Is there a nice reason why this method works?
Lagrangian Mechanics works because it is based in calculus of variations. Calculus of variations is basically a method of finding the most likely path by minimizing or maximizing the integral of the function on some interval. Calculus of variations can even give us snell's law!
Thus, the path that a particle will take on a certain time interval is the one that minimizes the time integral of the difference of the kinetic and potential energies.
Apart from finding K and U step in step and setting up those equations, in general, what problem is this Euler-Lagrange equation trying to solve?
Just learned this from one of my physics friends. Lagrangian equations only work on conservative forces. In English this means the potential energy must be a function of only position, not velocity, or acceleration, or anything like that.
Why don't we use K+U considering energy conservation that's widely known? I feel this is related to conservation of energy but the "-" is confusing...
@HelloWorld Yes! In this case, it comes from Hamilton's principle of stationary action, which applies the calculus of variations mentioned by @ljelenak. In general, there are many variational principles that can be used to characterize physical laws; a great book is The Variational Principles of Mechanics by Lanczos.
@echo Good question---there is in fact a whole other "dual" picture of Hamiltonian mechanics that is closely related to the Lagrangian formulation. There you work with position and momentum rather than position and velocity, and (roughly speaking) the Euler-Lagrange equations are replaced with Hamilton's equations. But the high-level story is the same: start with just the kinetic and potential energy; some standard calculations then give you the equations of motion.
@djevans Indeed. There are ways of incorporating things like dissipation and nonholonomic constraints into Lagrangian/Hamiltonian mechanics, but not using the very basic formulation shown here.