1 00:00:00,000 --> 00:00:02,020 Open a physics textbook on mechanics. 2 00:00:02,540 --> 00:00:03,760 You'll find two approaches, 3 00:00:04,340 --> 00:00:08,240 Newton's with forces and accelerations and geometric diagrams, 4 00:00:08,240 --> 00:00:13,260 and Lagrange's with energy and algebra and not a single picture. 5 00:00:14,200 --> 00:00:17,160 Joseph Louis Lagrange boasted that his masterwork, 6 00:00:17,440 --> 00:00:21,160 Mécanique analytique, contained no figures. 7 00:00:21,800 --> 00:00:25,100 He reduced all of physics to pure equations 8 00:00:25,100 --> 00:00:28,280 and made it easier to solve in the process. 9 00:00:28,280 --> 00:00:33,720 Last time we met Leonhard Euler, the most prolific mathematician in history, 10 00:00:34,140 --> 00:00:37,200 who invented graph theory, solved the Basel problem, 11 00:00:37,600 --> 00:00:42,520 and discovered the most beautiful equation in mathematics, all while going blind. 12 00:00:43,240 --> 00:00:45,000 Welcome back to Men of Mathematics. 13 00:00:45,740 --> 00:00:49,540 Today we meet Euler's successor, Joseph Louis Lagrange, 14 00:00:49,900 --> 00:00:53,460 the mathematician who transformed physics into pure algebra. 15 00:00:54,060 --> 00:00:58,120 Frederick the Great wanted the greatest mathematician in Europe, 16 00:00:58,280 --> 00:01:06,060 at his court, and got him. Joseph Louis Lagrange was born Giuseppe Lodovico Lagrangia on January 17 00:01:06,060 --> 00:01:14,280 25, 1736, in Turin, then part of the Kingdom of Sardinia. His father was a wealthy official who 18 00:01:14,280 --> 00:01:19,900 lost his fortune to speculation, which Lagrange later said was fortunate, since wealth might have 19 00:01:19,900 --> 00:01:27,140 kept him from mathematics. At 19, Lagrange was appointed professor of mathematics at the Royal 20 00:01:27,140 --> 00:01:33,820 Artillery School in Turin. His correspondence with Euler led to fundamental advances in the 21 00:01:33,820 --> 00:01:41,560 calculus of variations. In 1766, Frederick the Great invited Lagrange to Berlin to succeed Euler, 22 00:01:41,880 --> 00:01:47,260 who was returning to St. Petersburg. Frederick declared he wanted the greatest mathematician 23 00:01:47,260 --> 00:02:00,289 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 24 00:02:01,029 --> 00:02:05,929 Despite being foreign-born, his brilliance protected him during the terror. When his 25 00:02:05,929 --> 00:02:12,449 friend Lavoisier, the chemist, was guillotined, Lagrange mourned, it took them only an instant 26 00:02:12,449 --> 00:02:17,709 to cut off that head, and a hundred years may not produce another like it. 27 00:02:18,789 --> 00:02:25,049 Napoleon later made Lagrange a count and senator. Newton described motion using forces, 28 00:02:25,529 --> 00:02:33,489 F equals MA, with vectors and diagrams. Lagrange replaced this with energy. Write down the kinetic 29 00:02:33,489 --> 00:02:40,869 energy T and potential energy V, form the Lagrangian L equals T minus V, and apply the 30 00:02:40,869 --> 00:02:47,469 Euler-Lagrange equation. The equations of motion fall out automatically, no force analysis needed. 31 00:02:48,129 --> 00:02:52,409 For a pendulum, you get the equation of motion without ever drawing a force diagram. 32 00:02:53,289 --> 00:02:58,369 Newton's method requires choosing coordinates and analyzing forces in each direction. 33 00:02:59,149 --> 00:03:03,729 Lagrange's method works in any coordinate system, handles constraints automatically, 34 00:03:03,729 --> 00:03:09,869 and generalizes to quantum mechanics and relativity. The standard model of particle 35 00:03:09,869 --> 00:03:16,729 physics is written as a Lagrangian. Einstein's general relativity uses the same framework. 36 00:03:17,289 --> 00:03:21,929 How do you optimize a function when you're constrained to stay on a curve or surface? 37 00:03:22,869 --> 00:03:28,409 Lagrange's brilliant insight, at the optimum, the gradient of the objective must be parallel 38 00:03:28,409 --> 00:03:35,269 to the gradient of the constraint. The gradient of f equals lambda times the gradient of g. 39 00:03:35,269 --> 00:03:39,089 This introduces the Lagrange multiplier lambda, 40 00:03:39,669 --> 00:03:50,617 a technique used throughout economics physics machine learning and engineering Given n data points there exactly one polynomial of degree at most n 41 00:03:50,977 --> 00:03:52,157 passing through all of them. 42 00:03:52,797 --> 00:03:55,557 Lagrange gave an explicit formula to construct it. 43 00:03:55,557 --> 00:04:00,997 This is fundamental to numerical analysis, error-correcting codes, and cryptography. 