Lemma: Helmholtz lemma for linear operators

Independent of Input (S-bv). Let $E$ be a linear differential operator $h\mapsto E[h]$ with fixed smooth (or formal-jet) coefficients. Let ${\mathcal{L}}_E:=\frac{1}{2}⟨h,E[h]⟩$ with Gate V weights, and let $\mathrm{EL}$ denote the Gate V Euler operator. Then:

1. $\mathrm{EL}({\mathcal{L}}_E)=\frac{1}{2}(E+E^{†})$;

2. $E$ is the Euler derivative of some local Lagrangian if and only if $E=E^{†}$, and then $E=\mathrm{EL}({\mathcal{L}}_E)$, i.e.\ $E=\delta I_E/\delta h$ with $I_E[h]=\frac{1}{2}\int ⟨h,E[h]⟩$, modulo divergences and field-independent terms.

Proof. (1) For compactly supported $t$: $\frac{d}{dϵ}{|}_0\int {\mathcal{L}}_E[h+ϵt]=\frac{1}{2}\int ⟨t,E[h]⟩+\frac{1}{2}\int ⟨h,E[t]⟩$. Integrating by parts, with the boundary terms vanishing because $t$ has compact support and derivatives hitting the external coefficients retained, the second term equals $\frac{1}{2}\int ⟨t,E^{†}[h]⟩$. Hence $\int ⟨t,\mathrm{EL}({\mathcal{L}}_E)⟩=\int ⟨t,\frac{1}{2}(E+E^{†})[h]⟩$ for all $t$. (2) If $E=E^{†}$, then (1) gives $E=\mathrm{EL}({\mathcal{L}}_E)$. Conversely, if $E=\mathrm{EL}(\mathcal{L})$ for some local $\mathcal{L}$, the Fréchet derivative $D_E$ is formally self-adjoint (Helmholtz conditions; Olver, Thm. 5.92). Since $E$ is linear, $D_E=E$, so $E=E^{†}$.