Constraint Forces - Motion of a bead on a wire

Question

I am trying to understand the transition from Newtonian Mechanics to Lagrangian Mechanics. I have been looking at various examples of physical problems, starting from Newtonian Mechanics, and trying to 'naturally' transition to Lagrangian Methods. I'm currently looking at the motion of a bead on a wire with no applied forces or friction. So only the constraint force acts to keep the bead on the wire. The bead is started off with some velocity.

For simplicity, the wire is fixed in space.

Thinking about conservation of energy, there is nowhere the kinetic energy can reasonably go to - no fields, no friction, etc, so the absolute scalar speed should be constant. Also velocity is tangential to the bead, so the tangential velocity component should be constant. This tallies with the idea that constraint forces are perpendicular to a constraint surface - those constraint forces cannot change the tangential velocity at any point.

So the bead should just move along the wire at constant speed. i.e. if the wire had length intervals marked onto it, the speed in those units should be constant.

I find this counterintuitive - surely around a sharp bend in a wire, the speed should slow down? Or speed up? It's hard to accept that it just moves along at constant speed, the structure of the wire being more or less irrelevant to the problem. i.e. the bead can experience unlimited amounts of force, but because it is perpendicular to the wire, its tangential speed never changes.

Edit: The root cause of my confusion I think is this - it is often said that movement along a wire is caused by a constraint force which is always perpendicular to the wire. Is there any clear argument to support this?

Answer 1

The root cause of my confusion I think is this - it is often said that movement along a wire is caused by a constraint force which is always perpendicular to the wire. Is there any clear argument to support this?

You have already stated that the only force in this system is caused by interactions with the wire. Saying the force is always perpendicular to the wire equivalently means that the force never has a component parallel to the wire. It might be easier to see why this is the case. Any force component parallel to the wire on the bead caused by the wire would mean the wire is somehow pulling on the bead due to some sort of friction force, or pushing on the bead along the wire due to some mechanism (not sure what that would be exactly). There is no mechanism (ideally) where the wire would have this capability. Therefore, the force must always be perpendicular to the wire.

Addressing your faulty intuition, you might be thinking about how when you go around a sharper curve you have a larger magnitude of acceleration. And this intuition is true. However, as you have correctly reasoned this acceleration is always perpendicular to the velocity, so the speed along the wire cannot change.

the bead can experience unlimited amounts of force, but because it is perpendicular to the wire, its tangential speed never changes.

I wouldn't say "unlimited amounts of force". If your wire does have a kink in it to require an infinite acceleration then I think we are going to be in trouble in thinking about an infinite force. If instead you mean you could contrive in your mind a wire with an infinite number of bends, then that is fine. The motion of the bead will not remember how many turns it has encountered, and it does not know how many more turns it will encounter. You have set up a system where kinetic energy can never change, so it should not be surprising that you have found the only force present cannot do any work on the bead.