[nh23423fefe]

I found this playlist informative when I was having the same thoughts

https://www.youtube.com/playlist?list=PL2ym2L69yzkamORF9DGWR...

It isn't L that is differentiated and set to zero. Instead the variation of the action is zero. So L isn't minimized or maximized, the variation of the action is zero. This implies that the solution path y(x) when varied by dy is a stationary point of the action. So for this path all nearby paths have the same action.

ok but then

> What is the Lagrangian and why should it be minimized

the form of the lagrangian is derivable from the d'alembert principle, principle of minimum potential, and then hamilton's principle.

It seems to me the principle is that real system behave in such a way that is characterized by hamiton's principle (the variation of the action is zero for real paths), and then we operationalize that principle by the calculus of variations to get real paths which have the properties established by the principles (use the Euler-lagrange equations to find paths for a particular system)