Calculus of Variations

The life of Joseph-Louis Lagrange

Timeline 1736Born on 25 January in Turin, capital of the Kingdom of Sardinia, as Giuseppe Lodovico Lagrangia. 1753At 17, reads an essay by Edmond Halley and decides to devote himself to mathematics. 1755Writes to Euler with the method that will give rise to the calculus of variations. In September he is appointed professor at the Artillery School in Turin, aged 19. 1757Helps found the scientific society that will become the Turin Academy of Sciences. 1764Wins the Paris Academy prize by explaining why the Moon always shows the same face. 1766Moves to Berlin to take Euler’s place at the Academy of Frederick the Great. 1770Proves that every whole number is a sum of four squares and begins studying the permutations of the roots of an equation. 1772Shares the Paris prize with Euler for the three-body problem, where the Lagrange points appear. 1787After Frederick’s death, moves to Paris, where he lives in the Louvre. 1788Publishes the Analytical Mechanics. 1790Joins the commission that will create the metric system. 1794Lavoisier is guillotined. Lagrange becomes a professor at the new École Polytechnique. 1797Publishes the Theory of Analytic Functions, with the notation $f'(x)$. 1808Napoleon makes him a count of the Empire. 1813Dies on 10 April in Paris and is buried in the Panthéon. ◀ Previous Follow the journey Next ▶ 1736–1766 TurinAt 19 he is a professor at the Artillery School. He writes to Euler the method that will give rise to the calculus of variations. 1766–1787 BerlinHe takes Euler’s place at the Academy of Frederick the Great. He proves the four-square theorem and finds the Lagrange points. 1787–1813 ParisHe lives in the Louvre, publishes the Analytical mechanics, helps create the metric system and teaches at the École Polytechnique. He is buried in the Panthéon. Enable JavaScript to see the map. The stages are listed below. Lagrange in three capitals, Turin, Berlin and Paris. Map: © OpenStreetMap contributors

October 14, 2026 Read

The calculus of variations: a letter from a 19-year-old

The calculus of Newton and Leibniz can find the highest point of a curve: it is where the derivative vanishes. But there are questions of another kind. What is the shortest path between two points? What shape should a slide have so that a ball runs down it fastest? What curve does a hanging chain make? In these questions the answer is not a number but a whole curve, chosen from infinitely many possible curves.

October 14, 2026 Read

The Analytical Mechanics: a book without figures

Open Newton’s Principia of 1687 and you will see figures on almost every page: triangles, ellipses, arrows, tangents. Open Lagrange’s Analytical Mechanics, published a century later in 1788, and you will see none. The author warns of this himself in the preface, with pride, and adds: Those who love analysis will be pleased to see mechanics become a new branch of it, and will be grateful to me for having thus extended its domain. — Joseph-Louis Lagrange, preface to the Analytical Mechanics (1788)

October 14, 2026 Read

The Lagrange points: where gravity balances

On 25 December 2021 a rocket lifted off from French Guiana carrying the James Webb Space Telescope. It was not placed in orbit around the Earth, like Hubble. It travelled for a month to an empty point in space, 1.5 million kilometres from here, always on the side opposite the Sun, and stayed there. The address of that point comes from a prize-winning paper by Lagrange, from 1772. It is called L2.

October 14, 2026 Read

Four squares are enough

Take any positive whole number and try to write it as a sum of perfect squares, $0, 1, 4, 9, 16, 25, \ldots$, using as few terms as possible: $$\begin{aligned} 5&=4+1,\\ 6&=4+1+1,\\ 7&=4+1+1+1,\\ 30&=25+4+1,\\ 2026&=2025+1=45^2+1^2. \end{aligned}$$Some numbers are squares, others need two, others three. The number 7 needs four: there is no other way. Could some number need five? In 1770 Lagrange proved that no number does: Every positive whole number is a sum of four squares of integers.

October 14, 2026 Read

Shuffling the roots: Lagrange's theorem

In 1770 algebra had a problem that had been stuck for more than two centuries. There were formulas for solving equations of degree 2, the quadratic formula, and also of degrees 3 and 4, published by Cardano in 1545. For degree 5, nobody could find anything. The best mathematicians had tried, and every method that worked up to degree 4 jammed at 5. Lagrange did not try yet another method. He asked a different question: why do the old methods work?

October 14, 2026 Read

The derivative without infinitesimals and the mean value theorem

At the end of the eighteenth century, calculus worked very well and nobody could properly explain why. Leibniz’s $dx$ were “infinitely small” quantities: sometimes they were treated as if they were different from zero, so that one could divide by them, and right afterwards as if they were zero, so that one could discard them. Bishop Berkeley had mocked this in 1734, calling them “ghosts of departed quantities”. Lagrange was bothered by it. In 1797 he published a book whose subtitle is a whole programme:

October 14, 2026 Read

Lagrange multipliers: optimising under a constraint

Almost every real decision is a maximum or minimum problem with a constraint. A factory wants the greatest profit but has a limited amount of raw material. A can must hold 350 ml using as little aluminium as possible. A pendulum moves however it likes, as long as the bob stays attached to the string. Lagrange found a single method for all these problems. It appears in the Analytical Mechanics of 1788, to deal with the ties of mechanical systems, and in the Theory of Analytic Functions of 1797, as a general rule of calculation.

October 14, 2026 Read

Lagrange's legacy

Lagrange is the lofty pyramid of the mathematical sciences. — Napoleon Bonaparte, according to his contemporaries Lagrange left no anecdotes of precocious genius like Pascal, nor a body of work the size of Euler’s. He left something else: a way of doing mathematics. Where others drew, he calculated; where others found a trick for each problem, he looked for the rule that explained all the tricks. Almost everything he did is still in the textbooks under his name: the Euler–Lagrange equation, the Lagrangian, the Lagrange points, Lagrange’s theorem, the Lagrange remainder, Lagrange multipliers.

October 14, 2026 Read

Image credits for Lagrange

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Joseph-Louis Lagrange, engraving by an unknown artist: source page · public domain. Medallion with Lagrange’s profile, bronze by Pierre-Jean David d’Angers, Musées d’Angers: source page · CC0. Title page of the Méchanique analitique, Paris, 1788: source page · public domain. Statue of Lagrange in Turin, photo by Twice25, 2009: source page · CC BY-SA 3.0. Blog figures The cover, the diagram of the five Lagrange points and the graph of curves of equal area with the constraint line are original illustrations, with versions in each of the site’s languages. The diagram of the points is not to scale: L1 and L2 are in reality a hundred times closer to the Earth than the Sun is.

October 14, 2026 Read