It may be that we have made too little use of symmetry up to this point in our search for patterns. If so, it certainly could not be because symmetries are difficult to observe. Almost nothing is easier to notice or easier to test than when one part of the world can be exchanged for another without effecting the behavior of the objects that move through it. Rather, summetries are so simple that they seem almost trivial, and it is surprising that they could yield useful insight into the compex behavior of objects.
That Locality as a princple is useful is less surprising. In its most approximate form, it gave us the division of the object from the external world that made sense of Dimensional Analysis, and in more complete forms, as for the spring, it gave a clear division between those parts of the external world that should be considered and those that could be ignored, which made construction of the ``'' part of easy. Then, in the most complete form we have yet seen, that of the rubber band, locality was the principle that tied together all the parts of the band's path into a coherent whole, but in a way that worked independently of which part of the path we considered.
Now, with the insight afforded by Noether's theorem, we find that it is symmetry, together with locality, that provides the connection between minimization rules and equations of motion, and shows us what are the momenta and when they are conserved. So we will now look at these two ideas more closely, and see how important they have been, all by themselves, in determining what the action could have been.
There is an interesting question lurking at the back of all of this analysis. We have found that a minimization principle may be a good candidate for the more fundamental principle that underlies both Newton's laws and the descriptions of other systems, like light, for which they cannot be formulated as such. Yet unlike the fundamental, albeit seemingly empirical, role that plays in Newton's fomulation, where it is simply the thing that one puts on one side of a standard equation, it has no obviously equivalent importance in the Action principle. After all, the whole equation of motion simply arises as whatever slope the action has in the space of all imaginable paths. It is not obvious why that slope should keep having the same term in it. This appearance of a consistency for which there is no obvious reason is the kind of a thing that would tend to bother a physicist, and the search for the causes of such consistencies is a large part of the source for direction in modern research. So it would be good to keep this curious fact in mind in what follows.
Returning to the question of how important or unimportant symmetry actually is, suppose that, knowing nothing about a system except its symmetries and the basic characteristics encoded in its dimensional analysis, we were forced to guess an action for the system, for the purposes of inserting it into a minimization principle and deriving the system's dynamics. How likely is it that we would guess the correct action? That is, once we had guessed an action, how likely is it that the equations of motion we derived from taking its slope in the space of paths would wind up being the same as the obtained by whatever means?
We look at the case of the falling rock near the earth's surface. We know, from observing it over and over, that how a given part of the path looks does not depend on whether we had just thrown the rock a moment before looking or had thrown it from some other place much longer ago, so that it reached the same place going the same direction. This is an important and nontrivial observation, because it underlies our basic premise that whatever are the controlling dynamics of this system, they should at least be local in the position along the path. This is simply a recapitulation of our earlier observation that the path of the rock is the same, whether we consider it as a whole actual path or as a subpath of some other path. If this had not been the case, then we would have had to know the whole history of every object to predict its future behavior, which would be much more difficult if not impossible to do. And in fact, this notion that the paths of objects evolve in a local way is so common and so universal that we have come to just assume it as a basic premise.
How is this reflected in the form we must choose for the action? For the rubber band we know. It was precisely such a piece-by-piece locality that caused the kinks to even out independently. And for that system, it happened because the thing that was minimized, namely the length of the whole band, could be written as a sum of little lengths of bits of the band. Thus, for our action, we require the same form. Whatever it is that is to be minimized over the whole path, analogous to the length of the rubber band, it must be possible to write it as a whole sum of little parts, each of which comes from some little piece of the path and must be minimized separately. This is simply a familiar statement about lengths. If some big number is a sum of a lot of other little numbers, then there is no way that the sum can be minimized unless each of the little numbers is minimized as well. Otherwise, we could always make one of the little numbers smaller, which would decrease the value of the sum as well.
This idea is important enough that it is given a special name. It is common, though it will not always be necessary, to label the little bits of the path by the times at which the object passes through them. Then in that way, we can label each little piece of the path by the little bit of time it takes the object to pass through it. A slightly longer piece composed of two shorter pieces comes labeled by a time that is just the sum of the other two. (See fig. 6.10).
Since the action we want is supposed to be a property of the path, where it is and what it is doing at that place and moment, the action should include no reference to how we choose to divide the time into little pieces to talk about it. Therefore the action that is associated with the combined little pieces of path should also just be the sum of the action associated with each of the little pieces independently. For very short pieces of path, so short that in fact we cannot notice either a change in the position or a change in the behavior of the path, presumably the little bits of action should also be the same, since it is supposed to be determined entirely by the characteristics of the system and the path. In that case the piece of the action simply becomes proportional to the length of time labeling the pieces, since by cutting a very short piece of path in half, we should end up with two little equal pieces, and by cutting the little length of time in half, we also end up with two little equal pieces of time (see fig. 6.10).
The factor of proportionality between the bit of time and the bit of action is then the number that tells about where the bit of path is and what it is doing as it passes through there.
That special number is called a Lagrangian, after J.L. Lagrange, the mathematician responsible for the invention of much of this way of looking at physical laws. The relation is this:
(As an aside, it turns out to be very interesting that the action has the units it does. These units, , are also known as an angular momentum, because they are a measure of how ``hard'' it is to stop something that is spinning by breaking its rotational symmetry (i.e., by grabbing it). This momentum was one of the additional cases that we didn't treat in the text but which is treated in some detail in the problems. It was mentioned in the chapter on the Fundamental Dimensionful Constants, that from seemingly a totally different origin in atomic physics, it was discovered that for very small systems that can be cleanly observed because they are especially simple, all of the particles have amounts of this angular momentum that only come in certain multiples of some dimensionless number that is always the same. That number is h/, and explaining this observation requires the development of the quantum theory. We will find much later when we do this, that precisely because it is the quntum theory that ultimately underlies the principle of least action for objects, the important object, the action, has the same units as the fundamental constant h/.)
