I am confused as to how to reconcile the conservation laws of linear and angular momentum. Are they independent (and how can this be, if $L = r$ x $p$?) Does one supersede the other? In short: how are they related?

Moreover, the notion of conserving angular momentum seems a bit arbitrary to me, for $L$ itself seems to 'appear' upon us specifying a reference point. In other words, if I have a body moving in a straight line, then it has linear momentum, for it has both mass and velocity... this is a given. But, it will not have angular momentum, lest I specify a reference point. This seems a bit at odds with the theory that angular momentum cannot be created nor destroyed.

Lastly, if I wish to quantify the total momentum of a system $S$, how can this be done (maybe, $S$ = $L_{body}$ + $p_{body}$)? And will this value not change in accordance as to whether I have specified a reference point (paragraph 2)?

  • $\begingroup$ note $\vec L + \vec p \rightarrow \vec L - \vec p$ under a parity transformation, so that would cause more problems than it would solve. $\endgroup$
    – JEB
    Feb 5 '19 at 2:03

Linear momentum and angular momentum are independent conserved quantities for a closed system. The fact that $\mathbf{p}$ happens to appear in the definition of $\mathbf{L}$ does not matter.

For non-closed systems, you might have linear momentum conserved but angular momentum not conserved, or vice versa, or neither conserved, or both. Try to think of such examples with external forces and torques, and you will be convinced that linear momentum and angular momentum are indeed independent.

Angular momentum depends on the choice of origin. This is no more strange than linear momentum (and angular momentum) differing for observers in relative motion. The value of many physical quantities depends on the reference frame in which they are measured. Two inertial reference frames can differ by origin, 3D orientation, and relative motion.

The fact that these quantities take different values in different reference frames (i.e., they are relative, not absolute) has nothing to do with whether they are conserved (i.e., constant in time) in each reference frame.

You cannot add $\mathbf{p}$ and $\mathbf{L}$ together; they don’t even have the same units!

Later, in more advanced courses, you may learn that linear momentum and angular momentum are conserved because of two entirely different symmetries of the laws of physics. Linear momentum is conserved because the laws have the same form when the origin of the reference frame is changed. Angular momentum is conserved because the laws have the same form when the 3D orientation of the reference frame is changed. Sometimes people say “linear momentum is conserved because space is homogeneous” and “angular momentum is conserved because space is isotropic”.


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