Ah. What you have here is a ‘poorly posed question’, begging your pardon. If there is a fastest curve, then it seems natural to ask whether there is a slowest curve, but the question doesn’t end up anywhere interesting.
Let’s deal with a similar problem - what’s the shortest curve between two points? You can put a piece of string between the two and pull on it. This is mechanically minimising the length of the string between the two points. The maths effectively does the same thing: it finds the curve where any deviation makes the curve longer, just as it found the slope where any deviation makes the ball take a longer time to roll down it under gravity.
Now what is the longest curve between two points? You don’t find a limiting solution because there is always another curve that can have an extra wiggle in it and be longer.
What are the solutions for the makristochrone? If we only allow solutions where the ball rolls all the way and doesn’t get stuck, then we have to have some gradient at the start. But there will always be a curve with half the gradient at the start. The final solution would be a straight line with an infinitesimal gradient, followed by a sheer drop.
Hooray! A right angle (a flat bit and a sheer drop) is a solution! Well, actually, no it isn’t, really. What we have is a strategy for getting incredibly close to the limiting rule that the ball can’t get stuck. But the solution lies on the limiting rule, and not just close to it, and mathematicians are very particular about that sort of thing. It may seem like a lot of fuss over nothing but the rules are important.
If you were to be making something out of wood, something just short of a right angle would be the solution to all practical purposes. The maths does converge toward a single shape, unlike the ‘longest curve between two points’ which has infinite solutions.
Thank you for this. Likening it to the Newtonian bowling ball/feather drop as in the OP is a bit misleading because that’s not really what’s being demonstrated here. We have friction at play, so this is really demonstrating that forces resisting gravity can be close to the same in shapes that look very different.
For a practical example of rolling marbles down tracks, this reminded me of the Marble Machine X, (which is a musical instrument, even if it doesn’t much look like one at first glance):
In this case, the angle of each ramp is fixed, so to vary the speed Martin is varying the shape of the channel (the profile). Steeper channels move the points of contact closer to the edge of the sphere, which means it has to rotate more to cover the same distance. Sort of like a CVT gearbox.
The true fastest curve is the one that begins straight vertical and follows a cycloid through to the end. If the intended slope is shallow enough – a slope of less than 2/pi ~= 0.637 – then this fastest curve must dip down and then come back up!
The curved path in the demonstration rig never goes above the level of the straight path.
This means that the component of gravitational acceleration on a ball on the curved path is always the same or greater than that of the straight path (you know all that stuff about non-testicular balls on inclined planes).
Therefore, the ball will get to the bottom sooner on the curved path.
If the mean path of the curved path were to be shifted upward to match the slope of the straight path, both balls would reach the bottom at the same time.
Huygens wanted to make pendulum clocks more accurate. A weight that swings in an arc of a circle will have a period that depends on the amplitude of the swing. A weight that travels in a cycloid will have a constant period. You can get a weight on a string to travel in a cycloid by trapping the string between two cycloids. Neat!
Glorious maths, but it didn’t work for Huygens. It turned out the swing amplitude was not the limiting factor, and the side-effects when you actually build one make things worse.
The first part is correct but the rest is not. The shortest transit curve is a tradeoff between high velocity and short path. It is possible to construct a path entirely below the straight line which takes longer due to longer path length and you can’t computer the transit time just by looking at the average height.
I would suggest you look up any treatment on the calculus of variations which invariably includes the time integral for a ball to roll along a path. The old Thomas book (4th Ed) on calculus has this on p. 385. I must admit that my explanation may have been less than perfect but the math is correct and a tilt of the cuved path will yield the same time to reach the bottom.
I’m not 100% sure what you mean, but If you fix the horizontal distance L it is possible to construct a curved path that is faster than any possible straight line path regardless of the starting height (over the same horizontal distance). Or it is possible to construct a curved path that starts and ends at the same points as the straight line but is everywhere else below the line and still takes longer to transit.
