Acceleration only due to conservation of angular momentum

[Soren4]

I don't understand why the conservation of angular momentum can imply an acceleration, in absence of a force.

Consider for istance planetary motion. The angular momentum $\vec{L}$ of the planets is conserved and that means $\mid \vec{L} \mid=mr^2 \dot{\theta}=mrv_{\theta}$ is conserved too.

Consider the acceleration in polar coordinates
$$
\left( \ddot r - r\dot\theta^2 \right) \hat{\mathbf r} + \left( r\ddot\theta+ 2\dot r \dot\theta\right) \hat{\boldsymbol{\theta}} \ $$

The second term is zero since $\vec{L}$ is constant. In fact the second term can be rewritten as $a_{\theta}=\frac{1}{r}[\frac{d}{dt}(r^2 \dot{\theta})]=\frac{1}{r}[\frac{d}{dt}(\frac{L}{m})]$.
This means that there is no acceleration in the direction of $\hat{\boldsymbol{\theta}}$, which is clear since the gravitational force is a central froce.

But if the distance $r$ decreases $v_{\theta}$ (i.e. the velocity in the direction of $\hat{\boldsymbol{\theta}}$) must increase in order to keep $\mid\vec{L} \mid$ constant.

How can $v_{\theta}$ increase if there is no acceleration in the direction of $\hat{\boldsymbol{\theta}}$?

I understood that it happens because of the conservation of angular momentum but if there is an acceleration, necessarily a force is needed. I don't see where do this force come from.

 

[A.T.]

Speed changes because the force is not perpedicular to velocity, when the distance changess.