WHY THIS WAS THE RIGHT DIRECTION TO LOOK

We start by understanding why the role of constants of the motion as the excluders of incompatible pieces of path was a useful one to pursue, and why the suggestion of a minimization rule was the right clue to draw from it. Again we look at the simple case of the freely falling rock.

For this discussion we can either re-draw the figure of the first sections, showing the motion in vertical and horizontal directions of space, or we can draw the other picture of paths in a space-time diagram, showing for simplicity time and the vertical direction of space. The two are useful for different reasons. Recall, we started with the question drawn in both spatial directions, how we could tell directly that some combinations of little pieces of path were parts of the same actual path, while others were not fig. 6.8.

In particular, how could we label the little pieces of the path to identify that the parts of the smoothly arching path have something common that allows them to be parts of the same larger path, while the parts of the other path we could imagine, which falls first straight down, and then starts to move rapidly sideways, have something different about them that precludes their being parts of any single actual path.

We can draw another, and importantly different version of the same question in the proper diagram, the space-time diagram that is the proper place to talk about paths and minimization. It happens that the example we have chosen looks the same in the picture, though in this case the ``sideways'' motion of the smooth path is the ``motion forward in time'', and the uneven motion of the imagined path that doesn't really occur represents an object that instantaneously travels some part of the way down (in effectively no time), and then suddenly slows down and falls much more slowly, so that its evolution forward in time carries it the small remaining distance. Both pictures will be useful.

First we look at the space-space picture to discover a relevant point. We ask, why, from experience, would it be strange if falling objects took the broken path? In a sense, it is because the broken path looks like a ``bounce'' sideways, but if the object is falling through empty space, by assumption there is nothing special at that place off of which it can ``bounce'' . Being careful, we see that, even beyond the sense in which is an infinite collection of statements, one for each moment along the actual path, it is also three statements at each point, because there are three dimensions of space, and has to tell how the object changes its location in each of the directions. The change we have been discussing involves motion in the sideways direction. We are familiar from experience with the earth's gravity that there is a ``Force'' in the downward direction, because we have to hold objects ``up'' to keep them from falling, but there certainly is no force sideways, because objects of themselves don't tend to move sideways, either on level tables or in empty space, because of gravity. Thus, our statement that there is nothing for the path to ``bounce off of'' is the statement that, in the sideways direction, there is no force, or the value of the sideways-directed part of is zero. And, we would be surprised to see sideways change in direction, which requires a sideways acceleration (a nonzero ma) where there was no to ``cause'' such a change. It is in this way that precludes such sideways bounces in empty space.

But now recall the point we discovered in the last section, when we looked at the Escher picture of water that always flows but never goes downhill. We said that such 's are never the correct ones to use in a principle of least action, because the Principle of Least Action provides the -part of the slope in path space as the slope of a real hilly surface. And we said that so far, we have always been able to find such descriptions if we were careful. It happens that for this case of the falling rock, even our first attempt gives such an , so no special effort was needed.

Therefore, recognizing that is the slope of some potential function that looks like a hilly surface in real space, we see that the statement that in some direction is zero is the same as the statement that for little changes in some direction, the value of the potential, (the height of the hill) does not change. All this, despite the new names, is very familiar. Water doesn't flow sideways across the face of a hill, because to do so would not take it any lower. But, in this seemingly simple observation, we first encounter what turns out to be a very important concept, the notion of symmetry.

Because we have discovered that it is in some sense natural to view as the slope of some potential hill, we discover that the absence of an in some direction is ``caused'' by an absense of in that direction, or equivalently, by the fact that the potential is the same for changes in the direction. In other words, we could have chosen another path that differs from the one we are considering, by having been moved sideways a little, and all of the values of the potential function that determine the new path would be the same as the values that applied to the old one fig. 6.8.

Such a change in the path we consider that results in no change in the important circumstances is called a symmetry. This use of the word is the obvious analog to its less specific conventional use. One thinks of a symmetry as some way a thing can be changed or moved, after which it looks the same as it did before. (perhaps figures with examples of symmetries).

So the fact that we could have considered a changed path that nonetheless doesn't change the circumstances that determine the motion is what tells us that in that direction of change must be zero. Really, so far we have just said a lot of words to describe the same thing. The important thing to keep in mind is that because something about the circumstances doesn't change in going from one place to another (i.e, because there is a symmetry), some characteristic of the response of the system, in this case one of the directions of ma, should remain zero.

But now we look more carefully at what is. Recall the definition of the acceleration as the rate of change in the velocity with time. From that definition, as long as we keep the object itself the same (i.e., keep its mass the same), we see that is then just the rate of change of with time.

