Analytic Physics
a website devoted to exploring the analytic aspects of physics

Physics has been an analytical science since the era of Laplace, Lagrange and Hamilton in the sense that its results are expressed using calculus and differential equations. The structure of physics also allows it to be an analytic science by continuous and complex extension of its components. On this website, spaces are allowed to be of arbitrary continuous dimension, exponents of functions are allowed to be arbitrary real numbers, and classical phase spaces are allowed to have complex extensions, all for the purpose of understanding more of the deep structure of physics.

• Recently Uploaded
  • Multivariable Hypergeometric Functions at Unity
  • A Simple First-Order Nonlinear System
  • Zeta Functions in Trigonometry
  • Visualizing Riemann Zeta Function Zeros
  • A Miscellany of Bernoulli Numbers
• Recently Updated
  • Chinese Phonetic Groups
  • The Essence of Quantization
  • The Schrödinger Equation for Power Potentials
  • Pairs of Power Potentials in the Schrödinger Equation
  • Analytic Delay Differential Equations
• Of Special Interest
  • The Essence of Quantization
  • 27 Lines on a Cubic Surface
  • Interactive Lotka-Volterra Equations
  • Simó’s Three-Body Choreographies in Action
  • Chinese Phonetic Groups / highlight
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• Higher Dimensions
  • Rotations in Higher Dimensions
  • Ellipsoids in Higher Dimensions
  • Superellipsoids in Higher Dimensions
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  • Perspective Projection to Lower Dimensions
  • The Visual Appearance of Higher-Dimensional Objects
  • How to See in Higher Dimensions
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  • 27 Lines on a Cubic Surface
  • Interactive Bianchi-Pinkall Flat Tori
  • Visualizing Calabi-Yau Manifolds
• Complex Variables
  • Visualizing Complex Functions with Three.js
  • Healing Branch Cuts in Three.js
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  • Coloring the Complex Logarithm by Argument
  • Complex Branches of Inverse Trigonometric Functions I
  • Complex Branches of Inverse Trigonometric Functions II
  • The Complex Gudermannian Function
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  • Exploring Complex Surfaces via Differential Equations
  • Testing Complex Identities
  • Analytic Step Functions
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  • Special Functions Explorer - mpmath
  • Special Functions Explorer - SageMath
• Differential Equations
  • Differential Equations and Transformations
  • Differential Equations and Singularities I
  • Differential Equations and Singularities II
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  • Fubini’s Method for Second-Order Differential Equations
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  • A Simple First-Order Nonlinear System
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  • A Simple Deliciously Nonlinear Equation
  • Interactive Lotka-Volterra Equations
  • An Analytic Solution for a Lotka-Volterra System
  • Exploring Logistic Functions
  • Analytic Delay Differential Equations
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  • Second-Order Slope Fields in WebXR
• Catastrophe Theory
  • Hauser’s Gallery of Singular Algebraic Surfaces
  • Hauser’s Individual Singular Algebraic Surfaces
  • A Gallery of Strange Attractors in WebXR
• Mathematical Methods
  • A Miscellany of Useful Integrals
  • A Miscellany of Bernoulli Numbers
  • A Miscellany of Integral Transforms
  • Fourier Series for the Jacobi Zeta Function
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  • Practical Inversion of Generalized Hyperelliptic Integrals
  • Complete Inversion of Generalized Hyperelliptic Integrals
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  • Real Roots of Non-Polynomials
  • Arbitrary Function of a 2x2 Hermitian Matrix
  • Experiments in Periodicity
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  • Multiple Derivatives of the Sigmoid Function
  • Surface Area of an Ellipsoid
  • Transforming a Torus into a Cylinder
• Special Functions
  • A Miscellany of Elliptic Integrals
  • Carlson Symmetric Elliptic Integrals
  • Inverse Elliptic Integrals of the First and Second Kinds
  • A Miscellany of Elliptic Functions
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  • Hypergeometric Orthogonal Polynomials
  • Multivariable Hypergeometric Functions at Unity
  • Visualizing Riemann Zeta Function Zeros
  • Zeta Functions in Trigonometry
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  • Visualizing the Lambert W Function
  • Applications of the Lambert W Function
  • Analytic Approximations to the Lambert W Function
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  • A Generalized Lambert Function of Two Arguments
