We did as an example of the Lagrangian method the simple pendulum,
where 
. Saw that this reproduced the accelerations we
derived previously for polar coordinates, with minimal work.
Argued for three other advantages of Lagrangian formalism: 1) it leads
to a simple and elegant physical principle, that particles always
choose the path that extremizes the action S = 
; 2) it
reveals symmetries of the system directly, and clearly links
symmetries with conservation laws; and 3) it gives ready solutions to
constrained problems, and determines the forces of constraint.
To see point 3, we considered a nontrivial constrained problem: the motion of a bead on a wire with height y = f(x). After iterated applications of the chain rule, we found the equation of motion. Deriving this from the local forces would be extremely difficult.
-- KB