.2: Derivation of Hamilton’s Principle from Newton’s Laws in Cartesian Co-ordinates- Calculus of Variations Done Backwards!
• 4.3: But Why?
• 4.4: Lagrange’s Equations from Hamilton’s Principle Using Calculus of Variations
• 4.5: Generalized Momenta and Forces
• 4.6: Non-uniqueness of the Lagrangian
• 4.7: First Integral- Energy Conservation and the Hamiltonian
• 4.8: Example 1- One Degree of Freedom- Atwood’s Machine
• 4.9: Example 2- Lagrangian Formulation of the Central Force Problem
• 4.10: Conservation Laws and Noether’s Theorem
• 4.11: Momentum Conservation
• 4.12: Center of Mass
• 4.13: Angular Momentum Conservation

5: Mechanical Similarity and the Virial Theorem

• 5.1: Some Examples
• 5.2: Lagrangian Treatment
• 5.3: The Virial Theorem

6: Hamilton’s Equations

• 6.1: A Dynamical System’s Path in Configuration Space and in State Space
• 6.2: Phase Space
• 6.3: Going From State Space to Phase Space
• 6.4: How It's Done in Thermodynamics
• 6.5: Math Note - the Legendre Transform
• 6.6: Hamilton's Use of the Legendre Transform
• 6.7: Checking that We Can Eliminate the q˙i's
• 6.8: Hamilton’s Equations
• 6.9: A Simple Example

7: Time Evolution in Phase Space- Poisson Brackets and Constants of the Motion

• 7.1: The Poisson Bracket
• 7.2: Interlude - a Bit of History of Quantum Mechanics
• 7.3: The Jacobi Identity
• 7.4: Poisson’s Theorem
• 7.5: Example- Angular Momentum Components

8: A New Way to Write the Action Integral

• 8.1: Function of Endpoint Position
• 8.2: Function of Endpoint Time
• 8.3: Varying Both Ends
• 8.4: Another Way of Writing the Action Integral
• 8.5: How this Classical Action Relates to Phase in Quantum Mechanics
• 8.6: Hamilton’s Equations from Action Minimization
• 8.7: How Can p, q Really Be Independent Variables?

9: Maupertuis’ Principle - Minimum Action Path at Fixed Energy

• 9.1: Divine Guidance
• 9.2: It's Not About Time
• 9.3: The Abbreviated Action
• 9.4: Maupertuis’ Principle and the Time-Independent Schrödinger Equation

10: Canonical Transformations

• 10.1: Point Transformations
• 10.2: General and Canonical Transformations
• 10.3: Generating Functions for Canonical Transformations
• 10.4: Generating Functions in Different Variables
• 10.5: Poisson Brackets under Canonical Transformations
• 10.6: Time Development is a Canonical Transformation Generated by the Action

11: Introduction to Liouville's Theorem

• 11.1: Paths in Simple Phase Spaces - the SHO and Falling Bodies
• 11.2: Following Many Systems- a “Gas” in Phase Space
• 11.3: Liouville’s Theorem- Local Gas Density is Constant along a Phase Space Path
• 11.4: Landau’s Proof Using the Jacobian
• 11.5: Jacobian for Time Evolution
• 11.6: Jacobians 101
• 11.7: Jacobian proof of Liouville’s Theorem
• 11.8: Simpler Proof of Liouville’s Theorem
• 11.9: Energy Gradient and Phase Space Velocity

12: The Hamilton-Jacobi Equation

• 12.1: Back to Configuration Space…
• 12.2: The Central Role of These Constants of Integration
• 12.3: Separation of Variables for a Central Potential; Cyclic Variables

13: Adiabatic Invariants and Action-Angle Variables

• 13.1: Adiabatic Invariants
• 13.2: Adiabatic Invariance and Quantum Mechanics
• 13.3: Action-Angle Variables
• 13.4: *Kepler Orbit Action-Angle Variables

14: Mathematics for Orbits

• 14.1: Preliminaries- Conic Sections
• 14.2: The Ellipse
• 14.3: The Parabola
• 14.4: The Hyperbola

15: Keplerian Orbits

• 15.1: Preliminary- Polar Equations for Conic Section Curves
• 15.2: Summary
• 15.3: Kepler’s Statement of his Three Laws
• 15.4: Dynamics of Motion in a Central Potential- Deriving Kepler’s Laws
• 15.5: A Vectorial Approach- Hamilton’s Equation and the Runge Lenz Vector

16: Elastic Scattering

• 16.1: Billiard Balls
• 16.2: Discovery of the Nucleus
• 16.3: The Differential Cross Section
• 16.4: Analyzing Inverse-Square Repulsive Scattering- Kepler Again
• 16.5: Vectorial Derivation of the Scattering Angle

