Variational/Symplectic and Hamiltonian Operators

Given a differential equation (or system) $\Delta$ = 0 the inverse problem in the calculus of variations asks if there is a multiplier function $Q$ such that
$$Q\Delta=E(L),$$
where $E(L)=0$ is the Euler-Lagrange equation for a Lagrangian $L$. A solution to this problem can be found in principle and expressed in terms of invariants of the equation $\Delta$. The variational operator problem asks the same question but $Q$ can now be a differential operator as the following simple example demonstrates for the evolution equation $u_t - u_{xxx} = 0$,
$$D_x(u_t - u_{xxx}) = u_{tx}-u_{xxxx}=E(-\frac12(u_tu_x+u_{xx}^2)).$$
Here $D_x$ is a variational operator for $u_t - u_{xxx} = 0$.
This talk will discuss how the variational operator problem can be solved in the case of scalar evolution equations and how variational operators are related to symplectic and Hamiltonian operators.