The brachistochrone has many other cool properties!

And those properties come from something called calculus of variations. Recently, Kate Uhlenbeck won an Abel prize for her work in this area!

@GWOMaths https://twitter.com/AMAZINGSClENCE/status/1326848538941161472

Broadly speaking, calculus of variations is 'infinite dimensional calculus.'

Single variable calculus deals in problems related to change and optimization of single variable functions.

Multivar calculus = multivariable functions.

But what if...

... we have an INFINITE number of variables? A good example is if we have a function F which actually depends not on a variable, but on an entire FUNCTION f! (this is called a 'functional')

Consider for instance the energy in a hanging rope. There are two contributions: the potential energy from gravity, and the energy from the tension in the rope itself.

Total energy depends on the position of the rope! BUT...

... the position of the rope can't be described by finite variables. It's actually itself a function h(x), the height of the rope as a function of x!

The energy in the rope E actually depends upon h(x)!

A simple example of a functional is the definite integral from a to b. This assigns a number to each (integrable) function! So it maps functions to numbers.

Moving back to the rope example: calculus of variations studies questions like 'What position for the rope will minimize the energy stored in the rope?'

Now, this question can be answered by looking for _stationary points_ for the energy function. Just like looking for critical points in a standard function to optimize it, we can do the same for these functionals.