
The Brachistochrone: The Problem of Quickest Descent - luisb
http://fermatslibrary.com/s/brachistochrone-curve-the-problem-of-quickest-descent
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santaclaus
Calculus of variations is one of those things you learn in undergrad classical
mechanics that still blows my mind. The derivation is some hand-wavey thing
that involves pooping a little epsilon in front of the space _of all the damn
functions in the world with all the properties you could ever want_ and now we
just exteremize like calculus 101? But then it works. And it works amazingly
well. And it works with constraints. And it works with whatever god awful
brewed in the deepest cauldrons of hell coordinates you find useful. Pretty
fun stuff!

Are there any mathematical treatments that get at calculus of variations in a
more principled way than the standard physics class that are still accessible
to one with a physicsy background?

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tjl
There's two basic approaches to calculus of variations. One is based around
the Gateaux Variation which basically is an extension of the normal
derivative. So the vector x becomes x + epsilon*v and you take the limit as
epsilon goes to zero. The second is due to Lagrange and uses the symbol lower
case delta. The both amount to the same thing, but the notation is different.
There's epsilon in the Lagrange approach, it's just buried.

For Lagrange's approach, I recommend Lanczos' "The Variational Principles of
Mechanics". It's published by Dover so if you can't find it in a library, it's
only about $10 last I checked. For the other approach, we used John Troutman's
"Variational Calculus and Optimal Control: Optimization with Elementary
Convexity" in my Applied Math course on calculus of variations. It's good, but
I don't think it's too approachable if you haven't had the lectures to go with
it.

I've done a lot of work with calculus of variations including working out the
math for volume integration of variational gradients which is something that
appears in beam deformation problems.

One day I'll sit down and write books on calculus of variations, finite
element analysis, and mechanics of deformable bodies. I was very unsatisfied
with the texts I used for all of these in my undergrad and grad programs.

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jacobolus
Gelfand and Fomin’s 1963 book also seems pretty highly recommended:
[http://amzn.com/0486414485/](http://amzn.com/0486414485/)

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tjl
Hmm, it's a Dover book. That alone makes me think it's probably a good one as
they have a habit of reprinting some excellent books that would otherwise be
out of print. I'll have to check it out. Thanks!

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Swizec
This is so cool! I used to read and recap one academic article per week. Was
going to do it for a year, but I fell off the wagon pretty quickly.

Fermat's library looks like the perfect excuse to get back into it.

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gurkwart
I wrote a program to let a ball roll down the brachistochrone and compare it
to other curves as my final school project some years ago. Might help some to
get a better idea of it.

[http://tobiasgurdan.de/facharbeit/Brachistochrone/Brachistoc...](http://tobiasgurdan.de/facharbeit/Brachistochrone/Brachistochrone.html)

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CarolineW
I like that you also have the ball sliding when the initial part of the track
is near vertical. My German isn't good enough to read easily - is there a way
of varying the coefficient of friction? Setting that 0 simulates the
traditional problem where the falling object slides, and setting it to a very
high value gets closer to the situation where some of the energy is taken into
the rolling, and not into the speed of the ball.

But nice work - thank you.

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CarolineW
It's to be noticed that this paper is considering the classic Brachistochrone
problem where the "ball" is _sliding_ , not rolling. In other circumstances
this makes a _huge_ difference - I'd be interested in seeing a treatment of
the Brachistochrone problem that takes that very real, physical difference
into account.

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alphydan
A nice paper on the Brachistochrone which dwelves a little more on J.
Bernoulli's approach is:

[http://www.math.umt.edu/tmme/vol5no2and3/tmme_vol5nos2and3_a...](http://www.math.umt.edu/tmme/vol5no2and3/tmme_vol5nos2and3_a1_pp.169_184.pdf)

