Explain the proof in goldstein

[pardesi]

Question:
Analyze the motion of a small bead attached to a wire which is rotating along a fixed axis?

Proof(Using Lagrangian formulation):
Clearly here the generalized coordinate is the distance of the particle along the wire.
so we have the formulae
$$\frac{d \frac{\delta T}{\delta r}}{dt} - \frac{\delta T}{\delta r}=Q$$
where $$Q$$ is the generalized force acting on the object ...
goldstein claims that is 0 here i don't get that how?

 

[marcusl]

First, correcting the typos, Lagrange's equation is

$$\frac{d}{dt} \frac{\partial L}{\partial \dot{r}} - \frac{\partial L}{\partial r}=Q$$

There is no potential so L=T in this example as you have stated. In the method described on p. 26, the generalized force Q is zero because the constraint of moving along the wire is built into the generalized coordinates instead, and there are no non-conservative forces (i.e., friction).