A Variational Principle for Nonlinear Systems

The equations describing a reactor system are sometimes nonlinear and do not admit a solution for the neutron density that is separable into a function of time only and a function of the remaining variables. An appropriate variational principle is given by demanding that the calculation of the observable nature of the reactor is insensitive to the value employed for the density, thus obtaining an equation for the optimum distribution of detectors to measure the observable behavior. This optimum weighting function is not identical with the conventional adjoint function or importance in the nonlinear range but the conventional treatment of linear systems is found to be a special case of our general principle. It is shown that the approximate treatment of nonlinear systems as eigenvalue systems is fundamentally unsound.