Differential Equations

Hamilton's mechanics

In 1788 Lagrange had turned Newton’s mechanics into a set of equations valid in any coordinates. Hamilton admired that book, which he called “a kind of scientific poem”. In 1834 and 1835, in the two essays On a General Method in Dynamics, he took the idea further, bringing into mechanics what he had learned from rays of light. Position and momentum Lagrange described a system by its positions $q$ and velocities $\dot q$. Hamilton replaced the velocity with the momentum $p$ (for a simple particle, mass times velocity, $p = m\dot q$) and put at the centre of everything a single function, the Hamiltonian $H(q, p)$, which in most cases is the total energy, kinetic plus potential:

October 27, 2026 Read

Laplace's equation

To know how a planet attracts a point, one must add up the attraction of each of its pieces. Each piece pulls in a different direction, and adding forces is laborious: they are vectors, with three components each. Eighteenth-century mathematicians discovered a shortcut, and Laplace turned it into a tool. Adding numbers instead of arrows For a mass $M$ concentrated at a point, consider the number $$V = \frac{M}{r},$$where $r$ is the distance to the mass (we leave out the constant $G$). This number, which later received the name potential, holds all the information about the force: the components of the attraction in each direction are the derivatives of $V$ in that direction. And the great advantage is that, for several masses, the potentials simply add like numbers. First one adds the potential of all the pieces of the planet; then, with a derivative, one gets the force.

October 30, 2026 Read

The inverse method of tangents

Differential calculus answers a question: given a curve, what is its tangent at each point? The mathematicians of the seventeenth century soon realized that the inverse question was harder and more useful: given a property the tangents must have, what is the curve? This problem was called the inverse method of tangents, and it is the subject of the fourth and last book of the Instituzioni analitiche. Today we call it solving a differential equation.

October 29, 2026 Read

Morphogenesis: the leopard's spots in equations

An embryo starts as a ball of almost identical cells. Even so, it ends up with five fingers on each hand, stripes on a zebra and spots on a leopard. Where does form come from, if at the start everything is uniform? In 1952, in the last major paper he published, “The chemical basis of morphogenesis”, Turing proposed a mathematical answer that biologists took decades to confirm. Two substances, two speeds The idea is to imagine two chemical substances, which Turing called morphogens, spreading through a tissue. Today they are usually described like this:

October 18, 2026 Read

The Laplace transform

Logarithms turn multiplications into sums: to multiply two numbers, one adds their logarithms and comes back. The Laplace transform does something similar with differential equations: it swaps derivatives for multiplications. One solves an algebraic equation and then comes back. The origin Laplace came to this idea through probability. In the Analytic Theory of Probabilities, of 1812, he organized his calculations with generating functions: instead of working with a sequence of probabilities, he worked with a single function that stored them all. The continuous version of this idea is an integral that Euler had used earlier, but which Laplace explored systematically:

October 30, 2026 Read

The D that factorises: the derivative as an operator

Write $D$ for “differentiate with respect to $x$”. Then $D(x^3)=3x^2$, $D(\sin x)=\cos x$, and differentiating twice is $D(D(f))$, abbreviated $D^2f$. So far it is just an abbreviation. Things get interesting when you take the shape of the notation seriously: if $D^2$ looks like a square, why not treat $D$ as a number? A differential equation becomes a polynomial Consider the equation $$y''-3y'+2y=0.$$With $D$, it is written $D^2y-3Dy+2y=0$, or, taking $y$ out as a factor,

October 18, 2026 Read

Differential equations: the law in an anagram, ∂ and ∇

In 1676 the Englishman Robert Hooke published, at the end of a book on helioscopes, instruments for observing the Sun, a list of discoveries he did not yet want to reveal. One of them read: $$\textit{ceiiinosssttuv}$$It was an anagram: the letters of a Latin sentence, shuffled and put in alphabetical order. It was a way of recording priority for a discovery without handing it to rivals. Two years later, in 1678, Hooke revealed the sentence: ut tensio, sic vis, “as the extension, so the force”. The more a spring is stretched, the harder it pulls back. Today we write it in four signs:

October 10, 2026 Read