Open a physics textbook on mechanics.
You'll find two approaches,
Newton's with forces and accelerations and geometric diagrams,
and Lagrange's with energy and algebra and not a single picture.
Joseph Louis Lagrange boasted that his masterwork,
Mécanique analytique, contained no figures.
He reduced all of physics to pure equations
and made it easier to solve in the process.
Last time we met Leonhard Euler, the most prolific mathematician in history,
who invented graph theory, solved the Basel problem,
and discovered the most beautiful equation in mathematics, all while going blind.
Welcome back to Men of Mathematics.
Today we meet Euler's successor, Joseph Louis Lagrange,
the mathematician who transformed physics into pure algebra.
Frederick the Great wanted the greatest mathematician in Europe,
at his court, and got him. Joseph Louis Lagrange was born Giuseppe Lodovico Lagrangia on January
25, 1736, in Turin, then part of the Kingdom of Sardinia. His father was a wealthy official who
lost his fortune to speculation, which Lagrange later said was fortunate, since wealth might have
kept him from mathematics. At 19, Lagrange was appointed professor of mathematics at the Royal
Artillery School in Turin. His correspondence with Euler led to fundamental advances in the
calculus of variations. In 1766, Frederick the Great invited Lagrange to Berlin to succeed Euler,
who was returning to St. Petersburg. Frederick declared he wanted the greatest mathematician
in Europe at his court Lagrange stayed for 20 years producing his finest work After Frederick death he moved to Paris Lagrange arrived in Paris just before the French Revolution
Despite being foreign-born, his brilliance protected him during the terror. When his
friend Lavoisier, the chemist, was guillotined, Lagrange mourned, it took them only an instant
to cut off that head, and a hundred years may not produce another like it.
Napoleon later made Lagrange a count and senator. Newton described motion using forces,
F equals MA, with vectors and diagrams. Lagrange replaced this with energy. Write down the kinetic
energy T and potential energy V, form the Lagrangian L equals T minus V, and apply the
Euler-Lagrange equation. The equations of motion fall out automatically, no force analysis needed.
For a pendulum, you get the equation of motion without ever drawing a force diagram.
Newton's method requires choosing coordinates and analyzing forces in each direction.
Lagrange's method works in any coordinate system, handles constraints automatically,
and generalizes to quantum mechanics and relativity. The standard model of particle
physics is written as a Lagrangian. Einstein's general relativity uses the same framework.
How do you optimize a function when you're constrained to stay on a curve or surface?
Lagrange's brilliant insight, at the optimum, the gradient of the objective must be parallel
to the gradient of the constraint. The gradient of f equals lambda times the gradient of g.
This introduces the Lagrange multiplier lambda,
a technique used throughout economics physics machine learning and engineering Given n data points there exactly one polynomial of degree at most n
passing through all of them.
Lagrange gave an explicit formula to construct it.
This is fundamental to numerical analysis, error-correcting codes, and cryptography.
In celestial mechanics, Lagrange discovered five equilibrium points,
where a small object can maintain a stable position relative to two larger bodies.
These Lagrange points in the Sun-Earth system are now used for space telescopes.
The James Webb Space Telescope orbits L2, about 1.5 million kilometers from Earth.
Every positive integer can be written as the sum of at most four perfect squares.
Fermat claimed this.
Euler worked on it for decades.
Lagrange proved it in 1770.
42 equals 1 squared plus 4 squared plus 5 squared, which is 1 plus 16 plus 25.
Lagrange made number theory rigorous.
His study of polynomial equations, asking when they can be solved by radicals,
directly inspired the work of Abel and Galois on group theory.
Lagrange's theorem in group theory states,
the order of a subgroup divides the order of the group.
Though Goulois developed group theory, this fundamental theorem bears Lagrange's name.
Lagrange's masterwork, published in 1788, reformulated all of mechanics using pure algebra.
He boasted that it contained not a single figure, the ultimate triumph of analysis over geometry.
The book influenced all subsequent physics.
Hamilton would extend it.
Quantum mechanics would adopt it.
Particle physics would speak its language As chair of the Weights and Measures Commission during the Revolution Lagrange helped design the metric system The decimal approach we take for granted emerged
from this committee. Despite his achievements, Lagrange periodically lost interest in mathematics.
Once a discovery was made, the pleasure was gone. When we have found a truth, he said,
the pleasure of discovery is over, and we must go on to seek another.
Joseph Louis Lagrange transformed physics into pure algebra with his Lagrangian mechanics.
His multipliers solve constrained optimization throughout science and engineering.
His work on equations inspired group theory. His celestial mechanics still guide spacecraft.
And he did it all without drawing a single picture. On his deathbed, Lagrange reflected,
I have had my day. I have gained some reputation in mathematics. I never hated anyone. I have done
no ill. I am going to die. A gentle exit for a gentle genius, one who found perfection in pure
algebra and loss of interest in every theorem once proved. As E.T. Bell wrote, Lagrange's
mathematical work has a perfection of form that is truly classical. He is the supreme mathematical
architect, concerned with the harmony and symmetry of the whole. Next time on Men of
Mathematics, we meet the Newton of France, Pierre-Simon Laplace. He completed Newton's
work on celestial mechanics, founded modern probability theory, and articulated the ultimate
deterministic vision, an intelligence that could predict the entire future from perfect knowledge
of the present. Thank you for joining us on this journey through mathematical history.
If you appreciate the elegance of pure algebra, subscribe and hit the notification bell.
New episodes release every week. I'll see you next time.