But look! We have the kind of statement we have been looking for, albeit so far only in this special case. To say that the change of over time is zero is to say that stays the same from one time to the next, or equivalently from one part of the path to the next. So all along, the sideways part of is what were were looking for as our constant of the motion, and the fact that it is constant (i.e., that it does not change from one part of the path to the next), comes from the fact that changing our question to consider a path slightly to the side of the one we did is a symmetry, because the value of the potential function doesn't change under such a switch.

Constants of the motion, like (also called conserved quantities, because the ``amount'' of them neither increases nor decreases along the path) are very important, so they are given a special name. In general any such quantity is called a momentum, though there are often special cases in which it is given a special name to refer to what kind of a momentum it is. In the case we have examined, the sideways direction of the momentum was conserved. But what if we had looked at the downward direction? Definitely there is a downward force, which says that the potential function does change if we look instead at paths that are further down than our first one. In this case, we still can identify a downward component of , and we still refer to it as a momentum (this is a downward-directed momentum), though in this case since changes, is not a conserved quantity.

So while we have discovered that momenta are important because when there is was a symmetry it was a momentum that was conserved, the notion of momentum is useful more generally, and we can recognize it even when it is not conserved. This makes it important to clearly define what we mean by momentum, so we take a moment aside and do that.

Too often, especially in beginning classes, physicists are told that ``momentum = '', long before they are even told why momentum is a useful thing to care about. One of the huge results that we will discover when we look at special relativity is that in general, momentum is not . If we were to assert here that it were, then for consistency we would have to make all sorts of crazy statements later about how momentum could still be , as long as you allow the mass to change, after which time we would have no idea what we meant by mass, and so forth.

All that morass is simple to avoid, by realizing right at the start what momentum is, which follows from why it is important. In all of what we will do from now on, momentum will be defined as the thing that would be conserved if there were a symmetry. (Of course, if there happens indeed to be a symmetry, then the momentum is conserved, and is a constant of the motion).

So we go back now and check that this momentum, functioning as a constant of the motion, really does what we want, by excluding little pieces of paths that can never be parts of a single actual path. For the space-space picture, everything works fine. For the smooth actual path, the sideways velocity is indeed the same at all points along the path, and so is the same for all of them, which is how we can recognize them. On the other hand, the broken path describes first a segment for which the sideways is zero, and a second segment for which it has suddenly been changed to something nonzero, even though there was nothing about the empty space to create such a change. And that is precisely what does not happen in nature.

Comments here on momentum in experience, changing it by bouncing things, relation to destroying symmetries, etc. Standard examples from class.

Several disclaimers need to be made here. The critical reader will note that, apart from the motivation that we should consider as the slope of a potential, which we happened to draw from the minimization principle, all of what we have said about symmetries, conserved quantities, and the particular discovery that a momentum equal to in the sideways direction was the relevant constant of the motion, has been drawn just from simple arguments and from . We really have not made any significant use of the minimization principle, and so far have not even made a very good case that symmetries are the relevant concept at the center of all this. All we have really done is show that, because a momentum equal to sideways is the conserved quantity in this case, it does indeed function as an excluder of incompatible parts of paths.

So indeed, what has been gained by building all the apparatus of minimization? That can be seen by trying to use the hueristic arguments above to learn anything more, which happens not to work. Everything we have said so far, about the fact that the sideways momentum, , is the constant of the motion, because the force is zero, which in turn is because the potential is unchanged by shifts of the path sideways, is true, and it has served to introduce these tremendously important ideas in a case where they are already familiar from experience. But the fact that we could get these correct answers from such direct intuition was a happy consequence of our choice of example.

What would have happened if we had asked if there is a conserved quantity that applies to the downward motion? Certainly, we know from experience that falling rocks don't suddenly bounce sideways. And we have the momentum that encodes that pattern. But we also know that objects don't fall initially very fast, and then suddenly slow down, as for the imagined path in fig. 6.8.

But in this case, there is not a symmetry in the downward direction, because there is indeed a force that way. So is there a constant of the motion that encodes this other pattern? Well, we know that is constant for gravity, so maybe that is all we need.

But what about the example of the spring? In that case there are big motions and little motions, and we know that for all of them, there is not a symmetry for sideways shifts, because the spring pushes more or less as it is moved to further or less distant from its resting place. And in this case, the acceleration is not constant as it was for gravity. So clearly, while we have discovered useful notions of symmetry and the conservation of certain momenta, we have not derived or even clearly understood them, or even been able to answer such questions as when they exist or how to find them in general. It is for all these that we really need minimization and the Principle of Least Action, which give us one of the most important theoremae (?!) in all of modern physics.