The End of Time: The Next Revolution in Physics

Unfortunately for Maupertuis’s theological aspirations, it was soon shown that in some cases the action would not have the smallest but greatest possible value. The claims for divine economy were made to look foolish. However, the principle prospered and was cast into its modern form by the Irish mathematician and physicist William Rowan Hamilton a little under a hundred years after Maupertuis’s original proposal. A wonderfully general technique for handling all manner of mechanical problems on the basis of such a principle had already been developed by Euler and, above all, Lagrange, whose Mécanique analytique of 1788 became a great landmark in dynamics.

The essence of the principle of least action is illustrated by ‘shortest’ paths on a smoothly curved surface. In any small region, such a surface is effectively flat and the shortest connection between any two neighbouring points is a straight line. However, over extended areas there are no straight lines, only ‘straightest lines’, or geodesics, as they are called. As the idea of shortest paths is easy to grasp, let us consider how they can be found. Think of a smooth but hilly landscape and choose two points on it. Then imagine joining them by a smooth curve drawn on the surface. You can find its length by driving pegs into the ground with short intervals between them, measuring the length of each interval and adding up all the lengths. If the curve winds sharply, the intervals between the pegs must be short in order to get an accurate length; and as the intervals are made shorter and shorter, the measurement becomes more and more accurate. The key to finding the shortest path is exploration. Having found the length for one curve joining the chosen points, you choose another and find its length. In principle, you could examine systematically all paths that could link the two chosen points, and thus find the shortest.

This is indeed exploration, and it contains the seed of rational explanation. There is something appealing about Leibniz’s idea of God surveying all possibilities and choosing the best. However, we must be careful not to read too much into this. There does seem to be a sense in which Nature at least surveys all possibilities, but what is selected is subtler than shortest and more definite than ‘best’, which is difficult to define. Nothing much would be gained by going into the mathematical details, and it will be sufficient if you get the idea that Nature explores all possibilities and selects something like a shortest path. However, I do need to emphasize that Newton’s invisible framework plays a vital role in the definition of action.

Picture three particles in absolute space. At one instant they are at points A, B, C (initial configuration), and at some other time they are at points A*, B*, C* (final configuration). There are many different ways in which the particles can pass between these configurations: they can go along different routes, and at different speeds. The action is a quantity calculated at each instant from the velocities and positions that the particles have in that instant. Because the positions determine the potential energy, while the velocities determine the kinetic energy, the action is related to both. In fact, it is the difference between them. It is this quantity that plays a role like distance. We compare its values added up along all different ways in which the system could get from its initial configuration to its final configuration, which are the analogues of the initial and final points in the landscape I asked you to imagine. The history that is actually realized is one for which the action calculated in this way is a minimum. As you see, absolute space and time play an essential role in the principle of least action. It is the origin of the two-and-a-bit puzzle. Now let us see how it might be overcome.

DEVELOPING MACHIAN IDEAS