Angular Momentum Question: M_z from Landau-Lifgarbagez p21

[Piano man]

I've got a question about angular momentum arising from Landau-Lifgarbagez p21.

Firstly, I'm not sure where this equation comes from:

M_z=\sum_a \frac{\partial L}{\partial \dot{\phi}_a}

and from that, how do you get

M_z=\sum_a m_a(x_a \dot{y}_a-y_a \dot{x}_a)

Thanks for any help.

 

[Meir Achuz]

The partial derivative of the Lagrangian with respect to the time derivative of a space variable is the canonical momentum for that degree of freedom.
The second equation comes from a change of variables from r,phi to x,y.
You may want to study the Lagrangian in a good mechanics book.

 

[Piano man]

Ok, I still don't really follow.
When you say 'canonical momentum', is that derived from somewhere or is it empirical?
And I've been trying to change the variables to get the second equation, but I'm not getting anywhere. How does it work?
Thanks for your help.