When walking, I know that my aim Is caused by the ghosts with my name. And although I don’t see Where they walk next to me, I know they’re all there, just the same. -David Morin (Introduction to Classical Mechanics, 2008) Sounds eerie, right? But, in a certain interpretation of the notion of action of a path in physics, this is actually true, and the math behind it, calculus of variations, is truly fascinating in itself. Taylor Series: An overview For those of you who know calculus, it's taught late into differential stuff that we can find minimas and maximas of a function, say $f(x)$ by looking for the points where $f'(x)$ turns zero. Later on, in the same course, we get an intuition for why this is true, in the $\text{Taylor series}$, a powerful way to make sense of any ( differentiable) function, by using simple polynomials. So for any random differentiable function $f(x)$, the Taylor series, of $f(x)$ around the input $x=a$ looks like: $$f(x-a)=\sum_{n=0}^{\i...
