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The Brachistochrone: The Problem of Quickest Descent (fermatslibrary.com)
52 points by luisb on July 22, 2016 | hide | past | favorite | 11 comments



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?


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.


> finite element analysis, and mechanics of deformable bodies

This sort of irks me, too. Once you get to the continuum world everyone seems to forget about the Lagrangian reformulation. These are (mostly) classical systems so the machinery of Euler-Lagrange is perfectly valid, but I've met Mech E PhDs who vaguely recall Lagrangian mechanics as a footnote. Instead of all the obtuse walls of text and hand waving you get in some of these books, it would be hella rad to get at elasticity (and plasticity? via virtual work?) variationaly, in a mainstream text.


Gelfand and Fomin’s 1963 book also seems pretty highly recommended: http://amzn.com/0486414485/


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!


The HN crowd might enjoy Structure and Interpretation of Classical Mechanics: https://mitpress.mit.edu/sites/default/files/titles/content/... . For the more mathematically minded, the short little book by Gelfand and Fomin is (I think) a pretty good intro.


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.


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...


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.


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.


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...




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