Today during my classical mechanics class, I got into a discussion with my professor about a certain asymmetry of spherical coordinates. Expressing the square magnitude of velocity in a spherical coordinate system yields $$ \dot{r}^2 + r^2 \left( \dot\theta^2 + \dot\phi^2 \sin^2{\theta} \right), $$ where $\theta$ denotes the angle from the $z$ axis and $\phi$ from the $x$ axis.
My question is why there is an explicit $\theta$ dependence in the expression, considering the choice of coordinates is spherically symmetric. (More accurately, my question is why the Lagrangian for a point mass in a central field has an explicit dependence on $\theta$. My instinct was that $\theta$ should be a cyclic coordinate due to spherical symmetry.)
We agreed that such dependence arose from the essence of spherical coordinates, specifically the different natures of the $\theta$ and $\phi$ coordinates. And we have enumerated some other consequences of this:
- $\theta$-translation is not a well-defined rotation in a specific sense. While $\phi$-translation is always a rotation about the $z$ axis, there is not a single axis about which all rotations occur due to $\theta$-translation.
- $\theta$-translation is not one-to-one. A point on the positive $z$ axis is mapped to many different points while having no point associated to it, and vice versa for the negative $z$ axis.
- While a consistent transformation can be given such that only the $\phi$ coordinate of a point is translated (rotation about the $z$ axis), the same cannot be said for $\theta$.
What we could not conclude on was how to formally state why such things happen. We have dismissed the explanation that quantifying $\theta$ requires a certain $z$ axis, as we also need to specify an $x$ axis to quantify the $\phi$ coordinate. I tried to look for concepts like some (a)symmetries of a curvilinear coordinate, but to no avail. Is there a concept related to the phenomena I described? Thanks.