Great Problems and the People Who Solved Them

Differential equations did not arrive as a finished theory. Each method in this course was invented to answer a concrete question — how a bead slides down a wire, how the planets stay in their orbits, how a heated rod cools, how a rotating fluid mass changes shape — and the general theory was abstracted afterward. Reading the subject forward from its problems, rather than backward from its theorems, explains why the parts fit together the way they do. The people and the problems run from Newton to Poincaré.

Newton and Leibniz: the calculus and its quarrel

The subject begins with the tool that made it possible. Isaac Newton (1642–1727) did his foundational work in two plague years of rural solitude, 1665 to 1667, when Cambridge closed and he returned to Woolsthorpe. In that stretch, between the ages of 22 and 24, he developed the binomial series for fractional exponents, both branches of calculus, universal gravitation as the mechanism of the solar system, and the analysis of white light into the spectrum. He was secretive and disliked publishing; his mathematical discoveries circulated by conversation and letter, and his masterpiece, the Principia (1687), was dragged out of him by Halley over eighteen months of concentration. It deduced Kepler's laws from the inverse-square law, treated wave motion and tides, and computed the masses of the sun and planets.¹

Gottfried Wilhelm Leibniz invented calculus independently in Germany; his papers of 1684 and 1686 were the first publications on the subject. The two systems were built from the same idea but different notation, and Leibniz's differential notation proved the more fruitful. What began as mild insinuation escalated into the most damaging priority dispute in the history of science.

• Who was first. Newton had the calculus eight to ten years earlier but did not publish; Leibniz published first.
• The cost. British mathematicians, loyal to Newton's geometric methods and clumsy notation, cut themselves off from the Continent. Leadership in analysis passed to the Bernoullis, Euler, Lagrange, Laplace, and Gauss, and British mathematics stagnated for most of the eighteenth and nineteenth centuries.

The analytic notation of the Continent, not the geometric style of the Principia, became the language in which differential equations were actually solved.

The Bernoulli challenge

In 1696 Johann (John) Bernoulli posed a problem to the mathematicians of Europe: given two points at different heights, find the shape of wire down which a bead slides from the upper to the lower in the least time. He named it the brachistochrone, from the Greek for shortest time. Galileo had guessed the circular arc; the true answer, a cycloid, is far from obvious, and the problem became the historical source of the calculus of variations.²

The challenge drew solutions from both Bernoulli brothers, from Leibniz, and from Newton, who — by then Master of the Mint and out of the habit of research — solved it in a single evening and published anonymously. Bernoulli reportedly recognized the author at once: I recognize the lion by his claw. The Bernoulli family was to mathematics what the Bachs were to music, producing eight mathematicians in three generations:

• James Bernoulli (1654–1705). Self-taught in the new calculus; worked on infinite series and special curves, introduced polar coordinates and the Bernoulli numbers, and stated the law of large numbers.
• John Bernoulli (1667–1748). James's younger brother; mastered calculus and applied it to geometry, differential equations, and mechanics; posed the brachistochrone and taught the young Euler at Basel.
• Daniel Bernoulli (1700–1782). John's son; his Hydrodynamica founded fluid mechanics and gave the earliest kinetic theory of gases.

John's solution was the more ingenious, James's the more general, and the resulting quarrel between the brothers was conducted in language better suited to a street brawl. The rivalry was as productive as it was bitter: the general method James reached for is the one Euler would systematize.

Euler: the flood of methods

Leonhard Euler (1707–1783), a native of Basel and a student of John Bernoulli, was the most prolific mathematician of any era, averaging some 800 printed pages a year across a long life and continuing to increase his output after going blind in his last seventeen years. His imprint on differential equations is everywhere.³

• Calculus of variations. He founded the subject in 1744 on the basic differential equation for a minimizing curve, the Euler equation of the previous lesson.
• Notation. The symbols $e$, $\pi$, $i$, $\Sigma$, the functional notation $f(x)$, and $\sin x$, $\cos x$ are his, along with the identity $e^{i\theta }=\cos \theta +i\sin \theta$ that ties the exponential and trigonometric functions together.
• Special functions. He discovered the integral defining the gamma function, $\Gamma (x)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}\mathbf{d}t$, and Euler's constant $\gamma$; he met the Bessel functions in the vibration of a stretched circular membrane, and applied Laplace transforms to solve differential equations — all before Bessel and Laplace were born.
• Series methods. He contributed much of the theory of second-order linear equations and the method of solution by power series.

Euler's reach ran past differential equations into number theory, the Königsberg bridges (the origin of graph theory), and the polyhedron formula $V-E+F=2$. Simmons ranks him, with Gauss and Riemann, among the three greatest mathematicians of the modern era.

