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I understand that for a functional $J[f]$ on the space of differentiable functions $f$ on some domain, setting $\delta J[f]|_{f=f_0} = 0$ yields a (possibly nonlinear) partial differential equation in $f_0$, the $f$ that minimizes $J$. Solutions are non-degenerate if extrema of $J$ are locally unique, and the uniqueness of the PDE is related to the number of extrema of $J$.

My question is if, given a known (nonlinear) partial differential equation with solutions $u$, there is a general method to construct a functional that the $u$ will minimize?

I am not interested in functionals of the form $(f-u)^2$ and other functionals that contain the explicit form of the solutions $u$, and would like to know if there is a method to find $J$ given the terms in the PDE. I'm aware that this can be done for all linear equations such as $Lu=v$ in a straightforward way; namely, for $L\cdot u=v$, $J[u] = \langle u\cdot L\cdot u \rangle - \langle v \cdot u \rangle - \langle u \cdot v \rangle$; but don't know of any extensions to nonlinear PDEs.

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    $\begingroup$ This is definitely not easy. Have a look at chapter 8.2 (Existence of minimizers) of Evans' Partial Differential Equations. $\endgroup$
    – Funzies
    Jun 1, 2013 at 16:45
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    $\begingroup$ Answers are essentially going to be duplicates of physics.stackexchange.com/q/20298/2451 and links therein. $\endgroup$
    – Qmechanic
    Jun 1, 2013 at 17:21

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The problem of determining a Lagrangian (and so the action functional) such that the corresponding Euler-Lagrange equation equal a given PDE is known as the inverse problem of variational calculus. See wikipedia (http://en.wikipedia.org/wiki/Inverse_problem_for_Lagrangian_mechanics) for an introduction and nlab (http://ncatlab.org/nlab/show/variational+bicomplex) for a mathematical generalization of this idea (especially the references in the later are useful: Takens/Zuckermann/...).

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When I've tried this in the past, I've just sat down and stared at the problem for a long time. But frequently the opposite happens, and a physics PDE is derived from some conservation law (such as the Maxwell-Vlasov equation or more generally the BBGKY hierarchy) or are actually derived from some action principle that itself is derived from some symmetry considerations. Why would you go the other direction?

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  • $\begingroup$ I have several nonlinear PDE models of a phenomenon which is known to explicitly modify conservation laws (essentially via asymmetric constraints and dissapation), each of which is experimentally plausible at the moment. I'd like to take advantage of direct methods, especially Rayleigh-Ritz minimization, in constructing solutions directly, and comparing how the underlying structure of the models differs. $\endgroup$
    – Chay Paterson
    Jun 1, 2013 at 21:44
  • $\begingroup$ "asymmetric constraints" meaning? As for dissipation, I'd be surprised if you could derive a dissipative system from an action principle -- those things usually have some total integral as a conserved quantity just by virtue of coming from action minimization. There are a few special cases, but I can't think of a terribly general way to incorporate dissipation. $\endgroup$
    – webb
    Jun 4, 2013 at 16:41
  • $\begingroup$ Primary constraints that are not invariant under the symmetry that generates the conserved current. As for the second point, classical electrodynamics and some similar looking scalar field theories coupled to a simple matter term are clearly dissipative in some sense. But no, I can't think of a general way either. The specific system I have in mind isn't strictly speaking dissipative, though; I'm not sure it can be represented as a minimization problem as in classical mechanics, but some limits do reduce to boundary value problems. $\endgroup$ Jun 4, 2013 at 19:10

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