Undergrad Changing spherical coordinates in a Lagrangian

[Jaime_mc2]

In order to compute de lagrangian in spherical coordinates, one usually writes the following expression for the kinetic energy: $$T = \dfrac{1}{2} m ( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2 )\ ,$$ where $\theta$ is the colatitud or polar angle and $\phi$ is the azimuthal angle.

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?.

 

[kuruman]

Well, the trig identity says $\sin(\pi-\alpha)=\sin\pi\cos\alpha-\cos\pi\sin\alpha=\sin\alpha$. So your kinetic energy would be $T = \dfrac{1}{2} m ( \dot{r}^2 + r^2 \dot{\alpha}^2 + r^2 \sin^2 (\alpha) \dot{\phi}^2 )$. What's the difference? Answer: Just a name change.