TITLE: Example of ODE not equivalent to Euler-Lagrange equation
QUESTION [17 upvotes]: I am looking for an explicit (preferably simple) example of an ODE with time-independent coefficients in $\mathbb{R}^3$ such that there does not exist an Euler-Lagrange equation
$$\frac{\partial L}{\partial q^i}=\frac{d}{dt}\frac{\partial L}{\partial \dot q^i}, \, \, i=1,2,3,$$
with the same solutions.
I prefer to distinguish two cases: either $L$ depends explicitly on time or not.

REPLY [2 votes]: If you're willing to consider PDEs, I coauthored an explicit example of a PDE in a gauge field $A$ that fails to be the Euler-Lagrange equation arising from any gauge-invariant lagrangian involving $A$ alone in arXiv:1501.07548.
The PDE in question is formula (12), which is a deformation of the (topologically massive, for $\mu\neq 0$) 3D Yang-Mills equation of motion. The relevant term is the one involving the mass parameter $m$; for $m\to\infty$, we find (topologically massive) Yang-Mills. To avoid reproducing most of that (short) paper here, I will only sketch the proof that this is not an EL equation for any gauge-invariant lagrangian: the PDE is gauge invariant under standard Yang-Mills type gauge transformations, but fails to satisfy a Noether identity. However, gauge-invariant PDEs arising as EL equations always satisfy Noether identities; contradiction.