Brachistochrone. Why a curve at all?

[Gersty]

Why is the solution to the brachistochrone problem a curve at all? If the idea is to get from a higher point to a lower point under the influence of gravity alone, why is a straight line not quicker than a cycloid? It seems counter-intuitive that the shortest time would be along a curve and not a straight line. Even more so if that curve contains a segment where the object is moving upwards, and being slowed by gravity.

 

[SteamKing]

It all has to do with the gravitational energy which is available to the body traveling between A and B.

http://mathworld.wolfram.com/BrachistochroneProblem.html

 

[A.T.]

Consider the extreme cases:

endpoint is exactly below startpoint : Here the quickest way actually is a straight line

endpoint at same height as startpoint : Here a straight line will need infinite time, because you will never start moving. The only way to reach the endpoint is to go down and up again.

So in general, you need a compromise, that will somehow get you going initially, but won't be too far away from the straight line.

In other words :

time = path_length / avg_speed

Minimizing path_length alone doesn't help if avg_speed goes to zero. You have to optimize both.

 

[sophiecentaur]

Why is the solution to the brachistochrone problem a curve at all? If the idea is to get from a higher point to a lower point under the influence of gravity alone, why is a straight line not quicker than a cycloid? It seems counter-intuitive that the shortest time would be along a curve and not a straight line. Even more so if that curve contains a segment where the object is moving upwards, and being slowed by gravity.

This is a reasonable question - if you want Science to be 'reasonable' and 'acceptable' to your intuition. But why should you ever assume that Science works that way?
Throughout history, people have tried to approach things intuitively and it has led to many unsatisfactory results. Life became more predictable and things started to 'work' for us, once we started to apply the 'Scientific Approach'. Greek Philosophers sat down an pontificated about the way the Universe works and they didn't do Experiments. They got it wrong. Such a shame, because their maths was actually quite good.
The reason that the solution to the brachistochrone is not a single straight line path can be found by applying very straightforward maths to some very basic principles. The fact is that there is a Faster solution than a straight line - you could easily find a faster one by choosing two straight lines instead. So your intuition could then, possibly, accept that an even better solution could be obtained with an infinite set of short lines (a curve). It's a small step, then, to let the maths take over and give you a Cycloid curve.