

The Challenge of the Brachistochrone - xearl
http://www.jsoftware.com/papers/brachistochrone.htm

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Jun8
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](http://fredrickey.info/hm/CalcNotes/brachistochrone.pdf).

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zem
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.

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nkurz
[http://en.wikipedia.org/wiki/Tautochrone_curve](http://en.wikipedia.org/wiki/Tautochrone_curve)

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letstryagain
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!

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analog31
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.

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

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analog31
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.

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cabinpark
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.

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prezjordan
Was anyone else expecting it to be a catenary?

[http://en.wikipedia.org/wiki/Catenary](http://en.wikipedia.org/wiki/Catenary)

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

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gpcz
What are the "unmistakable signs of supreme genius?"

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

