Senior Projects 2: Johann Bernoulli
Jun. 3rd, 2006 04:14 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowContinuing 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.
Picture two pegs on a wall, at points A and B. For convenience, assume that B is below and to the right of A. Now place a ramp, of whatever shape, running from A to B. If you place a ball at A and let it go, it will take a certain amount of time to roll to B. (Assume that gravity is the only operative force; in particular, we're going to ignore friction.) Question: what shape must the ramp be, to minimize that length of time? That shape is the brachistochrone, and the problem is to identify it.
Your first thought might be that a straight-line ramp would be best. Now, it's true that that's the shortest-length path, but it turns out not to be the shortest-time path. To see why, envision a ramp that sags just a little below the straight line. At the beginning, then, the ramp is a little closer to vertical than the straight line; therefore, the effective acceleration due to gravity is a little bit greater. At the end, similarly, the effective acceleration is a little bit less. But - and this is the key point - the increased acceleration at the beginning has more time to work than the reduced acceleration at the end, and the net effect may be large enough to make up for the extra length of the path. Indeed, this is what happens.
So, what is the brachistochrone? Imagine nailing a horizontal straightedge to a wall, and placing a circular wheel against the wall, resting on the straightedge. Attach a pencil lead to the wheel, right at the rim, and let the wheel roll along the straightedge. The pencil lead will trace out a shape on the wall, known as a cycloid. (If you ever played with a Spiro-Graph, you probably drew a cycloid at some point.) The cycloid turns out to have many interesting properties - mathematicians have been studying it since the time of Galileo - but the one we're interested in here is that the brachistochrone is an segment of an upside-down cycloid.
The blockquote contains technicalities, which the math-averse may skip.
This idea of Johann Bernoulli developed into a general technique for identifying shapes - curves, surfaces, etc. - which are optimal in some respect. Other examples include the problem of enclosing the maximum area with a curve of fixed length, and the problem of determining the shape a soap-film will take if you dip a wire frame into soapy water. (If the frame is flat, the soap-film is just a plane, but if it's bent the shape is more complex; it's what's known as a "minimal surface", and the study of minimal surfaces is an active subfield of mathematics today.) The calculus of variations is also extremely valuable in physics, especially in dynamics, both classical and quantum.
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.
Picture two pegs on a wall, at points A and B. For convenience, assume that B is below and to the right of A. Now place a ramp, of whatever shape, running from A to B. If you place a ball at A and let it go, it will take a certain amount of time to roll to B. (Assume that gravity is the only operative force; in particular, we're going to ignore friction.) Question: what shape must the ramp be, to minimize that length of time? That shape is the brachistochrone, and the problem is to identify it.
Your first thought might be that a straight-line ramp would be best. Now, it's true that that's the shortest-length path, but it turns out not to be the shortest-time path. To see why, envision a ramp that sags just a little below the straight line. At the beginning, then, the ramp is a little closer to vertical than the straight line; therefore, the effective acceleration due to gravity is a little bit greater. At the end, similarly, the effective acceleration is a little bit less. But - and this is the key point - the increased acceleration at the beginning has more time to work than the reduced acceleration at the end, and the net effect may be large enough to make up for the extra length of the path. Indeed, this is what happens.
So, what is the brachistochrone? Imagine nailing a horizontal straightedge to a wall, and placing a circular wheel against the wall, resting on the straightedge. Attach a pencil lead to the wheel, right at the rim, and let the wheel roll along the straightedge. The pencil lead will trace out a shape on the wall, known as a cycloid. (If you ever played with a Spiro-Graph, you probably drew a cycloid at some point.) The cycloid turns out to have many interesting properties - mathematicians have been studying it since the time of Galileo - but the one we're interested in here is that the brachistochrone is an segment of an upside-down cycloid.
The blockquote contains technicalities, which the math-averse may skip.
Johann Bernoulli's analysis of the brachistochrone begins with the following insight. If P and Q are two points along the brachistochrone, in between A and B, then the piece of the brachistochrone from A to B lying between P and Q must be the brachistochrone from P to Q. (If it weren't, you could replace it with the real P-Q brachistochrone, and get a faster path.) Now, let f(x) be a function describing a path from A to B, and define T(f) to be the elapsed time associated with that path. (T is describable as an integral with respect to x - not the integral of f itself, but of something related to it.) Let f0 describe the brachistochrone. Pick a different path from P to Q, and let f1 be the result of inserting that path into the brachistochrone. For any t between 0 and 1, let ft be the path t of the way between f0 and f1 (so f.5 is halfway between, f.25 is a quarter of the way from f0 to f1, and so on). For t between 0 and -1, let ft lie similarly on the other side of f0 from f1. Then we can regard T(ft) as a function of t. Since the brachistochrone minimizes T, if we take the derivative of T(ft) with respect to t and evaluate at t=0, we should get 0. That is: we take an integral with respect to the variable x, and then differentiate with respect to the variable t. By carefully studying the relationship between the integral and the derivative, we can extract information about f0; and this is what Johann Bernoulli did.
This idea of Johann Bernoulli developed into a general technique for identifying shapes - curves, surfaces, etc. - which are optimal in some respect. Other examples include the problem of enclosing the maximum area with a curve of fixed length, and the problem of determining the shape a soap-film will take if you dip a wire frame into soapy water. (If the frame is flat, the soap-film is just a plane, but if it's bent the shape is more complex; it's what's known as a "minimal surface", and the study of minimal surfaces is an active subfield of mathematics today.) The calculus of variations is also extremely valuable in physics, especially in dynamics, both classical and quantum.
