It may seem to be a strange and perhaps impossible thing we are asking our new rule to do. What kind of rule can it be that recognizes, independently of which little pieces of path we choose to consider, whether they are mutually compatible as parts of a single greater path. And how can such a rule possibly be stated in a way that is simple enough to be at all useful, much less offer an advantage over ? It turns out that if we think only about paths of objects, this answer may not be obvious, but happily, mathematics is no respector of applications, and if we allow ourselves to think of other systems, an example of such a rule is part of common experience.

Suppose we stretch a rubber band across the surface of a table. As illustrated in fig. 6.3,

there are several different ways the rubber band can be held, and consequently several different ``paths'' across the table that it can occupy. Yet, for a given way we hold the ends, only one path is selected as the real one. And, like the parts of the trajectory of our falling object, the different segments of the rubber band only lay in certain combinations of ways. As emphasized in fig. 6.3,

the little bits of rubber band do lay in ways that are parts of a single path, but never lay in the other ways shown, drawn from different and mutually incompatible other actual paths. The similarity between the pictures, of the way the rubber band lays and the trajectories of the falling rock, turns out to be more than superficial.

Physically, certainly the two systems are not the same. The rubber band is a long string of material that occupies all of the parts of its ``path'' at the same time, whereas the falling object, at least as we have drawn it so far, only occupies one of the points on its trajectory at any given moment represented by the picture. Yet in its own way, whatever rule ``tells'' the parts of the rubber band where they can come to rest performs exactly the kind of identification of little segments of path that we would like to exploit. It orients each little segment of band in only the way that is compatible with the orientations of all the other little segments of the band, yet it does so in a way that is independent of which little segment we consider, as can be seen in fig. 6.3.

If we put a kink in some little part of the band, being careful to hold all the rest of the band as it was, the rubber obeys its rule so that the kinked part of the band returns to exactly the same configuration as it would have if we had disturbed the band in any other way. So somehow, the rule always leads to the same result, regardless of which part of the path we may choose to consider, or even to disturb. Thus, while the particular physics of the application of the rubber band may differ from that of the motion of the falling rock, the mathematics contains precisely the features we want.

But of course we understand the rules for rubber bands exceedingly well, and in fact they are some of the simplest rules we know how to state. What does a stretched rubber band do? It becomes as short as it can, and moves to whatever position makes that possible. That is the whole point of being stretched rubber. Its preferred state is unstretched, and if we pin its ends, the rest of the band moves in whatever way brings it closest to that preferred state. All of this happens locally, according to a rule that applies in each little bit of the rubber, independently of what the other segments are doing. Each bit of the rubber is moving in such a way as to become shorter, if possible, and whether all of the other bits are already as short as they can be or not does not matter to its dynamics. That is why the single little kink in the band straightens to the same place and the same orientation on the table as that part of the rubber would have if we had disturbed the band in any other way.

The utter simplicity of this kind of a rule, ``It becomes as short as it can'', and its familiarity, may mask how remarkable it truly is. It ties together the remotest segments of the actual orientation, so that all of them are compatible parts of a single whole, and yet it does this with dynamics for the pieces that are completely local. Each bit of the rubber finds a position that is related to the positions of all the other bits of the rubber, but it does so by following rules that contain no explicit reference to anything but the other bits right around it, to which it is directly connected.

This combination of global consistency and local simplicity will be very important to us. Not only is it the trademark of minimization principles, useful for rubber bands and perhaps promising for paths, but it will be the underlying motivation that will lead us to the formulation of another of the most important ideas created since the introduction of physics itself, the idea of local field theories, which we will explore in later chapters. But for now, having guessed that some sort of minimization rule may lead us to our desired constants of the motion, we try to discover how such a rule can be formulated for the paths of isolated objects.

If the reader thinks that even at this point we are still on rather shaky and speculative ground, he is not far wrong. Certainly, minimization is an efficient and elegant way to state the rules that determine the resting places of stretched rubber bands, and we may not be too surprised if it also offers a useful description of the actual paths of objects in some cases. But, we have yet to show that any such rule actually even exists for the paths of objects, and we have certainly not made a convincing case that minimization is the correct way to uncover the origin and meaning of constants of the motion. All we have done so far is argue that the notion of a constant of the motion is that of some criterion that identifies a particular actual path by excluding any combinations of little pieces of paths that cannot be parts of a single bigger path. The fact that such a mutual exclusion prescription is so much like what the minimization rule achieves for the little bits of the track occupied by a stretched rubber band makes this a plausible direction to pursue.

Now it turns out that, indeed, drawing our clue toward minimization from the mutual-exclusion function of conserved quantities is exactly the right thing to do, but this is much easier to see in retrospect once we have learned something more about the principles themselves. So for now, rather than try to justify this assumption further, we simply look at a minimization rule that has been successfully used for hundreds of years, because it was discovered even before Newton's laws, and which in fact applies to a system for which there is no formulation of an . This is Fermat's Least Time Principle for light, a phenomenon which has no property that can be called an , which fact has been measured experimentally to phenomenally high accuracy. Thus, even in our first example, we will discover a use of minimization principles that carries us further than we could reach with , and in the subsequent sections we will show how it also leads to as a consequence when the equation of motion exists.


Thu Aug 31 12:01:42 CDT 1995