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.
