More than one variable
This page explains how the method handles systems with many moving parts. It uses no mathematics. The technical pages it goes with are named above.
One thing moving is not enough
So far the volume has followed a single path: one thing, moving.
Real systems are not like that. A double pendulum has two arms. A bridge has millions of parts. The air in a room has more moving pieces than anyone would want to count.
The method extends to all of these. What follows is how, and where the extension needs care.
Many parts, one at a time
Suppose a system has three moving parts. Write down the cost rule for the whole system, taking all three into account.
Now nudge the first part only, leaving the other two exactly as they were. Run the same argument as before. You get a rule for the first part.
Now do the same for the second part, and the third.
You end up with one rule per part. Nothing new had to be invented. The trick of nudging one thing at a time did all the work.
The catch
That trick relies on something, and it is worth stating plainly.
You have to be able to nudge one part without disturbing the others.
Often you cannot.
Think of a pendulum: a weight on a rigid rod. Describe it by the weight's sideways and vertical positions. Those two are not free of each other. The rod has a fixed length, so moving the weight sideways forces it to move vertically as well.
Nudge one and the other has to follow. The trick fails.
There are two ways out, and both are useful.
Choose better descriptions. Instead of two positions, use the single angle the rod makes with the vertical. Now there is one quantity, it is free to take any value, and the difficulty vanishes.
Keep the description and pay for it. Sometimes the tidy description does not exist, or you want information the tidy one throws away. In the pendulum case, the angle description never tells you the tension in the rod. If that is what you are after — and an engineer usually is — you need the other method.
That other method is at Constraints and Multipliers.
Free choice of description
One of the real advantages of this whole approach shows up here.
The method does not care how you describe the system. Angles, distances, heights, anything convenient. Write down the cost rule in whatever terms suit the problem and turn the handle.
Doing the same thing with forces is much harder. Forces are arrows in space, and converting arrows into angles is fiddly and error-prone. The cost rule is a single number, and numbers do not have directions to keep track of.
For a spinning top, or a robot arm, or a chain of linked rods, this is the difference between a manageable problem and an unmanageable one.
From many parts to a continuous thing
Now push the idea as far as it goes.
A guitar string is not three parts or a hundred parts. Every point along it can move. There are infinitely many moving parts.
The method still works, and the way to see it is to sneak up on the answer.
Replace the string with a row of beads on a light thread. That is an ordinary system with a definite number of parts, and everything above applies. Now use more beads, and lighter ones. Then more still.
As the beads get closer together, the answer settles down to something. That something is the behaviour of the real string.
The result is the wave equation, and it describes waves running along the string at a fixed speed. Sound, light and ripples on water are all governed by close relatives of it.
Doing the calculation this way is worth the trouble. You could simply assert a cost rule for the string and turn the handle. Sneaking up on it from the beads shows that the rule you assert is the right one, rather than asking the reader to take it on trust.
Fields
When every point of something can move, physics calls that thing a field.
A string is a field along a line. The air in a room is a field in space. The electric and magnetic fields are fields in space, and so is the gravitational field.
Everything on this page applies to all of them. Write down a cost rule for the field, nudge it, and out come the equations. That is how modern physics is written, and it is explained at Fields.
One thing that changes
Moving to fields changes conservation laws in a way worth knowing.
For a few moving parts, "energy is conserved" means a number stays the same.
For a field it means something stronger and more useful. Energy cannot vanish here and appear over there. It has to flow — leaving one region by crossing the boundary and entering the next.
That is why you can talk about energy travelling along a wire, or sunlight carrying energy across space. The conservation law is local, and locality is what makes it possible for energy to have a route.
Where to go next
The machinery is now complete. The next question is the one the volume has been deferring: when is a stationary path actually the best one? That is at Is it really a minimum.