Our goal, for now, is to describe the motions of objects, like freely falling rocks and blocks that bob back and forth on the ends of springs. Presumably we are capable of imagining any behavior at all for such objects, any trajectory in space, no matter how strange, and any wild pattern of behavior from one time to another (and indeed, stranger things than that, when our imaginations have been well trained, including motions forward and backward in time, though the last chapter on Quantum Field Theory will be required to give that expression meaning). From this limitless collection of all imaginable paths, we learned from Dimensional Analysis how to generate a few numbers that selectively identify only those paths that actually happen in nature. We gave these numbers the name ``Constants of the system'', because they are the special numbers that apply the same way to all of the possible motions that the system can have. For falling rocks, we can always relate their times elapsed in fall and distances traveled by the pattern a=g, where is always the same number. For springs, once we have noticed that they have a cyclic motion, from the proposition that ``it does what it does because it is the way it is'' we are able to predict what the time of each cycle is from the characteristics of the system, the mass of the block and the number that characterizes the stiffness of the spring.
This represents tremendous progress, and indeed probably every thinking physicist uses Dimensional Analysis more than any other level of physical description, to get a general feel for ``how much, how big, how long, how often'' as it applies to a problem at hand. But this level of description left much unspecified. A minor problem is that it could not provide the factors of proportionality that describe any of the possible actual paths, and a more major one was that while it could tell (e.g., for the case of the spring) what characteristic time applied to the motions of the system, if there was one, it could not tell us what the significance of that characteristic time was. We had to put that in by hand.
Perhaps more significant was that, even if we knew the qualitative behavior of all the possible paths, and the meaning of all of the scales predicted by dimensional analysis, these numbers apply equally to all of the possible actual paths. They thus do nothing at all to tell us, in a given circumstance, which of its possible paths the object actually takes.
All of these problems were quite soundly solved with the introduction of the equations of motion, of which was the kind which Newton first figured out how to construct in general circumstances. Such an equation provided a complete specification of the particular path an object takes, once we have said how that path is started. It tells all of the qualitative behavior and includes all of the relevant scales provided by Dimensional Analysis, even filling in the factors of proportionality. In that way the equations of motion very effectively take us, from the collection of all possible actual paths selected by Dimensional Analysis, to a unique identification of a particular actual path.
But in some sense, we suffered a rather severe loss of efficiency in going from Dimensional Analysis and constants of the system to the equations of motion. The first step reduced us from the huge set of all imaginable paths to the much smaller set of possible actual paths with just a few numbers. Now that is a powerful idea! But then, to make the further step to the particular path through the equations of motion, we had to generate the whole path by building it, one infinitessimal piece at a time, as we trace out the evolution encoded in the rule .
It would represent a great progress in our understanding if we could somehow encode the same information as we get from in just a few numbers, as we did with Dimensional Analysis. Of course, these numbers could not be the same for all of the possible actual paths, because their whole function would be to further distinguish, within that set, one possible actual path from the others, and so to enable us to identify which one happens in a particular instance. And, of course, they would at the same time have to respect all of the results from Dimensional Analysis, because these must apply to all of the actual paths.
There are very good reasons to believe that such a simple set of numbers can be constructed, as we can see by thinking about the motion of a falling rock.
Fig. 6.2 shows the trajectory of a rock thrown slightly sideways and allowed to fall freely near the surface of the earth, as it would look in a long-exposure photograph. Certainly this is a path that a rock can follow, started when it leaves our hand and ended when it hits the ground. But there are other ways it could trace this same path as well.
Fig. 6.2 shows the paths that would result if, instead of throwing it sideways from a height, we threw it upward and sideways from the ground so that it happened to reach that height going at the same sideways speed, and simply as the continuation of its motion traced out our previous path as well. Or, alternatively, we could dig a hole so that it does not hit the ground right away, but keeps on going past where it stopped in the previous figure (6.2).
The fact that real rocks can and do travel in all of these different paths points out an interesting relation. In some sense, we should not consider a particular actual path as an isolated predictable occurrence or notion. At least part of what specifies the path is where we have chosen to start it and stop it, and under other circumstances, it might not have been a whole actual path at all, but merely a part of some larger actual path started or ended in some other way. In this sense, what the equations of motion do is not identify particular actual paths whole, but rather tie together all of the little pieces of path that make up an actual path, as many or as few of them as we allow by the way we start and stop the motion.
But now that we are looking at sub-paths as potentially parts of larger actual paths, we realize that there are some collections of subpaths that do occur together as parts of whole actual paths, and others that never do, as represented in fig. 6.2.
The first picture shows a path a falling rock might take ``of its own accord'', after being thrown. The second picture shows a different impossible path, made of two parts. Separately, each is a motion a rock could be made to take, by being thrown in a certain way. But they never occur together as two parts of any single path taken by a rock ``of its own accord''. It is quite arbitrary which little parts of the single smoothly sideways-moving path we choose to examine; the equations of motion can provide any sequence of them as parts of an actual path. But we never see any actual path composed of the other alternative, parts of a path that leads straight downward, and then other parts from a path going rapidly sideways.
Just the fact that such patterns exist, and that they exist independently of which particular little segments of path we choose to consider, suggests that there is something that the different subparts of a single possible actual path have in common, and that the subparts of different actual paths do not. Somehow the equations of motion always connect together only those pieces of path that ``can'' be parts of larger paths, and never those that cannot. What we want is to recast that association in a form that will not require us to fill in all the intervening motion between the parts to know if they could be parts of the same actual path.
We can give a name to the concept we trying to find. We suspect that there is a pattern, because the fact that the dynamics defines the path in a way that is independent of when we start or stop it tells us that the dynamics is associating all the potential subpaths that could make a single path, and excluding those that could not. When we found a pattern before, in Dimensional Analysis, we found that we could describe it by some numbers that, no matter what we did in imparting motion to the system, were always the same. In some sense, that is the essence of what existence of a pattern is. So again this time, if there is a pattern that picks out these possible compatible sub-paths, we would like to represent it as some small collection of numbers that all of them have in common, but which are different for the sub-paths of one actual path from those of another, and so serve to distinguish those that are compatible from those that are not. The fact that the numbers should be the same for all of the little parts of a single path means that they are the same at every instant along any real actual path. So they are the natural refinement of our earlier concept of Constants of the System, except that instead of being the same for all possible actual motions of the system, they are only the same for all points along a single motion, somewhat like the way the true statement was, as provided at each moment by the equation of motion, but different for one actual motion than for another. Thus the numbers that represent the idea we are searching for could naturally be called Constants of the Motion.
Remarkably, such numbers do indeed exist, and the tool that will enable us to show this fact and find out what they are also turns out to be the window into a whole new way of stating physical laws. We proceed to look for it by asking, ``what kind of a description of paths could do this job we want, of independently identifying the little sub-paths that can be parts of a single actual path, and excluding all those combinations that cannot.