I will Illustrate the question using an example problem:
We have a mass $m$ connected to a mass $m$ by a rod of length $l$, and also to a mass $4m$ by another rod of length $l$. The rods are massless, the masses are pointlike. The outer masses are positioned on a flat, slippery plane in such a way that the rods make a right angle, so one of the masses is in the air. What is the acceleration of the mass $4m$ immediately after the system is released?
I solved this problem successfully using Newton's laws (the answer is $g/9$). I then solved it by setting up the Lagrangian, using the Euler-Lagrange equations, and finally plugging in some special values (namely, the initial velocities are zero). I noticed it took significantly more effort, because I had to calculate the general equations of motion before plugging in the initial values; this was not necessary in the Newtonian approach.
Is there a way to solve problems like this using the Lagrangian formalism, and avoid doing more work than necessary (i.e. to avoid calculating the equations of motion explicitly)? I find the Lagrangian approach very useful for deriving the equations of motion, but many problems do not really require that.
ASCII sketch: (the arrow denotes the direction of the desired acceleration)
m 4m -->