As has already been remarked, the first thing we will learn when trying to use minimization to describe the motions of ``objects'' is that we have to use a more careful description of paths than we have so far. This is because there is an important difference between the motions of objects through space and the position of a stretched rubber band on a table.
A stretched rubber band is a long extended thing, and its various points lay in a certain track on the surface of the table that doesn't change from one time to the next as long as we don't change the way we are holding the band. Thus, time is really not a consideration in the description of the band, and we can concern ourselves only with the spatial extent it occupies.
In a somewhat different way, Fermat's least time principle chose a spatial path that was the same always, because light can be made to shine continuously in a way that makes it look less like a moving object than like a streaming current, so it makes sense to talk of the spatial path as a thing that doesn't change from one time to the next. Yet this path was chosen on precisely a criterion of time, as if the light flowing along the path were composed of a little stream of moving objects, each of which carried a little clock that was somehow important in determining which path it would travel.
All of our previous descriptions, though, of falling objects or objects on springs, involved not streams of many objects or extended things like rubber bands, but isolated individual blocks that traveled through space along some trajectories. All of our description by was taylored to such systems. Thus, to relate to minimization, we have to be somewhat more general and careful in our descriptions than we were for either rubber bands or light, and identify the movement of the objects as paths traced out in space and time. Certainly, we know from experience that it is not only what spatial path a rock would fall through, as it would be seen in a time-lapse photograph of its motion, that is important in determining the pattern we recognize, but also the succession of speeds at which it passes through each part of that trajectory. Thus, for the first time, we are required to invent the notion of a space-time diagram. (When we require much the same tool in the later chapters on special relativity, we will refer to it there as a ``spacetime diagram'', because in the course of our later investigations, we will learn that there is a much deeper relation between the two than we have any grounds to suspect so far.)
The pictures we have drawn of motion so far, largely without explanation because they are ``self-explanitory'' have been of the downward and sideways motion of falling rocks as we would see them if we were onlookers standing in front of the person who dropped the rock or threw it to his side. It happens that these really are not at all self-explanitory, but on the contrary make a rather sophisticated use of our memory to be familiar at all. We never ``see'' the whole path of anything that falls. At different moments of time, we see the object in successively different places. But, because we can remember where it was a moment ago, and we can see where it is now, we can project the two onto the same background in our imaginations and ``see'' the whole path as a single image, or recognize the intent when someone draws us a cartoon of it. To create the same picture with a camera, we really have to work rather hard, so that all of the moments can be traced on the same frame of film.
The much more natural thing to do with a camera, if one wishes to describe moving things, is to make a movie, so that, even though the same background appears in every frame and a great redundancy is suffered, we have a different picture of where that falling object is relative to the background in each frame, describing each moment of time when we took a picture.
Usually, one simply looks at movies by re-showing the frames in order, giving the same impression as the original motion. But another thing we can certainly do is cut all of the frames of the movie apart and stack them, one on top of another, so that now the whole two-dimensional image seen by the camera is a slice laying flat, and the succession of moments of time is displayed in the succession of frames of film up the page, as in fig. 6.5.
Since all of the points we need to make can be very nicely made for motion in one spatial direction, rather than show the whole two-dimensional images, we look at them ``edge-on'', so that we see only one direction of the motion through space, and see the successive moments of time stacked up the page. This way of drawing the paths of objects in space and time is called a space-time diagram. In what follows, we will turn the picture sideways, so that time goes across the page, because we will be describing objects falling freely near the surface of the earth. For such motions, the ``interesting'' direction is the vertical one, so it is intuitively convenient to draw it up the page.
Fig. 6.5 shows the motion of a rock that is thrown upward from the surface of the earth at some place and allowed to fall back down. (The reader will note that the path drawn this way, in space and time, looks just the same way it did when we drew the picture of vertical space and sideways space. This point is an accident that characterizes this particular system, but nonetheless an accident of great significance which will be very useful to us, and to which we will return in the following sections.) But though this path happens to look the same, there is a great difference between the two ways of drawing, which we see if we draw several paths, as in fig. 6.5.
