# Techniques for constructing an energy functional from a system of PDEs

Having an energy functional one can write the Euler-Lagrange equations. I want to do the reverse. Having a system of PDEs, I want to find an energy functional that "best" matches those. It is clear that not every system of PDEs is a system of Euler-Lagrange equations, and in the specific cases I am considering this needs not to be the case. Nevertheless I would like to find a functional whose solutions best match the solutions from my system of PDEs (wrt some metric - for example L2). Are there any references on this topic?

• I looked for exactly the same thing about a year ago. I was looking to derive an set of energy functionals for a system of reduced PDEs that I had derived. Alas, the only concrete thing I ever found was this, which is not much to go on. Commented Jun 19, 2020 at 14:47
• @mattos Yeah, I was thinking about something more similar to minimizing a functional. What you linked is specific to FEM and simply rewrites the equations in their weak form. Commented Jun 19, 2020 at 14:54
• I think this Q&A would give you some nice references for your problem: the topics you are investigating is called "inverse problem of the calculus of variation", and I advice you also to have a look at the web site of Enzo Tonti (linked in the Q&A above) in order to have a quick introduction to his general method of solving the problem. Commented Jul 3, 2020 at 5:43
• @DanieleTampieri The paper from Tonti is currently a bit over my head, but this resource was nevertheless the most helpful thus far. Feel free to make your comment an answer so I cam accept it. Commented Jul 3, 2020 at 12:40

I think this Q&A gives you some nice references for your problem: the topic you are investigating is called "inverse problem of the calculus of variation", and I advice you also to have a look at the web site of Enzo Tonti. In particular, recapitulating briefly, Tonti's main result (in its more recent formulation [1]), given a general (linear/nonlinear, local/nonlocal etc.) operator equation $$\mathsf{N}\big(x,u(x)\big)=f(x)\label{1}\tag{1}$$ one can always find a functional $$F$$ so that its functional derivative $$\left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon}F(u+\varepsilon\varphi)\right|_{\varepsilon=0}=0\;\text{ for all \varphi \in domain of \mathsf{N}}\iff u\text{ is a solution of \eqref{1}}\label{2}\tag{2}$$ Moreover, $$F$$ can be chosen in such a way that $$u$$ is a minimum for it. However, it is worth to remember that the equation(s) resulting from imposing the vanishing of the functional derivative of $$F$$ may have not always the form of Euler-Lagrange equations: it depends on the structure of \eqref{1} if this happens or not (see the reference by Filippov in the above linked Q&A for more details on such matters).
• Are you aware of a text that goes into more practical details regarding the procedure described in Tonti's paper? Namely, more examples with different kernels $k$ for different PDEs, as well as examples with the degenerate kernels. A notable practical disadvantage is the mentioned doubling of the integrals. Commented Jul 3, 2020 at 14:18