The Brachistochrone problem


I was struck by this GiF recently, at:


This little video shows three possible paths – the straight line (the shortest), a path that descends quickly and then becomes flat as it gets to the destination, and one in between, looking more elliptical* in shape.

Ever wondered what downhill profile of road will get you most quickly to the bottom of the hill freewheeling from a given point T (the top) to a given point P (the pub)? Now you know – it’s a cycloid, appropriately enough. Now I just need to think about the best shape of road from the pub to the top for the fastest ascent for a given a) wattage, and b) for a given pedal pressure…

…The classical problem is well-known as the difficult to solve Brachistochrone problem. See…/m231f08/m231f08brachistochrone.pdf for a solution – minimising a quantity over a famiły of functions, or…/nishiyama/math2010/cycloid.pdf which is perhaps a little more accessible and talks a little more about the cycloid (the resolution of the problem).

Even more arcanely (I was going to say more interestingly, but I’m realistic about reading endurance!) the cycloid, as the second article says, is isochronous. It means that even if you start freewheeling from a rest position R half way down the hill, it still takes the same time to get to the pub. Why? Briefly, because starting higher up, at the intermediate point R you are moving, whereas starting from R you are at rest. Weird, huh? It’s the same for any point R, no matter how close to the pub.

*the elliptical path is nearly as fast as the cycloid – but not quite!!

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