All of the actual paths advance forward in time, so there are in fact no ``straight vertical'' paths possible in the diagram. Moreover, all of the actual paths drawn have the same parabolic shape, which is the way the space-time diagram shows the first relation we discovered from dimensional analysis for this system, the constancy of the downward acceleration.
In showing paths this way, we have a description that is suitable for use in a minimization algorithm, because now we have a description that is the same as that for the rubber band. While the actual object only exists in one place at any time that we observe it, by drawing all the instants of our observation at once on the page, all of the moments of the object's history are included in the same picture. In some sense, in this diagram a part of the object's history exists ``at every point of the path at once'', so the history has become like all the bits of a stretced rubber band, if we draw the picture the right way.
To make use of the minimization rule for rubber bands, we needed two things. We needed to know how to measure distance to know what was minimized, and thus how actual paths were ``chosen'', and we needed to tell between which two points the band was stretched, to specify which particular path it took.
Measuring distances on the surface of a table is an old and by now well-understood exercise, but there is no obvious familiar notion of ``distance'' in space and time drawn together. In fact, our entire search for the action is the search for the thing that is minimized, which will for the first time provide us with a notion of such a distance. We will have much more to say about this from a different perspective in the chapters on relativity, though through the beautiful consistency of nature, our answers will be the same, and the new understanding provided by the theory of relativity will simply teach us simpler and more elegant ways to write it and recognize it. For now, we will take the notion of the action we discover as our first definition of something that behaves like a distance for paths drawn in space and time.
A much easier problem is the specification of the ``points between which the band is stretched''. In this case, they are the points in the space-time diagram at which the path was started and ended. Such a point, specified in both space and time, is called an event, and is an extremely useful and important tool for all that we will do. It should be clear that in some logical sense, there was no way we could really formulate a minimization algorithm for paths that was the analog of the minimization of distance for rubber bands, without first describing the paths in space-time diagrams. This is because, while a rubber band exists at all the points of its ``path'' in space at once, a moving object only exists at all the points of its history ``at once'' in a spacetime diagram, so the beginning and ending events in a spacetime diagram are really the only eligible candidates for the starting and ending ``points'' of the path. To simply work with the starting and ending points of the spatial trajectory would have been a different thing to do. As it turns out, it would have also usually been a wrong thing to do, as can be seen in fig. 6.5,
of several possible motions of a mass bouncing up and down on the end of a spring. Since the motion of the mass simply traces and retraces the same straight trajectory in space, a picture in space is all but useless for telling which path we would like to consider. And, since the mass can be made to bounce a little extent or a large one, the possible large trajectories entirely encompass the little ones as paths in space alone.
The specification of paths is infinitely better in the space-time diagram (next figure, with time drawn sideways, to exploit the vertical-vertical coincidence and the similarity to the Fermat pictures) fig. 6.5,
though interestingly, much like the case of the lens for Fermat's principle, still not unique. At least, because the motion in time is inexorably ``forward'', the paths do not lay on top of one another, and the large-extent paths are clearly distinct from the small-extent ones. And, for almost any starting and ending point we choose, these uniquely specify the path that connects them, with one class of exceptions.
Because all of the motions have the same cycle period, and because they all have the same middle resting place (as we learned in the chapter on dimensional analysis) there are many, even infinitely many motions, distinguished by their different magnitudes, all of which pass through the same middle resting position at the same sequence of times. Such ambiguity occasionally afflicts the minimization aspect of the principle of least action, though we will find that once we know the constants of the motion, even that is resolved very nicely.
Now we ask, supposing that we know the events that start and end a given path, how to construct the minimization algorithm that correctly identifies the actual path from among all those imaginable, for the particular cases where we already know how to describe the system with . To answer this question, we will have to look more closely at this set of ``all imaginable paths'' and propose a few of its characteristics, and make use of precisely what it means to minimize something, which will lead us to another remarkably simple but powerful notion.