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.

Research-level? $\endgroup$