Angular Momentum Question: M_z from Landau-Lifgarbagez p21

AI Thread Summary
The discussion centers on the derivation of the angular momentum equation M_z from Landau-Lifgarbagez, specifically the transition from M_z = ∑(∂L/∂dot(φ_a)) to M_z = ∑(m_a(x_a dot(y_a) - y_a dot(x_a))). The canonical momentum is defined as the partial derivative of the Lagrangian with respect to the time derivative of a generalized coordinate. The second equation is derived through a change of variables from polar coordinates (r, φ) to Cartesian coordinates (x, y). Participants express confusion about the derivation process and seek clarification on the concepts involved. Understanding these principles may require further study of Lagrangian mechanics.
Piano man
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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.
 
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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.
 
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.
 

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