Comparing paths
This page explains how physics compares paths. It uses no mathematics. The technical pages it goes with are named above.
A number for a whole journey
Think about driving from your home to a shop. There is more than one route. Each route takes a certain amount of fuel.
Fuel use is a number. A route is a whole path. So fuel use is a rule that takes a whole path and gives back one number.
Physics uses rules like this all the time. The rule might give the time a journey takes. It might give the length of a curve. It might give a quantity called the action, which this volume is about.
A rule that turns a whole path into one number has a name. It is called a functional. That is the only new word on this page, and it means exactly what the last paragraph said.
Why this is different from ordinary maths
A function takes a number and gives back a number. Put in 3, get out 9.
A functional is different. You do not put in a number. You put in a whole path — every point of it, from start to finish.
That makes the usual methods harder to apply. To find the lowest point of a hill you walk downhill. But a path is not a point on a hill, and there are infinitely many ways to change one: push it up in the middle, wobble it near an end, stretch it. Which of those counts as a small change?
Answering that is the first real task.
The rules of the comparison
Physics compares paths under strict rules.
All the paths start and end at the same two points. If you are asking which route to the shop is best, every route must actually reach the shop. Routes that end somewhere else are not in the competition.
All the paths are smooth. No sudden jumps. No sharp corners, at least to begin with. Later work relaxes this, but the simple version starts here.
Only the true path is real. The other paths do not happen. They are imaginary journeys, invented so that the true one can be compared against them.
That last rule matters more than it looks. When physics compares the real motion of a ball with a nearby path, that nearby path is not a motion the ball could make. It breaks the laws of physics. It is a made-up comparison, and it exists only to be worse than the real thing.
Hamilton said this himself, in 1834. He described the comparison paths as imaginable in geometry but impossible in dynamics.
Nudging a path
Here is the trick that makes the whole subject work.
Take the true path. Take any smooth nudge — a shape that is zero at both ends and whatever you like in between. Now add a small amount of the nudge to the path. How small is set by a single dial.
Turn the dial to zero and you have the true path back. Turn it up a little and you get a nearby path. Turn it up more and you get a path further away.
The dial is one number. So the problem has changed shape completely. Instead of asking about infinitely many paths at once, you can ask what happens to your functional as you turn one dial.
That is an ordinary question, of the kind school mathematics can answer.
What "stationary" means
Turn the dial and watch the number.
For most paths the number changes straight away. Turn the dial a little and the number goes up, or down.
For the true path something different happens. Turn the dial a little and the number barely moves. It changes far less than the size of the nudge would suggest. To a first approximation it does not change at all.
That is what stationary means. The number stops responding to small changes in the path.
You have seen this before. At the top of a hill the ground is flat. Take a step in any direction and your height barely changes. The same is true at the bottom of a valley. And it is true at a mountain pass, which is neither a top nor a bottom.
Flat is flat. The ground being level tells you that you are at a top, a bottom, or a pass. It does not tell you which.
The word "least" is wrong
This is why this volume does not say the action is least.
Being stationary means the number stops changing for small nudges. It does not mean the number is as small as it can be. Sometimes it is. Sometimes it is as large as it can be. Sometimes it is neither.
Here is a case anyone can picture. On a globe, the shortest way between London and Tokyo is the short way round. But you could also fly the long way round, the other side of the world, along the same great circle. That long route obeys exactly the same rule at every point of the journey. Locally it looks just as straight. It is stationary — and it is very clearly not the shortest.
Nothing you could measure while flying would tell you which of the two you were on. The difference is not in any single place. It is in the journey as a whole.
Working out exactly when stationary does mean shortest took mathematicians most of the nineteenth century. It is covered later in this volume.
Where to go next
The next step is to work out what actually happens to the number when you turn the dial. That gives a rule the true path must obey, and it is the most useful single result in this subject.
It is explained at What the Euler-Lagrange equation says.