Question

# A particle is confined to move on the surface of a circular cone with its axis on the z axis, vertex at the origin (pointing down), and half-angle α. The particle's position can be specified by two generalized coordinates, which you can choose to be the coordinates (ρ, φ) of cylindrical polar coordinates. Write down the equations that give the three Cartesian coordinates of the particle in terms of the generalized coordinates (ρ, φ) and vice versa.

Solution

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Relation of angle $\alpha$, $z$ and $\rho$ can be deduced from figure below and is given by:

\begin{align} \tan\alpha &= \frac{\rho}{z} \\ z &= \rho \cot\alpha \end{align}

We can now express Cartesian coordinates $(x,y,z)$ in terms of generalized coordinates $(\rho, \phi)$.

\begin{align} x &= \rho \cos\phi \\ y &= \rho \sin\phi \\ z &= \rho \cot\alpha \end{align}

We can easily inverty (3), (4) and (5) to express generalized coordinates in terms of Cartesian coordinates.

\begin{align} \rho &= z\tan\alpha \\ \phi &= \arctan(y / x ) \\ z &= z \end{align}

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