Hamilton's principle/Explainers/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
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
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
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.