The author is also a non-physicist, though he'd disagree with that characterization. He is a famous crackpot. Ask him to actually derive all of physics from these principles and he'd fail completely.
L is a path, and W its sum along time, furthermore dW = 0 means W is always minimized (things take the cheapest, lowest .. whatever metaphor for minimum makes sense in a context)
that's my understanding as a non physicist .. but dW means local minima or maxima.. so i'm confused :)
To be additionally pedantic as a physicist, dW = 0 means we have extramized the action. Technically, we are looking for a path of stable action (i.e. a small change to the path does not change the action), so both maxima, minima, and even saddle curves are allowed solutions.
If you want to learn, and potentially really understand, some of this, I would ignore the physics entirely and start with the calculus of variations [0]. It's fairly straightforward (at least insofar as anyone with some basic differential calculus background ought to be able to understand the formulation and follow the derivation of the Euler-Lagrange equation), and it's useful. You can use it to derive Snell's Law or to design gears or to make pretty curves or the fastest possible ramps. Or you can figure out what L=T-U means, and use the Euler-Lagrange equation to recover F=ma from L=T-U, and you have classical mechanics!
And then you can go off the deep end and figure out how this all applies to quantum field theory :)
(L isn't a path. It's a scalar-valued function mapping the state of a system to a scaler. In C, with one classical particle, it might be double L(double t, double x, double v) where t is time, x is position, and v is velocity. IMO the simplest example from physics that actually makes sense as a minimization problem is the principle of least time: light going from point A to point B takes the fastest path. This seems a bit silly at first glance -- light travels along without having any idea where it's going. But it really is self-consistent: light propagates through space and through materials such that, wherever it ends up, it gets there by a locally fastest possible route. From this, you can derive specular reflection as well as Snell's Law (refraction) and you can start to understand mirages, e.g. why hot roads look shiny in the distance.)
