principle of extremal action, Euler-Lagrange equations, de Donder-Weyl formalism?
The Helmholtz operator (going back to Helmholtz 87, Sonin, with modern version due to Vinogradov 78, Tulczyjew 80) is a map from partial differential equations (on sections of some bundle) to differential operators (on sections of that bundle) with the property that locally its kernel consists precisely of those PDEs which are variational in that there is a local Lagrangian density such that the PDE is the Euler-Lagrange equation which says that the variational derivative of this Lagrangian (equivalently: of the action functional that it induces) vanishes.
In modern terminology this says that together with the Euler-Lagrange operator the Helmholtz operator constitutes a chain complex of abelian sheaves (namely of differential forms on a jet bundle) which is locally exact. In fact this extends in both directions to a locally long exact sequence of forms on the jet bundle, called the Euler-Lagrange complex. See there for more.
A quick way to write the Helmholtz operator $H$ is as follows: If $\mathcal{E}$ denotes a partial differential equation and $L\mathcal{E}$ its linearization (evolutionary derivative), and $(L\mathcal{E}^\ast$ its formal adjoint differential operator, as a differential operator then
This means that the PDE $\mathcal{E}$ is locally variational precisely if its linearization is formally self-adjoint.
For proof that every Euler-Lagrange equation is in the kernel of the Helmholtz operator see geometry of physics – A first idea of quantum field theory this prop..
The Helmholtz operator originates in
Hermann von Helmholtz, Über die physikalische Bedeutung des Princips der kleinsten Wirkung, Journal für reine und angewandte Mathematik 100 (1887), 137–166
Sonin, …
where it is considered for linear ordinary differential equations. The modern general incarnation of the Helmholtz condition is due to
Alexandre Vinogradov, A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints, Soviet Math. Dokl. 19 (1978), 144–148.
W. M. Tulczyjew, The Euler-Lagrange resolution, in Lecture Notes in Mathematics No. 836, Springer-Verlag, New York, 1980, pp. 22–48.
Review includes
Ian Anderson, Aspects of the inverse problem to the calculus of variations.
Archivum Mathematicum, vol. 24 (1988), issue 4, pp. 181-202 (web)
Ian Anderson, The variational bicomplex, Utah State University 1989 (pdf)