Lagrangian mechanics are better when there are lots of constraints. The more the constraints, the simpler the Lagrangian equations, but the more complex the Newtonian become. Lagrangian mechanics is not very suited for non-ideal or non-holonomic systems, such as systems with friction. Lagrangian mechanics is also much more extensible.
What is the difference between Newtonian and Lagrangian mechanics in a ...
Now, someone tried to mark this question as a duplicate of this other one. From my point of view, having derived the Euler-Lagrange equations before even mentioning the least action principle or the action, it didn't seem too related. The point was, I wanted to have a physical interpretation of the Lagrangian, and leave the action and the principle as abstract constructions done for who knows ...
You can use Lagrangian and Hamiltonian formalism not only for physics but also for microeconomics, of course. As it has been already said here, the question is to treat the case as a functional (not as a mere function) and this will require to begin from a variational principle from an action. Most of these cases are not conservative, but eou can prove that inflation dp/dx>0 leads to value H ...
How is a Hamiltonian constructed from a Lagrangian with a Legendre ...
Summarizing, Lagrangian, Newtonian and Hamiltonian mechanics are different mathematical frameworks whose goal is to describe the same physics. The postulates of classical mechanics hold for all formalism, because they are facts of Nature, and we can use them to realize the passage from one to another scheme. Of course, one must treat mathematical objects with care in order to gain the equivalence.