Something about configuration manifolds in classical mechanics

[wrobel]

I think it could be interesting.

Consider a mechanical system

A circle of mass M can rotate about the vertical axis. The angle of rotation is coordinated by the angle $\psi$. A bead of mass m>0 can slide along this circle. The position of the bead relative the circle is given by the angle $\theta$.

It is interesting to note that if M>0 the Lagrangian of this system is defined on the tangent bundle $T\mathbb{T}^2$. But if M=0 then the Lagrangian is defined on the tangent bundle $TS^2$.

 

[aclaret]

how is define the set $\mathbb{T}$?

 

[wrobel]

$\mathbb{T}^n=S^1\times\ldots\times S^1,\quad \mathbb{T}^1=S^1$
but
$S^1\times S^1\ne S^2$!