Calculating forces efficiently in Lagrangian formalism

Question

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 / \ / \ m 4m -->

Answer 1

The power of the Lagrangian approach lies in its generality and clarity. First of all, it is much, much more widely applicable than Newton's laws. Moreover, it avoids the messiness of vectors since it involves only scalar quantities.

Furthermore, the thing of greatest interest in general will be the equation of motion of a system. Finding some specific solution to a somewhat trivial problem (no offence meant, I'm talking from a perspective of the grand scheme of physics as a whole) is really just a matter of plug-and-chug once you've got the system fully specified, as is the case once you formulate the Lagrangian. It doesn't really add any physical insights.

This is simply not the type of problem the Lagrangian method is 'designed for'. The Lagrangian quickly lets you understand all the important physics, which typically does not include the particular acceleration of a body given some particular initial conditions. If you personally think the method of Newton's equations is better suited for solving your problem, then you should just stick to them. There's a reason why they're still around after so many years :)