Editing Generalized coordinates (section)

===Bead on a wire===

[[File:Bead on wire constraint.svg|thumb|200px|Bead constrained to move on a frictionless wire. The wire exerts a reaction force {{math|'''C'''}} on the bead to keep it on the wire. The non-constraint force {{math|'''N'''}} in this case is gravity. Notice the initial position of the wire can lead to different motions.]]

For a bead sliding on a frictionless wire subject only to gravity in 2d space, the constraint on the bead can be stated in the form {{math|1=''f'' ('''r''') = 0}}, where the position of the bead can be written {{math|1='''r''' = (''x''(''s''), ''y''(''s''))}}, in which {{mvar|s}} is a parameter, the [[arc length]] {{mvar|s}} along the curve from some point on the wire. This is a suitable choice of generalized coordinate for the system. Only ''one'' coordinate is needed instead of two, because the position of the bead can be parameterized by one number, {{mvar|s}}, and the constraint equation connects the two coordinates {{mvar|x}} and {{mvar|y}}; either one is determined from the other. The constraint force is the reaction force the wire exerts on the bead to keep it on the wire, and the non-constraint applied force is gravity acting on the bead.

Suppose the wire changes its shape with time, by flexing. Then the constraint equation and position of the particle are respectively

:<math>f(\mathbf{r}, t) = 0 \,,\quad \mathbf{r} = (x(s, t), y(s, t))</math>

which now both depend on time {{mvar|t}} due to the changing coordinates as the wire changes its shape. Notice time appears implicitly via the coordinates ''and'' explicitly in the constraint equations.