# Joseph-Louis Lagrange

**Joseph-Louis Lagrange** (1736-1813) was a mathematician and astronomer born in Turin (Italy), as *Giuseppe Lodovico LaGrangia*. He worked for over twenty years in Berlin (Prussia), as the successor of Leonard Euler as a director of the Prussian Academy of Sciences. In 1787, he moved to France, where he stayed for the rest of his life.

## Calculus of Variations

Lagrange is one of the founders of the calculus of variations, a field of mathematics which was inspired by the *problem of the brachistochrone*, posed by Johann Bernoulli in June 1696.

The brachistochrone problem is that of finding the shape of a ramp that will get an object to drop, and move horizontally, in as short a time as possible. If no horizontal distance needed to be covered, we would just drop the object. Otherwise, we might set up some kind of ramp that the object slides down. What is the best shape of the ramp?

Anyone who has watched "half pipe" sporting activities might guess that the answer is a shape somewhat like a semicircle. By going down steeply at first, the object picks up speed early, which gets it to its destination faster. The actual solution is a *cycloid*.

In general, the calculus of variations solves this sort of problem: What is the shape of a function that maximizes or minimizes something that one might measure about the function? The problem of finding the shortest distance between two points is an example of this. It is sort of a generalization of differential calculus. Differential calculus tells us how to answer questions of the form "What is the value of x that maximizes or minimizes something that one might measure about x?" (That is, some function of x.) The calculus of variations takes this to a higher level, but the solution method is similar.