  • An Exact Solution of the Complex Kepler Equation
• Coding Methods
  • Special Topics in Three.js
  • Exploring Parametric Surfaces in WebXR
  • List of Exploratoria Exhibits
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  • Accessing SageMath as a Web Service
  • SageMath Online Integrator
  • Reverse Engineering Mathematica Colormaps
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• Classical Mechanics
  • Dynamic Constants for Additively Separable Hamiltonians
  • Angular Momentum Conservation for Additively Separable Hamiltonians
  • Sundman Transformations for Spherically Symmetric Potentials
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  • Potentials for Arbitrary Planar Orbits
  • Projectile Motion the Hard Way
  • Explicit Temporal Evolution for the Kepler Problem
• Runge-Lenz Vector
  • The Symmetry Corresponding to the Runge Vector
  • A Generalized Runge Vector for Spherically Symmetric Potentials
  • Visualizing the Generalized Runge Vector
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  • The Runge Vector and Angular Momentum
  • SO(n+1) and the Runge Vector
  • SO(n+1) and the Runge Vector Redux
  • The Runge Vector in General Relativity
• Power Potential Orbits
  • Pairs of Power Potentials in Classical Mechanics
  • Hamilton-Jacobi Theory for Power Potentials
  • Sundman Transformations for Power Potentials
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  • Closed Orbit Conditions for Power Potentials
  • Limiting Power Potential Orbits
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  • Orbit Integration for Power Potentials
  • The Analytic Approximations of Struck and Lynden-Bell
  • A Simple Analytic Approximation for Power Potential Orbits
• Gravitational Choreographies
  • How Gravitational Choreographies Are Possible
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  • Visualizing Simó’s Three-Body Choreographies
  • Simó’s Three-Body Choreographies in Action
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  • Visualizing Details of the Figure Eight
  • Fourier Analysis of the Figure Eight
  • Fourier Expansions for All Simó Choreographies
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  • Interpolating Euler and Lagrange Choreographies
  • Exploring n-Body Lagrange Choreographies
• General Relativity
  • A (Relatively) Simple Derivation of the Kerr Metric
  • Hamilton-Jacobi Separability for the Kerr Metric
  • The Schwarzschild Metric in Higher Dimensions
  • Visualizing Geodesics on Surfaces
  • Visualizing Aspects of the Kerr Metric
• Quantum Mechanics
  • The Essence of Quantization
  • Spin is Not a Relativistic Phenomenon
  • Interactive One-Dimensional Potentials
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  • The Schrödinger Equation in Higher Dimensions
  • The Schrödinger Equation for Power Potentials
  • Pairs of Power Potentials in the Schrödinger Equation
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  • Visualizing Continuity in Quantum Mechanics
  • Continuity in Quantum Mechanics via WKB Theory
• S-Matrix Theory
  • Dispersion Relation Integrals
• General Physics
  • Visualizing Sound Waves with Three.js
  • Visualizing the Principle of Least Action
  • An Interactive Swinging Atwood Machine
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  • n-Body Simple Harmonic Oscillators
  • n-Body Lagrange Choreographies for Pairwise Potentials
  • n-Body Simple Harmonic Oscillator Choreographies
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• Philosophical Context
  • How I Got Where I Am
  • Outline of Martin Heidegger’s Being and Time
• Language Tools
  • Chinese Phonetic Groups
  • Language Links
• Diversions
  • Assembly Language Programming for LEGO Mindstorms EV3
  • The Jansen Linkage in LEGO Mindstorms
  • Playing with Prime Numbers
  • Accessing Amazon Echo Data with JavaScript
  • An Alexa Skills Kit (ASK) JavaScript Cookbook

The mathematics on this website is encoded using MathML and is best viewed with a fully standards-compliant browser: that means Firefox. MathML support in Safari is not as complete as Firefox and has only recently returned to Chromium, so for all browsers other than Firefox MathJax is used for rendering.

A particular focus of this website is interactive mathematics rendered in the browser. This is achieved as much as possible using pure JavaScript, some of whose more modern features may not work in older browsers. Special functions and common numerical operations are processed using Math, while interactive cells are implemented with MathCell.

Here are some interesting and balanced comments on how one goes about learning physics, as well as a comprehensive set of links to publicly available resources. If you want to know about me, here you go.

Contact for questions or comments: paulmasson@analyticphysics.com

Paul Masson — Independent Physicist — San Francisco