Ah, I love CoV! The subject got off the ground historically with the
brachistochrone problem: given two points, one not directly above the other, in normal gravity, what is the shape of the curve of fastest descent? (The word "brachistochrone" is from brachistos and chronos, Greek words meaning shortest time.) One of the Bernoulli brothers posed this question during the time of Newton. Newton solved it, but didn't attach his name to the solution. When the Bernoulli brother saw the solution, he said, "I see the paw of the lion" - meaning Newton.
But Newton used methods that were not so capable of generalization. The brachistochrone problem involves finding the curve $y(x)$ that minimizes the integral
$$t=\int_{x_1}^{x_2}\sqrt{\frac{1+(y')^2}{2gy}} \, dx.$$
The answer is an inverted cycloid.
Euler and Lagrange made extremely important contributions, producing the Euler-Lagrange equation:
$$\pd{f}{y}-\frac{d}{dx} \left( \pd{f}{y'} \right)=0,$$
which is of fundamental importance in the subject. This differential equation produces the extremal for the functional
$$\int_{x_0}^{x_1} f(x,y,y') \, dx.$$
CoV has been applied ferociously to classical mechanics, both in the Lagrangian formulation and in the Hamiltonian formulation. One of the more interesting applications of CoV in classical mechanics is the problem of the spinning top. You can predict the precession properties of the top using CoV.
It's also used extensively in control problems - particularly optimal control problems. While in many engineering applications you simply slap a PID controller at the problem, such a controller is almost never going to be optimal. You might not
need the optimal controller, but if you do, there will likely be some CoV in your future.
It's a beautiful area of mathematics, with ongoing research.
If you're interesting in studying it, I would recommend
Troutman's book - it's a good intro and uses convexity very cleverly to get some important early results.
The background required varies depending on the level of the book you're studying. Troutman requires up through multivariable calculus and linear algebra, and I would recommend mathematical maturity as well. A book like Ewing requires functional analysis and graduate-level real analysis!
