Find the Lagrangian of this system The particle $m_2$ moves on a vertical axis and the whole system rotates about this axis with a constant angular velocity $\Omega$. Find the Lagrangian of this system. 
(Note that there are two $m_1$ in this system)
The solution provided by author is different from the conventional method:
Let the angular displacement be $\phi$, such that $\dot{\phi} = \Omega$
For the particles $m_1$, we have small displacement: $dl_1^2 = \sigma^2 d\theta^2 + \sigma^2sin^2\theta d\phi^2$
then we have $v_1 = (\frac{dl_1}{dt})^2$; therefore determine the kinetic energy of two $m_1$ particles.
Why this is the displacement of $m_1$? It seems that the author is trying to use spherical coordinate system?
 A: I am assuming by the centre you mean that at the level of the mass m₁ as that to me looks like the geometric centre.
Your co-ordinates may be workable but it will be very difficult to work with them as the centre isnt technically stationary, it is dynamic and hence non-inertial.
It will be exponentially convenient to assume the origin O of your system at the pivot like the author has done (You can chose it anywhere but this is intuivitively the most convenient point)
Now that the ambiguity of co-ordinate system has been solved lets move to your first question:
Yes the author has used spherical co-ordinates but observe that his definition of θ is actually the supplement of the convention
if you observe the diagram then from the definition of the angle
l₁=σ θ  ---->
dl₁=σ dθ
By carefull observation we can also find that (in case the circle isnt directly visible imagine yourself looking at it from the top)
l₂ =  σ sinθ φ------>
dl₂ = σ sinθ dφ (dont apply chain rule as you are not concerned with
that variation)
the infinitesimal displacement vector
dl =   σ dθ θ̂̂  +   σ sinθ dφ φ̂
Now simply dot this with itself
dl² = σ² dθ²  +     σ² sin²θ dφ²
Please let me know if there is any ambiguity
