Senior Projects 2: Johann Bernoulli

Continuing with the discussion of the senior projects I'm directing: student #2 took my Development of Modern Mathematics class last semester. In our initial discussion of possible projects, he mentioned that he had been an engineering student before switching to math, and that he was most interested in applied math. As it happened, a few days earlier I had described one of the more famous - and consequential - problems in the history of mathematics, the problem of the brachistochrone, and when I suggested a project on that topic he jumped at it.

The problem of the brachistochrone was first posed to Isaac Newton in 1696. Report has it that he solved it overnight. (Report is not always trustworthy when Newton is involved, especially if it's Newton who's reporting.) The story continues that a friend persuaded him not to reveal his solution for six months, so as to allow other mathematicians to take a crack at it. (Sounds suspiciously convenient to me, but who knows, at this late date?) Among the other people who solved it were Christiaan Huygens and both of the Bernoulli brothers. The last two worked independently; the rivalry between Jakob and Johann is legendary. Johann's first solution (as I understand the story) turned out to be flawed; after Jakob came up with a better answer, Johann tried again, and it's that second solution that is of interest. The methods Johann developed to solve the problem spawned the branch of mathematics known as calculus of variations, which has proven a rich and valuable field of study; my student's project is to master and present Johann's solution of the problem. There's more on the problem below the cut.

( The Brachistochrone )