In this article, I will be introducing a field in mathematics known as calculus of variations which I explored in my IB Mathematics Internal Assessment.
What is Calculus of Variations?
To understand the intuition behind calculus of variations, it may help to consider a simple scenario where it would be used: finding the shortest path between two points. While it is obvious that the answer is a straight line, calculus of variations provides a way of mathematically proving it.
Consider an arbitrary path y(x) that joins the points (x₁, y₁) and (x₂, y₂). Using our knowledge of calculus, it is known that the arc length of this path can be expressed as the following integral:
Hence, our goal is to find a function y(x) for which the above integral is a minimum. This is analogous to optimization in single variable calculus where the goal is to find a certain point for which the value of a function is stationary. However, in this case it is much more complicated because now we are attempting to find a certain function for which a functional (a function of functions) is stationary. While in single variable calculus this was simply done by finding the points at which the derivative of the function is equal to zero, a more complex approach must be taken for calculus…
