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.

 

[pmacias]

The solution to the brachistochrone problem being a curve is due to the fundamental principle of least action in physics. This principle states that the path taken by a particle between two points in space is the one that minimizes the action, which is the integral of the particle's kinetic and potential energies.

In this case, the particle is under the influence of gravity, and the potential energy is proportional to the height. Therefore, the particle will naturally follow a path that minimizes the change in height, resulting in a curve rather than a straight line.

Additionally, the curve chosen in the brachistochrone problem, known as a cycloid, has the property of being a tautochrone. This means that all points along the curve take the same amount of time to reach the bottom, regardless of their starting point. This is due to the cycloid's unique shape, which allows it to maintain a constant speed throughout the curve, even when moving upwards.

While it may seem counter-intuitive that a curve would be the fastest path, it is important to remember that the particle is not only moving downwards but also being accelerated by gravity. This acceleration allows the particle to travel faster along the curve, making it the quickest route.

In conclusion, the solution to the brachistochrone problem being a curve is a result of the fundamental principles of physics, specifically the principle of least action. The unique properties of the chosen curve, the cycloid, also play a crucial role in minimizing the time taken for the particle to travel between two points under the influence of gravity alone.