Hamilton's principle/Explainers/Multipliers - Revision history

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Published from the Principia Machina source

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	{{Technical\|Hamilton's principle/Mathematics/Lagrange multipliers}}		{{Technical\|Hamilton's principle/Mathematics/Lagrange multipliers}}
			{{Hamilton's principle map}}

	This page explains how to keep a constraint and still use the method. It uses no		This page explains how to keep a constraint and still use the method. It uses no

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	[[File:OPL-HP-F-19.png\|thumb\|right\|400px\|Tension in a swinging load.]]		[[File:OPL-HP-F-19.png\|thumb\|right\|400px\|Tension in a swinging load.]]

	[[Hamilton's principle/Explainers/Constraints]] left a problem. A constraint stops		[[Hamilton's principle/Explainers/Constraints\|Constraints]] left a problem. A constraint stops
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	The second kind has its own page, at		The second kind has its own page, at
	[[Hamilton's principle/Explainers/Fixed length problems]].		[[Hamilton's principle/Explainers/Fixed length problems\|Fixed length problems]].

	== A familiar version ==		== A familiar version ==
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{{Technical|Hamilton's principle/Mathematics/Lagrange 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 ==

[[File:OPL-HP-F-19.png|thumb|right|400px|Tension in a swinging load.]]

[[Hamilton's principle/Explainers/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
[[Hamilton's principle/Explainers/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
[[Hamilton's principle/Explainers/Fixed length problems]].