We notice a curious path of development so far. The problem with dimensional analysis so long ago, and the thing that limited its usefulness, was that by itself it lacked even rules for distinguishing objects from external worlds, and for rejecting the parts of the environmemt that were not important. Then Newton's laws provided a concrete way to express such distinctions once we decide how to draw them, and further provided us with a great increase in descriptive power by giving a point-by-point evolution law, called the equation of motion. And, in some cases, the division of the important from the unimportant parts of the external world was made very clear by the notion of locality. This was the case for the block on the spring, for which the only part of the external world that mattered was the end of the spring, the part actually touching the object. We felt that in approaching locality we were achieving ever better and less ambiguous refinements of our description of the motions of objects.
However, at the same time we suffered a great loss in efficiency in going from Dimensional Analysis to local equations of motion, because a whole path had to be computed rather than just a few numbers. Now we have resolved that by uncovering the existence of constants of the motion, but the only way to get to them was by the Principle of Least Action. Suddenly our most fundamental description, minimization of an Action over an entire path, is once again about as nonlocal as anything could be, though certainly through the locality of the Lagrangian it encodes the same local dynamics as was present in . So in gaining one simplicity and a new principle, symmetry, we appear to have lost another that we valued, which is locality as a part of the fundamental rules.
It will require the development of many new ideas before we can usefully return to this issue. Remarkably, there is a single set of principles, from which even the elegant Principle of Least Action follows as a consequence. That set of principles includes as its cornerstones symmetry and locality at the same time, and gives as its consequences the existence of conserved quantities and in some cases equations of motion of the form , and includes all of the results of dimensional analysis, but we will have to wait for the quantum theory to find it.
In the mean time, we have built a wonderful collection of tools for describing the motions of objects, and before we look at its limitations and the ways to develop it further, we will examine some applications and discover a host of wonderful surprises about the real structure of the natural world.