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Here are two versions of the conservation of angular momentum.

  1. The total angular momentum is constant if there is no external moment on the system
  2. The total angular momentum of a particle is constant if it is only under the influence of a conservative force with the potential function $V (\mathbf x)$ invariant under rotations. (i.e. $dV/d\theta=0$ in the case of two-dimensional space)

I am perplexed about how different those statements are. Of course "no external force" does not mean all forces are conservative. Are those two versions of angular momentum conservation related, or are they just two independent, unrelated statements?

Also please point out any inaccurate statements there.

What I have noticed is that both statements imply Kepler's 2nd law.

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  • $\begingroup$ Condition #1 is not the condition of conservation of angular momentum. #2 only applies if there is a Lagrangian associated to the dynamics of the system. $\endgroup$ – AndresB Jun 3 at 14:46
  • $\begingroup$ @AndresB I have corrected #1. Could you write an answer about #2? $\endgroup$ – Ma Joad Jun 3 at 14:53
  • $\begingroup$ I asked a related question not long ago, it was for conservation of linear momentum but it works the same physics.stackexchange.com/q/469471 , check Elio Fabri reply. $\endgroup$ – AndresB Jun 3 at 15:33
  • $\begingroup$ Either way, if (1) the system do allow a Lagrangian formulation and (b) the Lagrangian is invariant under raotations (or change just by a total derivative) then there is a quantity conserved and that quantity is the total angular momentum. $\endgroup$ – AndresB Jun 3 at 15:37
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These two statements refer to two different entities. Statement # 1 applies to a "system" which can be a complex entity of significant spatial extent. Statement # 2 applies to a "particle" which usually means a point particle or a system that can be approximated by a point for the purposes of the problem.

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  • $\begingroup$ So why both of them can be used to derive the same Kepler's 2nd law? $\endgroup$ – Ma Joad Jun 3 at 14:49
  • $\begingroup$ Statement # 1 can't be used to derive Kepler's laws because there is an external force (gravity) in the Kepler problem. $\endgroup$ – Lewis Miller Jun 3 at 14:56
  • $\begingroup$ But if you consider both the planet and the sun as one system, then there is no external force or torque. $\endgroup$ – Ma Joad Jun 3 at 22:09
  • $\begingroup$ But you referred to a particle in a central potential, not a two body system. $\endgroup$ – Lewis Miller Jun 3 at 23:16
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For an otherwise isolated system the total angular momentum is conserved. As to the a gular momentum if the particle statement 1 is correct but trivial. Statement 2 is only true for a spherical particle, or for a non spherical one in an orientation where the torque happens to be zero. It is then a special case of statement 1. Kepler's second law states conservation of momentum, hence the connection.

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