17: Small Oscillations

• 17.1: Particle in a Well
• 17.2: Two Coupled Pendulums
• 17.3: Normal Modes
• 17.4: Principle of Superposition
• 17.5: Three Coupled Pendulums
• 17.6: Normal Coordinates
• 17.7: Three Equal Pendulums Equally Coupled

18: Driven Oscillator

• 18.1: More General Energy Exchange
• 18.2: Damped Driven Oscillator

19: One-Dimensional Crystal Dynamics

• 19.1: The Model
• 19.2: The Circulant Matrix- Nature of its Eigenstates
• 19.3: Comparison with Raising Operators
• 19.4: Finding the Eigenvectors
• 19.5: Eigenvectors of the Linear Chain
• 19.6: Allowed Wavenumbers from Boundary Conditions
• 19.7: Finding the Eigenvalues
• 19.8: The Discrete Fourier Transform
• 19.9: A Note on the Physics of These Waves

20: Parametric Resonance

• 20.1: Introduction to Parametric Resonance
• 20.2: Resonance near Double the Natural Frequency
• 20.3: Example- Pendulum Driven at near Double the Natural Frequency

21: The Ponderomotive Force

• 21.1: Introduction to the Ponderomotive Force
• 21.2: Finding the Effective Potential Generated by the Oscillating Force
• 21.3: Stability of a Pendulum with a Rapidly Oscillating Vertical Driving Force
• 21.4: Hand-Waving Explanation of the Ponderomotive Force
• 21.5: Pendulum with Top Point Oscillating Rapidly in a Horizontal Direction

22: Resonant Nonlinear Oscillations

• 22.1: Introduction to Resonant Nonlinear Oscillations
• 22.2: Frequency of Oscillation of a Particle is a Slightly Anharmonic Potential
• 22.3: Resonance in a Damped Driven Linear Oscillator- A Brief Review
• 22.4: Damped Driven Nonlinear Oscillator- Qualitative Discussion
• 22.5: Nonlinear Case - Landau’s Analysis
• 22.6: Frequency Multiples

23: Damped Driven Pendulum- Period Doubling and Chaos

• 23.1: Introduction
• 23.2: The Road to Chaos
• 23.3: Lyapunov Exponents and Dimensions of Strange Attractors

24: Motion of a Rigid Body - the Inertia Tensor

• 24.1: Definition of Rigid
• 24.2: Rotation of a Body about a Fixed Axis
• 24.3: General Motion of a Rotating Rigid Body
• 24.4: The Inertia Tensor
• 24.5: Tensors 101
• 24.6: Definition of a Tensor
• 24.7: Diagonalizing the Inertia Tensor
• 24.8: Principal Axes Form of Moment of Inertia Tensor
• 24.9: Relating Angular Momentum to Angular Velocity
• 24.10: Symmetries, Other Axes, the Parallel Axis Theorem

25: Moments of Inertia and Rolling Motion

• 25.1: Examples of Moments of Inertia
• 25.2: Analyzing Rolling Motion
• 25.3: Rolling Without Slipping - Two Views
• 25.4: Cylinder Rolling Inside another Cylinder

26: Rigid Body Moving Freely

• 26.1: Angular Momentum and Angular Velocity
• 26.2: Precession of a Symmetrical Top
• 26.3: Throwing a Football…
• 26.4: Equations of Motion for Rigid Body with External Forces

27: Euler Angles

• 27.1: Definition of Euler Angles
• 27.2: Angular Velocity and Energy in Terms of Euler’s Angles
• 27.3: Free Motion of a Symmetrical Top
• 27.4: Motion of Symmetrical Top around a Fixed Base with Gravity - Nutation
• 27.5: Steady Precession
• 27.6: Stability of Top Spinning about Vertical Axis

28: Euler’s Equations

• 28.1: Introduction to Euler’s Equations
• 28.2: Free Rotation of a Symmetric Top Using Euler’s Equations
• 28.3: Using Energy and Angular Momentum Conservation
• 28.4: The Asymmetrical Top

29: Non-Inertial Frame and Coriolis Effect

• 29.1: The Lagrangian in Accelerating and Rotating Frames
• 29.2: Uniformly Rotating Frame
• 29.3: Coriolis Effect - Particle Moving near Earth’s Surface
• 29.4: Gyroscopes and Gyrocompasses in Navigation

30: A Rolling Sphere on a Rotating Plane

• 30.1: Introduction
• 30.2: Holonomic Constraints and non-Holonomic Constraints
• 30.3: D’Alembert’s Principle
• 30.4: Ball with External Forces Rolling on Horizontal Plane
• 30.5: Ball Rolling on Rotating Plane
• 30.6: Ball Rolling on Inclined Rotating Plane

Index

Glossary

Glossary

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