L = r X p - in Sphereical Coordinates

[Oijl]

L = r X p --- in Sphereical Coordinates

Homework Statement

Express in vector form angular momentum in spherical coordinates.

$$\rho$$ is the distance from the origin to the point
$$\phi$$ is the angle made in the Cesarean x-y plane counter-clockwise from the positive x-axis
$$\theta$$ is the angle downward from the z-axis

Homework Equations

The Attempt at a Solution

Is it as simple as

r = $$\rho$$$$\hat{\rho}$$ + $$\phi$$$$\hat{\phi}$$ + $$\theta$$$$\hat{\theta}$$
and
v = $$\dot{\rho}$$$$\hat{\rho}$$ + $$\rho$$sin($$\theta$$)$$\dot{\phi}$$$$\hat{\phi}$$ + $$\rho$$$$\dot{\theta}$$$$\hat{\theta}$$
?

 

[jbsteele]

So thenL = r X p = (\rho\hat{\rho} + \phi\hat{\phi} + \theta\hat{\theta})X(\dot{\rho}\hat{\rho} + \rhosin(\theta)\dot{\phi}\hat{\phi} + \rho\dot{\theta}\hat{\theta})= \rho\dot{\phi}\sin(\theta)\hat{\rho} + \rho\dot{\theta}\hat{\phi} - \phi\dot{\rho}\hat{\theta}