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I was studying the two body problem and came across the following statement, the two body system is a planar system. As,

$$L=\frac {\mu}{2} \left(\dot{r}^{ 2 }+r^2\dot\theta^2+r^2\sin^2\theta\dot{\phi}^2\right) -U(r)$$ $\phi$ is cyclic $\Rightarrow\,p_{\phi}=\text{constant}$ (i.e $\hat{k} \cdot\vec{l} =\text{constant}$).

Now, the explanation continues saying that $\vec{l}$, the total angular momentum is constant so the motion must be planar. But my doubt is, why should this be so? The symmetry of the Lagrangian just tells that the component of total angular momentum along the $z$ axis is constant.

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Right, but the Lagrangian $L$ is invariant under all 3D rotations, not just rotations around the $z$-axis. Phrased differently: Spherical coordinates could be generated relative to any direction, not necessarily the $z$-axis.

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The lagrangian tells so only after you have chosen the total angular momentum along the z-axis.

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  • $\begingroup$ Do you mean the z axis? $\endgroup$ – Abhikumbale Oct 23 '17 at 9:10
  • $\begingroup$ @Abhikumbale I mean the z-axis $\endgroup$ – c.p. Oct 23 '17 at 21:20

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