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Published from the Principia Machina source

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	{{Technical\|Hamilton's principle/Mathematics/Holonomic constraints}}		{{Technical\|Hamilton's principle/Mathematics/Holonomic constraints}}
			{{Hamilton's principle map}}

	This page explains what happens when the parts of a system are tied together. It		This page explains what happens when the parts of a system are tied together. It

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	[[File:OPL-HP-F-33.png\|thumb\|right\|400px\|D'Alembert's principle: dynamics rewritten as statics.]]		[[File:OPL-HP-F-33.png\|thumb\|right\|400px\|D'Alembert's principle: dynamics rewritten as statics.]]

	[[Hamilton's principle/Explainers/More than one variable]] handled systems with		[[Hamilton's principle/Explainers/More than one variable\|More than one variable]] handled systems with
	several parts by a simple trick: nudge one part, leave the others alone, and read		several parts by a simple trick: nudge one part, leave the others alone, and read
	off a rule for the part you nudged. Repeat for each part.		off a rule for the part you nudged. Repeat for each part.
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	The second method is at		The second method is at
	[[Hamilton's principle/Explainers/Multipliers]].		[[Hamilton's principle/Explainers/Multipliers\|Multipliers]].

	== The kind of constraint that behaves ==		== The kind of constraint that behaves ==
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	The method for keeping the description and getting the forces out is at		The method for keeping the description and getting the forces out is at
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{{Technical|Hamilton's principle/Mathematics/Holonomic constraints}}

This page explains what happens when the parts of a system are tied together. It
uses no mathematics. The technical page it goes with is named above.

== Things that are not free to move ==

[[File:OPL-HP-F-18.png|thumb|right|400px|A constraint force normal to the permitted motion.]]

Most systems are not free. A bead is threaded on a wire. A pendulum bob hangs on a
rigid rod. Two railway trucks are coupled together. A door swings on hinges.

In each case something restricts the motion. Physics calls such a restriction a
'''constraint'''.

Constraints are not a nuisance to be tidied away. They are usually the point. A
pendulum is interesting '''because''' the rod holds the bob at a fixed distance. A
door is useful because the hinges stop it wandering off.

== Why constraints break the method ==

[[File:OPL-HP-F-33.png|thumb|right|400px|D'Alembert's principle: dynamics rewritten as statics.]]

[[Hamilton's principle/Explainers/More than one variable]] handled systems with
several parts by a simple trick: nudge one part, leave the others alone, and read
off a rule for the part you nudged. Repeat for each part.

Constraints kill that trick.

Take the pendulum. Describe the bob by how far across it is and how far down. Now
try to nudge the across-position while leaving the down-position exactly as it
was.

You cannot. The rod is a fixed length. Move the bob sideways and it must rise or
fall to stay on the circle.

So you cannot test one part at a time, and the argument that gave a rule per part
no longer runs.

== What survives ==

[[File:OPL-HP-F-44.png|thumb|right|400px|Varying one degree of freedom at a time.]]

Something weaker does survive. The method still says the total cost does not
change under any nudge the constraint '''allows'''.

That is genuinely less information. Before, you knew nothing was changing
anywhere. Now you only know that nothing changes in the directions you are
permitted to look.

Picture standing on a hillside path that runs around the slope. Walk along the
path and your height does not change. That does not mean you are on flat ground.
The hill still slopes — steeply — but it slopes in a direction the path will not
let you go.

A constrained system is exactly like this. The cost is not flat. It is only flat
in the allowed directions.

== The forces you cannot see ==

There is a physical reading of that hillside, and it is the useful one.

Something is holding you on the path: the ground pushes back. In a pendulum the
rod pulls on the bob. In a bead on a wire the wire pushes on the bead.

These are '''constraint forces'''. They exist precisely to stop motion that the
constraint forbids.

And they have a special property. They always push at right angles to the
permitted directions — sideways into the hill, never along the path. So they do no
work on any motion the system is actually allowed to make.

That is why the two accounts fit together. The cost is not flat, because the
constraint force is pushing. But you can never detect that push by moving in a
permitted direction, because the push has no component that way.

== Two ways to proceed ==

'''Change the description.''' Instead of tracking the pendulum bob's across and
down positions, track the single angle the rod makes with the vertical. The
constraint is now built into the description and cannot be violated. One quantity,
free to take any value, and the original method works untouched.

'''Keep the description and account for the forces.''' Sometimes the tidy
description does not exist. And sometimes you specifically want the constraint
force, because it is what will break.

An engineer designing a crane needs the tension in the cable. The tidy angle
description of the swinging load never mentions tension at all. It gives the
motion and throws away the number the engineer came for.

The second method is at
[[Hamilton's principle/Explainers/Multipliers]].

== The kind of constraint that behaves ==

Everything above assumed a particular kind of restriction: one that limits '''where
things can be'''. The rod fixes a distance. The hinges fix an axis.

There is a second kind, and it behaves differently. Some restrictions limit not
where you can be but '''how you can move'''.

A coin rolling upright on a table without slipping is the standard example. The
rolling condition does not forbid any position: with enough manoeuvring you can
get the coin to any spot on the table, facing any direction. Nothing is out of
reach.

But at every instant, the way it may move is restricted. It cannot slide
sideways.

So the coin has full freedom of destination and restricted freedom of route. That
is a different kind of constraint, and it is called '''non-holonomic'''.

Parallel parking is the everyday version. A car can end up in any parking space,
facing any way. It just cannot get there by sliding directly sideways, which is
why the manoeuvre takes several moves.

== A genuine warning ==

The methods on the following pages are built for the first kind of constraint, the
kind that limits position.

Applied to the rolling kind, they give an answer, and the answer is '''wrong'''.
Not approximately wrong. It describes a different system from the one you asked
about.

This is unusual and worth flagging. Most methods in physics either work or visibly
fail. This one fails quietly, producing plausible equations for a body that does
not exist.

Getting rolling bodies right needs a different route, and this volume does not
take it. It notes the boundary so that nobody walks across it by accident.

== Where to go next ==

The method for keeping the description and getting the forces out is at
[[Hamilton's principle/Explainers/Multipliers]].

Hamilton's principle/Explainers/Constraints - Revision history

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