I'm reading over my lectures notes on rigid body dynamics and I'm trying to prove as many results as possible. However I can't seem to prove that $\tau = \frac{dL}{dt}$ for a rigid body, where $\tau = r \times F$ is the torque and $L = m\cdot r\times v$ is the angular momentum.

I could already prove that that's the case for a point-like object ( this can be done deriving the definition of $L$ and using the fact that $\frac{dr}{dt} \times v = 0$). However none of my (many) attempts was a success when considering a rigid body. What's more I couldn't find any proof of it on internet depsite a thorough search.

Here is what I tried :

For a rigid body, we have the following : $L = \int dL = \int r\times v \cdot dm$

Thus : $\frac{dL}{dt} = \int \frac{d(dL)}{dt} = \int r\times dF$ where $dF$ is the force applied to the tiny mass $dm$.

However I can't find any way of calculating the force $dF$ knowing the total force $F$, its point of application and all the caracteristics of the rigid body. To be honest I'm not much convinced by this approach but it's the only one that got me to the beginning of somewhere (although I imagine there must be some elegant way of proving this).

Thanks for any hint or full answer !

By the way I'm french, hope it might explain any english mistake : )