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?