The Calculus of Variations + Waterslides

People give up on maths. People give up on maths because it’s hard, but it doesn’t do itself any favours though. Maths is said to be a language that a person can never be fluent in, which is like playing the Zombies Mode on Call of Duty; There’s no completing it, you just keep hacking away at things. Maybe somebody will find a new glitch, maybe somebody will develop a new technique, but you’ll probably just end up running around like a headless chicken and relying on that one friend who actually knows what they are doing. When I was given ‘An Introduction to the Calculus of Variations’ as the topic of my dissertation, the headless chicken approach was inevitable. It seemed that I had been dropped into wave 50 without the help of friends, equipped only with a Springfield and a few Molotov cocktails. If you do not get the reference, just know that I seemed screwed.

A Functional is a mapping from a vector space X, usually of functions, to the real or complex numbers, often expressed as the definite integral. That may as well be Spanish to some of you, as it seemed to me at first glance. A mapping (or function) can be seen as a machine that eats stuff and spits it out again once that stuff has followed a rule or set of rules it has. Now imagine there is machine eating machine, a cannibal machine if you will; It chews up these machines whilst those machines chew on stuff too. Simultaneously, the cannibal machine and the smaller machine spit its contents out at the same time. That is the most exciting way to describe a functional, as a cannibal. In the field of the Calculus of Variations at least, This cannibal that we call a functional is an integral between two points and is essential to this whole area of maths.

Now we can finally define what the Calculus of Variations is. It is the area of mathematics that focuses upon maximising or minimising these cannibalistic functionals, similar to finding the maxima of the simpler f(x) functions, where the gradient equals 0. It gets a little more complicated than that but it’s along the same lines. I do aim to write this blog though such that you get to the end without falling asleep so I will gladly leave that stuff out.

The Brachistochrone Problem was something that mathematicians took a while to crack. ‘Brachisto’ meaning ‘shortest’ in ancient Greek, and ‘Chronos’ meaning ‘time’, this was the problem of shortest time. It aimed to find the fastest time for which a particle dropped down a slope under only gravity, no other forces like friction or air resistance, to any another point below it. Here is a diagram for it. I made this in Microsoft Word at 3am and it’s actually in my dissertation. Professionalism.

Turns out that after loads of tedious calculus, the answer to the curve between the starting point and any end point that minimises the time is found by a cycloid curve. A cycloid curve is one which is created when you trace from a point on a circle when it rolls along a flat surface.

Now imagine a cycloid curve is a picture in Microsoft Word. To find the Brachistochrone (shortest time) curve from our start point to another randomly given point, imagine lining up our start point with the start of our cycloid curve, but the cycloid curve is the opposite way round to the animation, we basically drag the corner of the picture of the cycloid curve to make it bigger or smaller while it stays in the same ratio of width and height and stop when our end point lies on the curve somewhere. What we are changing to find our specific section of the cycloid curve is the radius of the rolling circle, that’s it. It will always be fastest with a section of a cycloid curve (assuming only gravity is acting). Looking at the diagram below, when you find the first derivative of any cycloid curve, no matter what size, the extrema, where the gradient is zero, will always lye on the same line if it starts at the point (0,0).

This line is y=2x/π, or if you flip upside down to make things easier to visualise like we have above, it is y=-2x/π. This is important when designing the ideal waterslide. It doesn’t take a genius to know that a waterslide needs to end with a flat section for safety, but if we want the fastest possible safe waterslide, we must have a Brachistochrone curve slide that ends at the flat part of the curve. We need to have the endpoint and start point of the waterslide that lye on the same of y=-2x/π and draw a Brachistochrone curve between them. There are 3 other cases if you wanted the fastest possible slide (Brachistochrone slide) but the points don’t both lye on this line.

In Case 1, the user will be rapidly smashed into the ground. This is terrible for waterpark safety and the lawsuits will ultimately get your park closed. Would not recommend. Case 2 is like Case 1 but the user will get to enjoy a lovely view before any tragedy, catching some sweet air in the process. Would also not recommend. Case 3 is a little better, remember that we are not taking into account any forces other than gravity because on a waterslide there will be little anyway as you are also propelled a little by the water. Case 3 sees the user stuck in an infinite loop of going up and down, and an air ambulance would have to be called to rescue the user. Better than 1 and 2 but still not good for publicity.

The Brachistochrone curve is also an example of a Tautochrone (‘same time’) curve. This means that if many people used the slide at the same time from different starting positions, they would arrive at the bottom at the same time. This sounds nice at first, that they would all join together in some train at the bottom, but the reality is that they would all have different velocities at this moment, and all but one of these people would be kicked in the back. Again, would not recommend.

Thanks for reading.

Luke Bennett.