Angular velocity calculation from Schwarzschild metric

[fourvector]

Hello,

I need to find the angular velocity using Schwarzschild metric.
At first I wrote the metric in general form and omitted the co-latitude:
ds²=T*dt²+R*dr²+Φ*dφ²

and wrote a Lagrangian over t variable:
L = √(T+R*(dr/dt)²+Φ*(dφ/dt)²)

now I can use the Euler–Lagrange equations for φ variable and note that L does not depend on φ.
dL/d(dφ/dt) = const => Φ*(dφ/dt) / L = const => dφ/dt = const * L / Φ

The result is that dφ/dt depends on dr/dt because L contains dr/dt term.

But there is one more way that I can calculate the dφ/dt.
I can write the Lagrangian over new τ variable:
L = √(T*(dt/dτ)²+R*(dr/dτ)²+Φ*(dφ/dτ)²)

I can do the same calculations for dφ/dτ and dt/dτ variables:

dL/d(dφ/dτ) = const => Φ*(dφ/dτ) / L = const => dφ/dτ = const * L / Φ
dL/d(dt/dτ) = const => T*(dt/dτ) / L = const => dt/dτ = const * L / T

Now I can divide one over another and get
dφ/dt = const * T / Φ

The result is that the angular velocity does not depend on dr/dt.

Could someone help me what is wrong with one of my calculation?

 

[mfb]

The angular velocity of what?
Please give the full and exact problem statement.