We said several sections ago that the only proper way to refer to paths was in a space-time diagram, because only there does the object really occupy ``the whole path''. Therefore, while we may have guessed some correct answers from the space-space pictures of the last section, we cannot call these insights derivations. So we now return to the proper space-time diagram for the motion of the falling object.
If we were to draw a three-dimensional picture, with sideways space as well as vertical space and time, we would be including, in a proper way, all of the results pictured in the last section, and all of the arguments would still be true. So for simplicity, because we know that by making pictures with enough dimensions, we can always include any spatial directions we want, we will take that for granted and represent whichever one is of interest as ``space'' and draw simple space-time diagrams with only one direction of space and one of time.
We start this section with a problem, in which we work through a proper derivation of the ideas we introduced in the last section. In the course of these arguments, we will see how the momentum arises as the constant of the motion, and why this only happens when there is a symmetry.
It should be starting to become clear, by now, that it is no accident that all of these issues, of minimization, conserved quantities, absence of forces and symmetry are related. While it may have seemed farfetched at the beginning to introduce minimization ``out of the blue'', it keeps being related to things which stay the same all along the path. Recognizing that absence of forces in Newton's laws coincided with conserved momenta was one part of the hint. A much more compelling one, though, comes from a sublte piece of argument we used, without much note at the time, to prove a needed point in the section on Fermat's theorem. We now return to that, because it will show us how to make a more general argument, of much the same kind, which applies to all actions.
We were considering the equal-angle law for reflection from mirrors. Of course, once that is specified, it can just be derived (from some algebra) that the condition of equal angles for the two halves of the path is equivalent to the condition of minimum length for the whole path. We didn't do the algebra, though. Rather, we let a rubber band do the algebra for us. The reason this worked was that the tension at every point along the rubber band was the same. It followed, then, that the difference between one path and another was the way that tension could be eased by moving the ``bounce point'', where the rubber was pinned against the mirror.
It is important to note that there was nothing special about any position on the mirror itself, or indeed any point along the its path through space at all, that affected the rubber. Rather, it was the way the parts of the rubber path related to each other that created the preference for one position of the ``bounce point'' over others. Indeed, it is precisely the fact that no place was special that allowed the tension to be the same everywhere. The equivalence of all the positions in empty space, that the rubber could have occupied, or all bounce points on the mirror, is an example of a symmetry, and the tension in the band, the same at each point, is an example of a conserved quantity. It is precisely in those cases that no position for the rubber is favored over any other (by space or by the mirror), that the bits of the rubber are free to re-arrange themselves in whatever way distributes the tension evenly over the whole band; in fact, they must, in response to the symmetry of their circumstances.
A contrasting example would be the cable of a suspension bridge, hanging under its own weight in gravity. (We chose light enough rubber bands in the last example that we could ignore gravity to the accuracy we needed.) In this case, the cable would definitely prefer to fall, and only the way it is held at the endpoints prevents that downward motion. Again there is tension in the cable, but whatever the tension is in the middle, the tension at the ends must not only match that, but also provide the additional amount needed to support the weight hanging from them. The balance of these effects, from the ends to the middle, determines the shape of the hanging cable and the distribution of tensions in it.
Further, one can see why a constant tension is necessary in response to a symmetric environment. Consider a stretched rubber band in empty space. Suppose the tension were different at any place than at any other. Then the entire part of the band between those two places is free to move (remember: in empty space one place feels the same as another). This would allow the more tense part to shorten (relieving its tension somewhat) and cause the slack place to lengthen (increasing its tension) until the tensions in the two were the same and the pulling stopped. In the case of the hanging wire, movement of a big piece up or down in gravity is not a symmetry (those cables weigh tens of tons), and big segments are not free to move to make tensions equal.
We now generalize this to an observation about any action, for any path. Suppose the actual path (the place of least action) for some object is the straight line shown in fig. 6.9.
We always draw time, and suppose we have drawn whichever dimension of space is currently of interest. Whatever the action is, we have already said that it can only depend on properties of the path itself, and on how which positions the object occupies in its environment. That is all we need to know.
