# Lagrange’s Equations for a Tetherball

I’m trying to write down the equations of motion for a tetherball moving in 3D around a pole while the string is getting shorter.

I’ve started with lagrange equations:

$x(t)=l(t) \sin (\theta) \cos (\phi)\\ y(t)=l(t) \sin (\theta) \sin (\phi)\\ z(t)=h(t)+l(t) \cos(\theta)\\ \\ T = \frac{1}{2}m(\dot x^2 +\dot y^2+\dot z^2)\\ U=m g l(t)(1-\cos(\theta)) + mg(S-(h(t)+l(t)))$

where $l(t)$ is the length of the string. Here only I am assuming the radius of the pole is really small compared to $l$. The polar angle is $\theta(t)$. $h(t)$ is the change in height due to the string wrapping on the pole. $S$ is the length of string when unwrapped.

The change in length along is given by:
$\dot l(t) = -\frac{r\dot\phi}{ \sin(\theta)}$

where $r$ is the radius of the pole. And the sliding pivot point is given by:
$\dot h(t) = \frac{r\dot\phi}{ \tan(\theta)}$

After plugging those in $T$ I apply the Lagrange derivative to $L = T-U$ and solve for $\ddot \theta$ and $\ddot \phi$

Now when I simulate the results I get a linear velocity that is some how increasing which is not supposed to happen for a tetherball because no new energy has been introduced to the system and the angular momentum is not conserved.

I would appreciate some feedback

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