Before undertaking the major subject of variational principles in quantum mechanics, the present chapter is intended as a brief introduction to the extension of variational theory to linear dynamical systems and to classical optimization methods. References given above and in the Bibliography will be of interest to the reader who wishes to pursue this subject in fields outside the context of contemporary theoretical physics and chemistry. The specialized subject of optimization of molecular geometries in theoretical chemistry is treated here in some detail.

Linear systems

Any multicomponent system whose dynamical behavior is governed by coupled linear equations can be modelled by an effective Lagrangian, quadratic in the system variables. Hamilton's variational principle is postulated to determine the time behavior of the system. A dynamical model of some system of interest is valid if it satisfies the same system of coupled equations.