TITLE: Can the equation of motion with friction be written as Euler-Lagrange equation, and does it have a quantum version?
QUESTION [20 upvotes]: My (non-expert) impression is that many physically important equations of motion can be obtained as Euler-Lagrange equations. For example in quantum fields theories and in quantum mechanics quantum equations of motion are obtained from the classical ones only if a Lagrangian (or Hamiltonian) is known for the classical case. Is my understanding too oversimplified?
Are there examples of physically important equations which are not Euler-Lagrange for any Lagrangian?
More specifically, let us consider the classical motion of a particle in $\mathbb{R}^3$ with friction: 
$$\overset{\cdot\cdot}{\vec x}=-\alpha \overset{\cdot}{\vec x},\, \, \alpha>0,$$
namely acceleration is proportional to velocity with negative coefficient. 
Is this equation Euler-Lagrange for an appropriate Lagrangian? Is there a quantum mechanical version of it?
Added later: As I mentioned in one of the comments below, I do not really know how to make formal what is "quantum mechanical version".  As a first guess one could try to write a Schroedinger equation with general (time dependent?) Hamiltonian such that some version of the Ehrenfest theorem would be compatible with the classical equation of motion with friction.

REPLY [2 votes]: The answer appears to be yes.  See  Chad R. Galley, "The classical mechanics of non-conservative systems,"  Phys. Rev. Lett. 110, 174301 (2013) [Editors' Highlight]