I What is the definition of moment M_z in Arnold's book on classical mechanics?

[Yingnan Xu]

Hi guys, so in Arnold's mathematical methods of classical mechanics p43, he defined the moment M_z, or L_z, the angular momentum, relative to the z axis of vector F applied at the point r is the projection onto the z axis of the moment of the vector F relative to some point on this axis, M_z=(e_z,[r,F]), where this square bracket is cross product. I think this should be the time derivative of M_z otherwise the unit is not consistent. Or he actually means the moment of M_z is the torque?

 

[BvU]

Hello Yingnan, $\quad$ $\quad$ !

I don't have this particular book, but yes: he means torque. Alias moment, with dimension
mass $\times$length²/ time²

your

M_z=(e_z,[r,F])

notation is unfortunate; better write $$M_z = (\vec r\times\vec F)_z {\text { or, better }}\vec M = \vec r\times\vec F $$

 

[Andy Resnick]

Hi guys, so in Arnold's mathematical methods of classical mechanics p43, he defined the moment M_z, or L_z, the angular momentum, relative to the z axis of vector F applied at the point r is the projection onto the z axis of the moment of the vector F relative to some point on this axis, M_z=(e_z,[r,F]), where this square bracket is cross product. I think this should be the time derivative of M_z otherwise the unit is not consistent. Or he actually means the moment of M_z is the torque?

Good catch- I didn't notice that one. Since he defined M = [r,dr/dt] earlier on pg 42, M_z=(e_z,[r,F]) or M = [r,F] does seem to be inconsistent with regards to the units [L] and [T] (see also the theorem proof on pg. 44).