Multipliers
This page explains how to keep a constraint and still use the method. It uses no mathematics. The technical page it goes with is named above.
The trade
Constraints left a problem. A constraint stops you nudging one part at a time, and the whole method depended on being able to.
The method of multipliers fixes this by making a trade.
You give up nothing in your description. You keep all the original quantities, and you do not have to find a clever way of building the constraint in.
In exchange, you accept one new unknown for each constraint.
That sounds like a bad deal — you had too many unknowns already. It is not, and the reason is that the new unknown turns out to be something you wanted.
How it works, in words
Instead of forbidding paths that break the constraint, you charge for breaking it.
Add a penalty to the cost rule: a price, times the amount by which the constraint is violated. Now every path is allowed. Paths that obey the constraint pay nothing, because they violate it by zero.
The price is the new unknown. It is not fixed in advance. It is whatever it needs to be.
With no forbidden paths, you can nudge one part at a time again, and the original method runs.
Then you set the price. It has to be exactly high enough that the cheapest path happens to be one that obeys the constraint. Set it too low and cheating pays. Set it correctly and the winner obeys the rule without being forced to.
The price is the force
Here is why the trade is good.
The price is not bookkeeping. It is the constraint force — the pull in the rod, the push from the wire, the tension in the cable.
Come back to the crane. Describing the swinging load by a single angle gives you the motion and never mentions the cable. The multiplier method gives you the motion and the tension, in the same calculation.
The tension is what the engineer needs, because it is what determines whether the cable holds.
So the extra unknown is not overhead. It is usually the answer.
The pendulum, both ways
Take a weight on a rigid rod, swinging under gravity.
By clever description. Use the angle from vertical. Out comes the equation of the swing: how the angle changes with time, and hence the period. Nothing about the rod appears, and the rod's tension cannot be recovered from the result.
By multipliers. Keep the sideways and vertical positions, add the price for stretching the rod, and run the method. Out comes the same swing — and out comes the tension as well.
The tension has two parts, and both make sense. One is the share of the weight the rod carries, which depends on the angle. The other comes from swinging the weight in a circle, which depends on speed.
That is why a rope holding a swinging load is under most strain at the bottom of the swing, where the load is moving fastest and hanging straight down. Both parts are largest at the same moment.
An engineer sizing that rope needs exactly that number.
Two kinds of constraint, two kinds of price
There is a distinction here that trips people up, and it is easy to state.
Some constraints apply at every point. The rod has fixed length now, and in one second, and always. Such a constraint needs a price that can vary from moment to moment.
Other constraints apply once, to the whole thing. A chain has a fixed total length. That is one fact about the chain as a whole. It says nothing about any particular point.
A constraint of the second kind needs a single fixed price, not a varying one.
Getting these the wrong way round is the commonest mistake in this area. The test is simple: ask whether the constraint is one rule per moment, or one rule for the whole journey.
The second kind has its own page, at Fixed length problems.
A familiar version
You may have met this before, outside physics.
In economics, a firm minimising cost subject to producing a given output uses the same method, and the multiplier is called the shadow price — how much the cost would rise if the output requirement went up by one unit.
That is the same idea. The multiplier measures how much the answer would change if the constraint were loosened slightly.
For the pendulum: how much the energy would change if the rod were slightly longer. For the crane: how hard the cable is pulling.
The multiplier always measures the cost of the constraint. Whatever the setting, that is what it is telling you.
Where to go next
The single-price case, for constraints that apply to a whole path at once, is at Fixed length problems.