Proof of angular momentum conservation

[KT KIM]

This is from text [Introduction to Lagrangian and Hamiltonian Mechanics] on NTNU opencourse.
Annnnd... I don't use english as my primary language, so sorry for poor sentences.

I can't get two things in here.

First, at (1.12) I can't understand how L dot derivated like that.
Since I know differentiation of cross product should be done like

d/dt(AxB)=d/dt(A) x B + A x d/dt(B)

then, at (1.12), why it doesn't have the terms of d/dt(r) x p ?
I think it only has the terms of r x d/dt(p)

Second, I can't get how

were derived by using (1.13), How could ri X Fji = 1/2(rij X Fji)
is possible?

These might be dumb questions, but please help me.

 

[wrobel]

en, at (1.12), why it doesn't have the terms of d/dt(r) x p ?

##\boldsymbol p_i=m_i\dot{\boldsymbol r}_i,\quad \dot{\boldsymbol r}_i\times\dot{\boldsymbol r}_i=0##
by the way ##\sum_{ij}{\boldsymbol r}_{i}\times\boldsymbol F_{ji}=0##

 

[KT KIM]

Thank you for clear explanation. Got the first one.

Yes Σij rij x Fji = 0 so eventually it makes L dot = tau (torque)
But, what I want know is the mathematical manuever that makes ri X Fji = 1/2(rij X Fji)

 

[wrobel]

do the calculation for two particles directly and everything will be clear