The PDEs we have considered so far were derived by modeling a variety of phenomena in physics, engineering, etc. In this chapter we shall derive PDEs from a new perspective. We shall show that many PDEs are related to optimization problems. The theory that associates optimization with PDEs is called the calculus of variations. It is an extremely useful theory. On the one hand, we shall be able to solve many optimization problems by solving the corresponding PDEs. On the other hand, sometimes it is simpler to study (and solve) certain optimization problems than to study (and solve) the related PDE. In such cases, the calculus of variations is an indispensable theoretical and practical tool in the study of PDEs. The calculus of variations can be used for both static problems and dynamic problems. The dynamical aspects of this theory are based on the Hamilton principle that we shall derive below. In particular, we shall show how to apply this principle for wave propagation in strings, membranes, etc.

We shall see that the connection between optimization problems and the associated PDEs is based on the a priori assumption that the solution to the optimization problem is smooth enough for the PDE to make sense. Can we justify this assumption? In many cases we can. Moreover, even if the solution is not smooth, we would like to define an appropriate concept of weak solutions as we already did earlier in this book in different contexts.