I've read somewhere that the Lagrangian approach was too computationally intensive; leading to explosion of possible paths and dimensions. Anyone can confirm ?
The Lagrangian approach is kinda great. If you can describe the potential energy and the kinetic energy of a system as two functions of whatever variables, Lagrangian mechanics allows you to derive the differential equations that govern the evolution of that system for free.
There's absolutely zero fucking around with forces, torques, etc. to get yourself a set of equations with which to model system behavior. You do have to add one constraint equation for every constraint on the system, but this is way easier than trying to formulate a set of differential equations that just happens to satisfy an arbitrary set of constraints.
I don't know of a reason why Lagrangian mechanics would tie one to a particular algorithm or class of algorithm; pretty much no matter how you do it, if you're modeling a mechanical system, you're solving some differential equations in one way or another.
But doesn't it have a hard time with coulomb friction, since you can only work with conservative fields/forces? A quick search confirms you have to use a bolted-on "dissipation function".
Correct. In contrast to Newtonian mechanics, Lagrangians and Hamiltonians completely describe essentially all fundamental laws of physics -- including things like quantum and relativity.
However, they are cumbersome to work with for some complex, compound phenomena, such as friction.
In the case of classical mechanics, the Lagrangian approach leads to exactly the same equations. The reason to learn it is that the Lagrangian approach generalizes and is used in nearly all physics[1].
Discrete Lagrangians are actually a really fun way to derive various structure preserving integrators. Define your Lagrangian at discrete time steps, then just take some derivatives (no need for calculus of variations due to the discretization in time) and watch time integration methods fall into your lap. This is a nice approach as it makes it really obvious what discrete equivalents of your continuous system are conserved.