For the falling rock, since we have already factored out the time that measures the place along the path, the Lagrangian itself must be some simple number that depends only on the circumstances and the characteristics of the path. We know from dimensional analysis that there is only one pattern inherent in the circumstances, encoded by a constant number we have called g, with units of an acceleration. Because we have been careful to notice the symmetries, we are aware that the sideways position of the path cannot matter, because interchange of one place for another to the side of it is a symmetry. Thus the only characteristic of the circumstances that is left is the height above the earth's surface. And, recall, we are aware that the object has a mass, which we can denote by m, though it seems not to be important (a point to which we return in a moment). The only combination that we can form from these numbers with the correct units for an energy is , so we expect that the only dependence of the Lagrangian on the circumstances can be through some term .
The only other thing that the Lagrangian can depend on is the characteristics of the path itself. Of these, we have already taken account of the position by calling it the circumstances, so the characterists that are left are the velocity, the acceleration, and so forth. It happens that there is only one new term that we can create from the velocity that has units of an energy, and it is .
We might suppose that there could be also a term like , where is the downward acceleration, since the path is characterized by that acceleration and symmetry does not forbid us to assume dependence on . It turns out that, apart from specifying different starting conditions for the path, which are not contained in the action anyway, this term is redundant with the previous term , so having included one, we have in a sense already included the other.
To check out the few terms remaining, we make use of other conditions and restrictions that we have already discussed above. Recall that the whole use of noether's theorem depended on the property, that the action looked like the smooth surface of a hilly country when drawn on the map of all paths. There certainly is no mathematical theorem that says that the slope at the bottom of any arbitrary valley is flat. For sharp or peaked valleys, the slope does not really even have a meaning at the bottom, because it looks different as you approach it from different sides (fig. 6.10).
So if the action is to be useable in a minimization principle, it must be a smooth one. So as the condition to assume that what we are doing will generate any answer at all, we must restrict ourselves to smooth actions.
But this, together with other consistency conditions, rules out all remaining possibilities for the acceleration. We already noted that since, in general, acceleration describes motion in three directions, has three parts. Since the sideways directions are all the same by symmetry, neither of them can appear by itself in an action. We cannot include the absolute magnitude of the acceleration (throwing away the information about its direction), because that is not smooth, as shown in fig. 6.10.
So we have already looked at the only term that incorporates the acceleration, properly mimics the symmetries we have observed for the system, and has the required properties of smoothness.
Thus, the only two terms we can expect in the Lagrangian, on the grounds of locality, symmetry, smoothness and dimensional analysis are and . We already know that dimensional analysis is not enough to tell us where things like dimensionless factors of proportionality go, so we don't expect to get them right. But the fact that the whole action can be restricted to being made of only these two terms from such simple criteria is nearly unbelievable. Implicitly in what we have done, we have been assuming that the action has a minimum, and while that is not enough to find the dimensionless factors, translating that requirement into mathematics is enough to tell is what the relative signs of the terms should be. (appendix for calculus-based?) If we perform that check, we find that the only allowable Lagrangian must take the form:
and consequently the action becomes
If we calculate the slope of this action in the space of paths, we find that when the proportionality factor () is 1/2, we get Newton's relation exactly.
So now, in addition to the staggering realization that the whole dynamics of falling objects in earth's gravity can be predicted from only dimensional analysis, symmetry and locality, as encoded in an action principle, we also see why for so many systems equaled , and not something else. There are remarkably few terms that one can build into an action that depend on the characteristics of the path. One that can almost always be included is a term like , which is responsible for ``'' terms in the equations of motion, and the other terms that could be built often either violate some symmetry or are redundant with it, as in the case of the falling rock.
We also see that the fact, evident from our first dimensional analysis, that the mass does not seem to matter, is reflected in the form of the action. Since all of the terms are simply multiplied by the same m, it clearly does not affect the action's minimization, and we could presumably have ignored it for this application. In fact that is just so, though when we understand much later why action is important in the quantum theory, we will see why it is appropriate to include m, even as an overall multiplier, and consequently why large objects (meaning those with large m's) do not display many of the signature ``quantum'' behaviors that are so characteristic of smaller objects. For now, the action does precisely everything we have required it to do. In some sense, it is reassuring that, since there is no special role for dimensions in a minimization principle, we see no importance of including the m or omitting it. After all, while we have gotten an increadible amount of detailed prediction, equations of motion and their conserved quantities, from seemingly very little, the symmetries, we would feel very uncomfortable having gotten something for nothing. The power of a physical law is measured by how effectively it organizes our thoughts, and enables us to use our simplest observations to make our most powerful predictions. But if the law gives us predictions that don't follow from any of the things that we have observed, it can only be telling us about something we have put in that nature did not.
One last mnemonic can be installed, perhaps as an aid in future associations. Physicists tend to use a lot of letters to represent objects, because they are a lot easier to write in equations than the words that tell all about what they stand for. Nonetheless, sometimes the letters can seem obscure and distracting. In this case, they are curiously handy for remembering the point. S, the letter for action, could be thought of as referring to Symmetry, because it is really symmetry that gives the action its form, as well as all of its power. After all, only in those cases where there is symmetry is one of the momenta conserved, and only then is there a useable constant of the motion.
And, though L was obviously and appropriately chosen to respect Lagrange, it is interesting that the whole reason the action takes the form is does, as a sum of little bits of time multiplied by a Lagrangian, is that only in such a form does it guarantee that the resulting equations of motion will indeed be Local, like those that pull the kinks out of a rubber band. Therefore, it is locality that necessitates L at all, and that is represented by it.