We want to consider the case that moving the path to one side or another in space is a symmetry (that is why the actual path has been drawn as a straight line). That is defined to mean that the change of the path's position, in itself, contributes no change to the action. However, the action need not depend only on the path's position in space. It also can depend on properties of the path itself, like its velocity. This is exactly the same as the way the length of a rubber band is determined by the amount-of-stretch of each of its little segments. In empty space, the length of the rubber has nothing to do with which part of empty space it lies in, but it is completely determined by the shape (the sequence of amounts-of-stretch) of the path itself.
Next, we also choose to look at a different path, deformed from the actual path as shown in fig. 6.9,
and ask what the new action is. The reason to do this is to make use of the ``slope of a valley'' observation, by taking little steps around in the valley of the actual path, and seeing what can happen. In spite of the fact that moving part of the path sideways is a symmetry, we have also changed the shape of the path by changing the velocities at two places, and the action could certainly depend on this. However, if we still want to consider paths that start and end at the same events, we are not free to vary the velocity in any way we want. If we speed up by a certain amount at one place along the path, we have to slow down by just the same amount later, to arrive at the same final place at the same time. For this reason, though we are free to choose the locations of speed-up and slow-down, the amounts have to be exactly opposite. This is much like the way we could move a piece of rubber one way or another, but if it is pinned at the endpoints, the amount of displacement gained at one end of the change must exactly equal the amount lost at the other.
Finally, we recall the whole point of the action, which is that, like a smooth valley, its slope at the minimum is zero. The slope is the amount the action changes, per little change we make in the path. The virtue of the action is that, like the equations of motion, it is built entirely of pieces which depend on local properties of the path, in a way that we will see in the next section. Therefore, if the action even could change because of the change in velocity, if we have changed the velocity only at one little place, the change in the action must have come from properties of the path at that place alone. But we already said that, for little steps around the actual path, the total change in the action is constructed to be zero (recall, that is how we built actions from equations of motion). If the only change in the action comes from the velocities (because of the symmetry), and we have only changed the velocities at two little places, and the action is truly built up only of local properties of the path, then the change in the action must just be the sum of the two little changes that resulted from the two little local shifts in velocity. To be zero, these must cancel. That is what it means for the action to be minimized.
Recall, though, that the two changes in the action are not the only things to cancel. That fact is what will make this description useful. In general, the changes in the action should depend on how much we changed the velocity, which was ours to choose. This is like the way the total length of a rubber band depends on how much you choose to deform it away from its resting state. But, with the endpoints fixed, the two changes of velocity were forced to be equal and opposite, just like the two changes in the action were equal and opposite. Therefore, it must be that the changes in the action are just proportional to the changes in velocity, and that the proportionality factor is the same at both places. This way, whenever we are forced to make two equal and opposite changes in the velocity, we automatically create two equal and opposite changes in the action. Unlike either the changes in velocity or their resulting changes in the action, though, the proportionality factors, at least as accurately as we can resolve them with our ``small steps'', are just properties of the actual path, that don't depend on what our changes were.
This last point shows which quantity is interesting. The rate of change of the action, per little local change in velocity, is some number that is the same at both places on the actual path. Of course, since we never said where we chose to put the speed-up and slow-down places, this rate of change must be the same everywhere along the actual path, or the result couldn't be true.
Again, we illustrate with the rubber. There is some tension at each place in the rubber. If we grab a segment of the rubber and move it just a little bit sideways, by stretching it locally at one end and relaxing it at the other (this looks just like the space-time picture, interpreted differently), then we increase the tension in some little local region at one end and decrease it in a local region at the other. The rate at which we increase the tension, per bit of stretch, is a property of however the rubber was laying at the start. If it is different at one of our deformation-place than at the other, then we can always relieve more tension at one end than we create at the other, and hence reduce the overall tension, and thus the overall length, of the band. If we assume we started at the absolute minimum possible length, though, then there is no way we could have done this, which means that we always gain exactly as much tension (exactly as much length) at one end as we lose at the other, from any small deformation. Again, we tend to think of tension as the important thing about rubber, because it makes things move, but it is really just proportional to local amount-of-stretch. Therefore, it is the rate of change of total length, per bit of small deformation, that is kept the same (by the tension) at each place along the actual path of a stretched piece of rubber. Of course, all this is only true when there is a symmetry with respect to the displacements themselves.
In precisely the same way, it is the rate of change of the action, per small change in velocity, that defines what we have been recognizing as the momentum. When there is a displacelement symmetry in some direction, it is the same at all points along the actual path; that is, it is conserved. This is the result known as Noether's theorem. In the next section, when we see how to build actions explicitly in terms of local properties of the path, we will also see how the particular form of the momentum is related to the properties of the path at each point.
