Lagrangian vs Newtonian formalism

I have seen the lagrange equations derived from newtons laws in the special case, where forces were derivable from a potential.
Now with the introduction of hamiltons principle, I think my book wants to say this: We can always find a lagrangian such that the principle of least action holds. This lagrangian describes the entire system and the terms T and U, the potential and kinetic energy, need not be what we in Newtonian dynamics like to think of them of as - at least that is how I understood and I think it's correct since the lagrangian for an electrodynamic system for instance contains the term A*v (*=dot), which I don't see how you would interpret in terms of the classical definition of potential energy.
Now that means that the principle of action is regarded as a deep principle which is always true. Though we know, that the newtonian formalism is also always true for any coordinate frame and arbitrarily complicated systems. So how do you show that the two formalisms are equivalent? And how do you know that the lagrangian for a classically dynamic system always just contain the ordinary potential and kinetic energy of the system - unlike the electromagnetic case, where a new term has to be introduced.
Hope this made at least somewhat sense