44 00:04:01,737 --> 00:04:05,977 In celestial mechanics, Lagrange discovered five equilibrium points, 45 00:04:05,977 --> 00:04:11,357 where a small object can maintain a stable position relative to two larger bodies. 46 00:04:12,097 --> 00:04:17,377 These Lagrange points in the Sun-Earth system are now used for space telescopes. 47 00:04:18,497 --> 00:04:24,937 The James Webb Space Telescope orbits L2, about 1.5 million kilometers from Earth. 48 00:04:25,417 --> 00:04:30,117 Every positive integer can be written as the sum of at most four perfect squares. 49 00:04:30,817 --> 00:04:31,917 Fermat claimed this. 50 00:04:32,457 --> 00:04:34,077 Euler worked on it for decades. 51 00:04:34,697 --> 00:04:36,997 Lagrange proved it in 1770. 52 00:04:37,497 --> 00:04:45,097 42 equals 1 squared plus 4 squared plus 5 squared, which is 1 plus 16 plus 25. 53 00:04:46,417 --> 00:04:48,257 Lagrange made number theory rigorous. 54 00:04:49,137 --> 00:04:54,017 His study of polynomial equations, asking when they can be solved by radicals, 55 00:04:54,017 --> 00:04:58,537 directly inspired the work of Abel and Galois on group theory. 56 00:04:59,477 --> 00:05:01,977 Lagrange's theorem in group theory states, 57 00:05:01,977 --> 00:05:05,697 the order of a subgroup divides the order of the group. 58 00:05:06,637 --> 00:05:11,617 Though Goulois developed group theory, this fundamental theorem bears Lagrange's name. 59 00:05:12,357 --> 00:05:18,797 Lagrange's masterwork, published in 1788, reformulated all of mechanics using pure algebra. 60 00:05:19,477 --> 00:05:25,597 He boasted that it contained not a single figure, the ultimate triumph of analysis over geometry. 61 00:05:26,157 --> 00:05:28,717 The book influenced all subsequent physics. 62 00:05:29,357 --> 00:05:30,537 Hamilton would extend it. 63 00:05:30,937 --> 00:05:32,497 Quantum mechanics would adopt it. 64 00:05:32,917 --> 00:05:44,546 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 65 00:05:44,546 --> 00:05:50,606 from this committee. Despite his achievements, Lagrange periodically lost interest in mathematics. 66 00:05:51,446 --> 00:05:57,306 Once a discovery was made, the pleasure was gone. When we have found a truth, he said, 67 00:05:57,306 --> 00:06:01,626 the pleasure of discovery is over, and we must go on to seek another. 68 00:06:02,586 --> 00:06:08,006 Joseph Louis Lagrange transformed physics into pure algebra with his Lagrangian mechanics. 69 00:06:08,546 --> 00:06:13,326 His multipliers solve constrained optimization throughout science and engineering. 70 00:06:13,906 --> 00:06:19,746 His work on equations inspired group theory. His celestial mechanics still guide spacecraft. 71 00:06:20,646 --> 00:06:25,546 And he did it all without drawing a single picture. On his deathbed, Lagrange reflected, 72 00:06:25,546 --> 00:06:32,546 I have had my day. I have gained some reputation in mathematics. I never hated anyone. I have done 73 00:06:32,546 --> 00:06:40,426 no ill. I am going to die. A gentle exit for a gentle genius, one who found perfection in pure 74 00:06:40,426 --> 00:06:47,446 algebra and loss of interest in every theorem once proved. As E.T. Bell wrote, Lagrange's 75 00:06:47,446 --> 00:06:53,566 mathematical work has a perfection of form that is truly classical. He is the supreme mathematical 76 00:06:53,566 --> 00:06:59,046 architect, concerned with the harmony and symmetry of the whole. Next time on Men of 77 00:06:59,046 --> 00:07:05,486 Mathematics, we meet the Newton of France, Pierre-Simon Laplace. He completed Newton's 78 00:07:05,486 --> 00:07:11,206 work on celestial mechanics, founded modern probability theory, and articulated the ultimate 79 00:07:11,206 --> 00:07:17,446 deterministic vision, an intelligence that could predict the entire future from perfect knowledge 80 00:07:17,446 --> 00:07:21,646 of the present. Thank you for joining us on this journey through mathematical history. 81 00:07:21,646 --> 00:07:27,246 If you appreciate the elegance of pure algebra, subscribe and hit the notification bell. 82 00:07:27,826 --> 00:07:30,986 New episodes release every week. I'll see you next time.