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My question is what happens if I want to give a different meaning to ##\theta##, for example the latitude itself or even the opposite angle from the South Pole, ##\pi - \theta##.

Let ##\alpha## be such an angle ##\alpha = \pi - \theta##. Since ##\dot{\alpha}=\dot{\theta}##, is it valid to just perform the change of the kinetic energy as $$T = \dfrac{1}{2} m ( \dot{r}^2 + r^2 \dot{\alpha}^2 + r^2 \sin^2 (\pi-\alpha) \dot{\phi}^2 )\ ,$$ or do I need to rebuild the expression using the new variable from the beginning? Or maybe the same original expression still holds true?.