Problem of Two Fixed Centres (Part III)
Two constants of motion classify the orbits of the two-centre problem into satellite, lemniscate and planetary types, all reproduced in simulation.
Archive
Notes and short articles from 2022 to 2024, written while I was a student at the University of Edinburgh. Most are worked examples in physics and mathematics. In October 2026 I corrected the errors I found in them. Each post ends with a note on what was wrong and what changed; otherwise they are kept as they were.
A few I would still point people to:
- Proving the figure-eight orbit exists: Chenciner and Montgomery’s action-minimisation argument on the shape sphere.
- Sundman’s power series solution to the three-body problem: why a convergent series solution exists and why it is useless in practice.
- Symplectic integrators: why they keep the energy error bounded over long runs.
- Theorema Egregium: why every flat map is wrong.
- Plane crash problem: how long does it take to fall?
The posts on the three-body problem form the research log of my 2024 summer project, listed in order on Research. You can also browse everything by topic or level, or find the LaTeX course material.
2024
Classifying and Finding New Possible Orbits (Part I)
A brute-force scan of initial velocities for three equal masses: which starting conditions stay bounded, where the figure eight sits, and how to search better.
Proving the Figure-Eight Orbit Exists: Action Minimisation on the Shape Sphere
An outline of Chenciner and Montgomery’s variational proof: minimise the action over one twelfth of the orbit, rule out collisions, and assemble the figure e...
Sundman’s Power Series Solution to the Three-Body Problem (Part II)
Applying Sundman’s series to Lagrange’s equilateral solution shows why it is impractical: it needs more than 10 to the power 8 million terms for one digit of...
Sundman’s Power Series Solution to the Three-Body Problem (Part I)
Sundman’s 1912 series solves the three-body problem for non-zero angular momentum. How the regularising variable works, and why the series is useless in prac...
Problem of Two Fixed Centres (Part II)
Converting simulated trajectories from Cartesian to prolate spheroidal coordinates, and why the first results disagreed with the theory.
Understanding and Visualising the Shape Sphere
Reducing the planar three-body problem by translations, rotations and scaling: the Jacobi map, the Hopf map, and orbits drawn on the shape sphere.
Problem of Two Fixed Centres (Part I)
Euler’s three-body problem: a test mass attracted by two fixed masses. Prolate spheroidal coordinates, the Lagrangian, and Jacobi elliptic functions.
Ellipses and the Two-Body Problem
Polar equations of the ellipse, bound orbits from negative energy, and the Laplace–Runge–Lenz vector, which gives Kepler’s first law.
Symplectic Integrators
Why area-preserving (symplectic) integrators keep the energy error bounded, and how third- and fourth-order schemes improve on Verlet for the figure-eight or...
Plotting and Visualising the Three-Body Problem
Simulating three gravitating bodies with the velocity Verlet scheme, close encounters and slingshots, and a first look at the figure-eight orbit.
Linear Regression with OLS and SGD in Python
Fitting a straight line to world population data with ordinary least squares and with stochastic gradient descent in scikit-learn, and how to judge the fit.
The Discrete Fourier Transform and the FFT: A Worked Example
The radix-2 Cooley–Tukey idea: split a DFT into even and odd parts, then check it by hand on the sequence [1, 2, 3, 4].
Image Processing with Python and the Convolution Theorem
Blurring images and detecting edges by convolving each colour channel with a kernel, computed quickly with the FFT.
Thermodynamic Potentials and Maxwell Relations
Deriving the internal energy, enthalpy, Helmholtz and Gibbs free energies from the fundamental relation, and one Maxwell relation from the symmetry of second...
Solving Systems of Congruences: The Chinese Remainder Theorem
A number leaves remainder 2 on division by 5, 1 by 3 and 4 by 7. What is it? Solving simultaneous congruences with Bézout’s identity.
Wilson’s Theorem
A prime p divides (p − 1)! + 1, and only primes do: a short proof by pairing each residue with its inverse.
Vieta Jumping and Infinite Descent: IMO 1988 Problem 6
If ab + 1 divides a² + b², the quotient is a perfect square: a proof by infinite descent using Vieta’s formulas, with pictures of the jumping.
The Third Fundamental Form and How to Find It
Using the Cayley–Hamilton theorem for the shape operator to show that III − 2H II + K I = 0.
A Brief Introduction to Bose and Fermi Gases
Why the classical ideal gas violates the third law of thermodynamics, and how quantum statistics, for bosons and fermions alike, fixes it.
IMO 1990 Problem 3: Lifting the Exponent and Fermat’s Little Theorem
Find all integers n > 1 for which (2ⁿ + 1)/n² is an integer, using Fermat’s little theorem and the lifting-the-exponent lemma.
Evaluating Integrals with the Laplace Transform
Using the Laplace transform and the division-by-t rule to show that the Dirichlet integral of sin(x)/x equals π/2.
Euler’s Totient Function, the Carmichael Function and Carmichael Numbers
Why 3²⁰ ≡ 1 (mod 100): Euler’s theorem, its sharpening by the Carmichael function, and the composite numbers that fool Fermat’s primality test.
Theorema Egregium: Why Every Flat Map Is Wrong
Gaussian curvature is intrinsic, so a sphere and a plane cannot be isometric, and no flat map of the Earth can preserve distances.
Why Is Momentum Conserved?
In Lagrangian mechanics, a coordinate that the Lagrangian does not depend on comes with a conserved momentum, a first glimpse of Noether’s theorem.
Julia Sets and the Mandelbrot Set
Filled Julia sets, the Mandelbrot set, and the dichotomy theorem that connects them: c is in the Mandelbrot set exactly when its Julia set is connected.
Pendulum Models for Liquid Slosh
A group project modelling the sloshing of a confined liquid with a multi-pendulum model and a continuous forced-string model.
Overview of the Two-Body Problem
Separating centre-of-mass and relative motion reduces two interacting bodies to a single body with the reduced mass.
Tensors and Their Transformations
The transformation law for Cartesian tensors under a change of basis, why it reads T’ = L T Lᵀ for rank 2, and a worked inertia-tensor example.
2023
More on Multivariable Calculus
Areas from Green’s theorem, a check of the generalised Stokes theorem on the unit ball, and grad, div, curl and the Laplacian in orthogonal curvilinear coord...
Two Particles Falling Together Under Gravity
Two point masses start at rest a distance R apart. How long until they collide? A short calculation, checked against Kepler’s third law.
The Shell Theorem: A Gravitational Analogue of Gauss’s Law
Dropping a ball through a tunnel to the centre of the Earth, and why only the enclosed mass matters: Gauss’s law for gravity and the shell theorem.
Plane Crash Problem: How Long Does It Take to Fall?
Terminal velocity of a person and a plane under quadratic drag, first with constant air density and then with an exponential atmosphere.
Perturbations in Quantum Mechanics
First-order non-degenerate perturbation theory: the energy shift is the expectation value of the perturbation, with a worked infinite-well example.
Perturbations in Classical Mechanics
Small perturbations of a circular orbit: the radial equation, the condition for stable oscillation, and when a large kick lets a particle escape.
2022
Conservation Laws in Physics: Noether’s Theorem
A one-dimensional sketch of why translational symmetry implies conservation of momentum, followed by a derivation of angular momentum conservation for centra...
Kepler’s Third Law and the Virial Theorem
Two results about orbiting bodies: Kepler’s third law, from circles to general ellipses, and the virial theorem with an application to the Jeans mass.