# How are these two versions of the conservation of angular momentum different?

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.

• 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. – AndresB Jun 3 at 14:46
• @AndresB I have corrected #1. Could you write an answer about #2? – Ma Joad Jun 3 at 14:53
• 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. – AndresB Jun 3 at 15:33
• 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. – AndresB Jun 3 at 15:37