Lagrangian mechanics can handle dissipative systems. In many cases, friction included, it's as simple as adding a dissipative term to the Euler-Lagrange equatio

Lagrangian mechanics can handle dissipative systems. In many cases, friction included, it's as simple as adding a dissipative term to the Euler-Lagrange equations. This seriously complicates the process of deriving them, but physicists were always handwaving their way through it anyway. Hamiltonian mechanics faces more serious obstacles, though if you really want to you can engineer a noncanonical symplectic structure to capture the dissipation. Wouldn't recommend it though.