kinematics in spherical coordinates

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It is sometimes convenient to express the position and velocity vectors of a particle in terms of the spherical coordinates r\,,\theta\,, and \phi\,. The Cartesian components of the unit basis vectors \hat\mathbf{r}\,, \hat\boldsymbol{\theta}\, and \hat\boldsymbol{\phi}\, depend on position, and care must be taken when evaluating their time derivatives. Now

\begin{matrix} \mathbf{\hat r} & =& \sin\theta\cos\phi \hat\mathbf{x} + \sin\theta\sin\phi \hat\mathbf{y} + \cos\theta\hat\mathbf{z}\\ \boldsymbol{\hat\theta} & = &\cos\theta\cos\phi \hat\mathbf{x} + \cos\theta\sin\phi \hat\mathbf{y} - \sin\theta\hat\mathbf{z}\\ \boldsymbol{\hat\phi} & =& -\sin\phi \hat\mathbf{x} + \cos\phi \hat\mathbf{y} \end{matrix}

and with some calculation,

\frac{d\mathbf{\hat r}}{dt}\, = \dot\theta \boldsymbol\hat\theta + \dot\phi\sin\theta \hat\boldsymbol{\phi}\,,
\frac{d\boldsymbol{\hat \theta}}{dt}\, = -\dot\theta \hat\mathbf{r} + \dot\phi \cos\theta \hat\boldsymbol{\phi}\,,
\frac{d\boldsymbol{\hat \phi}}{dt}\, = -\dot\phi\sin\theta \hat\mathbf{r} - \dot\phi \cos\theta \hat\boldsymbol{\theta}\,.

Velocity

The time derivative of \mathbf{r} = r \hat\mathbf{r}\, is simply

\mathbf{v} = \dot r \hat\mathbf{r} + r \dot\hat\mathbf{r} = \dot r \hat\mathbf{r}+r\dot\theta \boldsymbol\hat\theta + r\dot\phi\sin\theta \hat\boldsymbol{\phi}\,.

Angular velocity

The angular velocity is

\boldsymbol{\omega} = \hat{\mathbf{r}} \times \frac{\mathbf{v}}{r} = \dot\theta \hat\boldsymbol{\phi} - \dot\phi \sin\theta \hat\boldsymbol\theta\,.

Acceleration

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