Lagrangians for self-adjoint and non-self-adjoint equations

It is relatively easy to obtain a variational formulation for a self-adjoint equation, while it is difficult or impossible to do this for a non-self-adjoint one. This work suggests an alternative approach to the derivation of the Lagrangian for both cases. A trial-Lagrangian is constructed, and the exact variational formulation can be obtained after the identification of the unknown function involved in the trial-Lagrangian.