The Challenge of the Brachistochrone | Hacker News Reader

The Challenge of the Brachistochrone

jsoftware.com

45 pointsxearl16 comments

Although "the lion's paw" thing has been much quoted note that the details of Newton's solution were not preserved, there's only a single paragraph that was published by Leibniz that gives a geometric solution. Assuming Newton did indeed followed a geometric argument to arrive at the solution, Bernoulli's approach that morphs the problem into the equivalent one of light traveling through a medium of varying index of refraction is much more original and more brilliant.

To see Bernoulli's solution and for some additional interesting information about the problem see http://fredrickey.info/hm/CalcNotes/brachistochrone.pdf.

as if not more interestingly, the same curve is also the solution to the isochrone problem - if the curve were a wire, a frictionless bead sliding down it under gravity would take the same time to reach the bottom, no matter where along the arc it started.

http://en.wikipedia.org/wiki/Tautochrone_curve

In my third year at university we covered this in an Applied Mathematics course - Calculus of Variations. At the time our lector explained that in the entire field of mathematics we've learned over the last 20,000 years or so how to solve about 10,000 different mathematical problems. And of those 10,000 problems, Calculus of Variations comprised about 10. This was one of those 10. I don't know how accurate his numbers are but the point was that this is an extremely narrow field!

I don't know what 10 problems your professor had in mind but his comment was either in jest or perhaps was made from a narrow mathematical viewpoint: variational principles form the basis of much of Physics, e.g. Hamilton's principle, Fermat's principle, etc.

As a side note, for those of you who, like myself, have puzzled how Euler-Lagrange Equations are drived, the answers given to this question are priceless: http://physics.stackexchange.com/questions/885/why-does-calc...

Unless you misunderstood your professor, what they said is entirely and utterly wrong. Calculus of variations is one of the guiding principles of modern physics. Pull out any textbook on quantum field theory or general relativity and there will be a section dedicated to using calculus of variations to derive the equations of motion. Heck, for a lot of theories, writing down the action and deriving the equations of motion from the Euler-Lagrange equations is the first thing you do.

Plus calculus of variations is closely related to differential geometry, which is a massive field. Same with control theory, another massive field with close ties to differential geometry and calculus of variations.

I have no idea what point your professor was trying to make.

Long ago, I taught a college math course for a semester. My office mate was teaching differential equations. I asked him if he taught any engineering applications of multivariate differential equations, and he confidently assured me that there were none.

What are multivariate differential equations? Is that another name for partial differential equations? If so, then I assume your office mate was joking?

Oops, I meant multivariate calculus. But he really was that far out of touch. Not to be too tough on him, he was a grad student teaching assistant with a pure math background.

That's a poor reflection on his education then. Even the purest mathematics majors I knew still were aware of the applications of most of what they learnt.

Was anyone else expecting it to be a catenary?

http://en.wikipedia.org/wiki/Catenary

This is actually a closely related problem, solved using the same mathematical tools - calculus of variations.

no, but it makes sense!

Actually in another link, someone is mentioned as having attempted to prove it was a catenary for months until Newton showed it wasn't.

What are the "unmistakable signs of supreme genius?"

i'm assuming the solutions involved proving that there was a unique curve that satisfied the constraint. seems pretty non-trivial though.

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