Theoretical Mechanics: Fall-2021
Curvilinear Coordinates Practice: Due Day 16

1. Speed Squared in Curvilinear Coordinates

   Show that the speed squared written in different standard coordinates systems is:

   1. in rectangular coordinates: \(v^2 = \dot{x}^2 + \dot{y}^2+\dot{z}^2\)
   2. in cylindrical coordinates: \(v^2 = \dot{s}^2 + s^2\dot{\phi}^2+\dot{z}^2\)
   3. in spherical coordinates: \(v^2 = \dot{r}^2 + r^2\dot{\theta}^2+r^2\sin^2\theta\dot{\phi}^2\)

   where:

   
   

   The algebra for these calculations is a little tedius, but it's good to do them once in your career to convince yourself that these expressions are true!

   See: GMM: Curvilinear Coordinates for more discussion of curvilinear coordinates. Note that the labels for the aximuthal and polar coordinates in different for math and physics (\(\phi\) and \(\theta\) are swapped.)