Now, possessing a proper understanding of the relation of symmetries to the conservations of momenta, we can return to the question that illustrated our ignorance. Is there a symmetry, and associated with it a conserved quantity, that encodes the pattern of uniformness in the downward rate of fall of rocks? We look all the way back to the original space-time diagram of the last section. It addressed precisely this issue. Recall that while it looked exactly like the space-space picture of the trajectory because of the way it was drawn, the unevenness of the wrong imagined path, compared to the smoothness of the actual path, referred not to changes sideways in space, but to changes of the path forward or backward in time.
But we are well aware that, if there is anything that stays the same, it is the behavior of the earth's gravity from one moment to the next, so we could have dropped the same rock now, or sooner, or later, and we certainly would have expected the same behavior. So it appears that this motion forward or backward in time is every bit as much a symmetry as the motions sideways in space. So, it should have, from the arguments we have just completed, an associated conserved quantity. And in fact it does. The conserved quantity associated with the symmetry of shifts in time is given the special name Energy. The understanding that it was conserved came long before it was realized that it was indeed just another momentum, and because time is special and in important ways different from space anyway, the designation of this special momentum by a special name has been retained. Nonetheless, its role as a momentum is just like that of any other, and it is or is not conserved based on whether there is or is not a symmetry of time shifts, just like any other momentum relates to its associated symmetry.
It happens that if one pieces together the action for falling rocks from , as described before, and then works through the application of Noether's theorem to find the form of the energy, one discovers that the potential, of which was the slope, is one term in the energy, and that there is another term that comes from the response by the path. This second term is a kind of ``energy due to motion'', which is called from its greek root a kinetic energy. The fact that the first term, the potential, is also a term in the energy, enables us to identify what kind of a potential it is. This potential that induces motion is indeed a potential energy. In this case, since the downward force (the slope of the potential energy in the downward direction) is just constant, , the potential energy obtained by patching together this constant slope over little steps is just a sloping plane. In other words, the potential energy is just , where is how high the object is above the surface of the earth.
The fact that the energy has these two terms, and that only the whole energy is conserved along the path, accounts for the fact that the downward motion is curved on the spacetime diagram, and not a straight line. In other words, stays the same all along the path because of the time-shift symmetry, so at the lower places on the path, where is smaller, increases correspondingly as ``energy is shifted from the potential for motion encoded in to the achievement of motion encoded in ''. It is in this way that the conservation of energy provides the same simple description of the various parts of the path as did the conservation of sideways motion for a different symmetry.
This set of momenta that we have discovered, sideways momenta and energy, is representative of all of the symmetries that this system can have. It happens that there are others, associated with rotations, but the analysis of those would look much like what we have already done. We note, just in passing, that we also now understand why, in spite of the fact that the two diagrams have such different meanings, it happened that the actual path looked the same in the space-space diagram and in the space-time diagram of the last section. It was because of the accident that for this system sideways momentum happens to be a conserved quantity, and it happens to equal the sideways component of . Therefore, along the actual path, the object covers the same amount of sideways distance in each little bit of time (that is what it means to have a constant sideways velocity) and it is a tautology that it covers the same amount of time in each amount of time, so the two pictures, of motion sideways in space and motion forward in time, look the same when one draws the actual path of the falling rock.
The entire last two sections have been devoted to showing that the Principle of Least Action can reproduce the results of when the latter exists, and to showing how to get from one description to the other. But recall that our first example of a minimization principle, Fermat's principle for light, described a system with no formulation. In that case the action was a time along a path, and it was discovered more or less by the insight of an astute man. We can ask whether this astuteness is always required, in which case it is largely a matter of luck that guides us to discovery of an action, or if there are other ways to choose an action besides extrapolation from , which could be used also in those cases for which the latter does not exist.
We will find that precisely those principles which have been important in determining the existence and form of the constants of the motion, symmetry and locality of the dynamics, are also fundamental in motivating the choice of an action. Indeed, these are the origin of the deepest power of the Action Principle, which make it irreplacable in descriptions of nuclear physics and elementary particle physics, and even in cosmology, where symmetry and locality are often the only fundamental properties we can observe.