Lagrange: mechanics made analytic

Joseph Louis Lagrange (1736–1813) was appointed professor at the Royal Artillery School in Turin at nineteen. His earliest and most important work was in the calculus of variations: in 1755 he sent Euler the method of multipliers for isoperimetric problems that had baffled Euler for years, and Euler, with notable generosity, withheld his own results so as not to diminish the younger man's credit.⁴

Lagrange's masterpiece, the Mécanique Analytique (1788), unified mechanics into what Hamilton later called a kind of scientific poem. It reduced the whole subject to analysis, with no diagrams, resting on Hamilton's principle and yielding:

• Lagrange's equations of motion, one per degree of freedom;
• generalized coordinates, freeing the analysis from any particular coordinate system;
• the concept of potential energy.

These are the tools that turn a mechanical vibration problem into a differential equation without ever drawing a free-body diagram. Lagrange also led the establishment of the metric system.

Gauss and Riemann: the age of rigor

Carl Friedrich Gauss (1777–1855) marks the boundary between the modern era in mathematics and everything before it. A prodigy from Brunswick, he decided for mathematics at eighteen on discovering which regular polygons are constructible by ruler and compass, and filled a terse scientific diary with results he never found time to publish.⁵

Two consequences reach directly into this course. Gauss's insistence that a proof must banish every doubt — $\frac{1}{2}$ proof $=0$ — reset the standard by which the existence and convergence theorems of the subject would later be judged. And his intrinsic geometry of curved surfaces, generalized by his student Bernhard Riemann into the geometry of higher-dimensional spaces, supplied the language that Einstein's general relativity would need. Gauss held many results back — non-Euclidean geometry, elliptic functions, Cauchy's integral theorem — remarking that no architect leaves the scaffolding standing after the building is done.

Laplace: the mechanics of the heavens

Pierre Simon de Laplace (1749–1827) was famous enough in his own time to be called the Newton of France. His lifelong subjects were celestial mechanics and probability.⁶

• Mécanique Céleste (1799–1825). Five volumes applying Newton's gravitation to the solar system as a whole, in which the planets perturb one another. Where Newton had thought divine intervention might be needed to keep the system from degenerating, Laplace proved the idealized system stable. Asked by Napoleon why God went unmentioned, he is said to have answered, Sire, I had no need of that hypothesis.
• Potential theory. His development of the potential is so thorough that the fundamental equation of the theory has carried his name ever since — Laplace's equation.
• Théorie Analytique des Probabilités (1812). Here he used Laplace transforms and generating functions freely, decades before the transform became a standard method for solving initial value problems.

Laplace's political suppleness — he changed principles with each change of regime and emerged each time with grander titles — has dulled his reputation, but his mathematical legacy anchors two full modules of this course.

Poincaré: from formulas to phase portraits

Jules Henri Poincaré (1854–1912) was recognized as the greatest mathematician of his generation, and probably the last to hold creative command of the whole of the subject. He worked in nearly every field and founded several.⁷

His decisive move for differential equations came in celestial mechanics. In Les Méthodes Nouvelles de la Mécanique Céleste (1892–1899), confronting the three-body problem, he found the equations unsolvable in closed form and changed the question. Rather than seek formulas, he studied the qualitative behavior of solutions — their stability, their periodic orbits, the geometry of their trajectories in the phase plane.

• Qualitative theory of nonlinear equations. The study of critical points, stability, and periodic solutions and limit cycles begins here.
• Topology. His memoir Analysis situs (1895) opened modern algebraic topology, the natural setting for the global structure of solution curves.
• Recurrence and asymptotics. The Poincaré recurrence theorem and his theory of asymptotic expansions grew from the same celestial-mechanics work.

The shift Poincaré made — from solving an equation to understanding its solutions without solving it — is the intellectual content of the nonlinear and numerical modules, where closed-form answers stop being available.

The lineage of the course

Read as a whole, the modules of this course are the sediment of named problems. The brachistochrone produced the calculus of variations; the motion of the planets produced celestial mechanics, perturbation theory, and eventually the qualitative theory of nonlinear systems; the vibrating string and the heated rod produced Fourier series and the special functions of Bessel and Legendre.

The methods look tidy in a textbook because the problems that forced them into existence have been quietly removed. Putting the problems back — the wire, the planets, the string, the three bodies — is what turns a catalogue of techniques into a history, and the history is what explains why the subject arrives, in the end, at a bead sliding down a curve and a rotating mass of fluid changing its shape.