Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation$$\frac{d}{dt}\frac{\partial L(q,\dot q)}{\partial\dot q}=\frac{\partial L(q,\dot q)}{\partial q},$$and these are critical points of the functional$$I(q)=\int L(q,\dot q)dt.$$Of course one needs to be precise with what considers a solution to both equations. This amounts to specifying regularity and a domain of the functional. This example is an ODE, but many PDE examples are possible as well (for example electromagnetism, or more exotic physical theories). Once one knows a variational description of the problem, many more methods are available to solve the problem.
Now I do not expect that any PDE or ODE can be viewed (even formally) as a critical point of a suitable action functional. This is because this whole set up reminds me of De Rham cohomology: "which one-forms (the differential equations) are exact (that is, the $d$ of a functional)?". The last sentence is not correct, but the analogy maybe is? Anyway, my question is:
Are there any criteria to determine if a given differential equation admits a variational formulation?
