Understanding Hamilton Mechanics - What is this "action integral" I am learning about Hamilton Mechanics and am reading the below material about it: 
Additionally, I reference: http://www.unige.ch/~hairer/poly_geoint/week1.pdf
I would like help in understanding where the "action integral comes from in the above image. Why are we talking about this integral? That is, why are we taking the integral of the Lagrange function?
 A: The idea is that you want to construct a functional $S:\mathcal{P}→\mathbb{R}$ where $\mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $\vec{F}$  and then the solutions of the equation $\vec{F} = m\vec{a}$ are equivalent to finding the natural evolution of the system under consideration.
There is no mathematical reason why you would want to pick any of these principles. In mechanics, they very often can be proven to be equivalent mathematically. But practically it turns out that the principle of least action has been easily extended to encompass more phenomena than the method with forces has been. Nowadays, most theories in physics, in particular field theories, are cast within the framework of the principle of least action.
