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Background Information

The lagrangian of a particle in a central force field $V(r)$ is $$ L=\frac12m(\dot r^2+r^2\dot\theta^2+r^2\sin^2\theta\dot\varphi^2)-V(r). $$ The particle must move in a plane, so the coordinate system can always be chosen such that $\dot\varphi=0$. The motion is restricted to be 2D. Then $$ L=\frac12m(\dot r^2+r^2\dot\theta^2)-V(r).\tag{1} $$ The Euler-Lagrange Equation for $r$ gives $$ m(\ddot r-r\dot \theta^2)+V'(r)=0.\tag{2} $$ With some trick one derives Binet equation, but the formula itself is not the point of this post.

Canonical variable $\theta$ doesn't appear in Lagrangian (1), so the canonical momentum is conserved, too. Set it to be $$ p_\theta=mr^2\dot\theta=l=\text{const}.\tag{3} $$ The Lagrangian becomes $$ L=\frac12m\dot r^2+\frac{l}{2mr^2}-V(r). $$ The problem is now 1D. Applying Euler-Lagrange Equation again, we will get $$ m\ddot r+\frac{l}{mr^3}+V'(r)=0.\tag{4} $$

Question

Eq (2) is not consistent with Eq (4), since Eq(3) holds. What's wrong? Thanks for answering.

Qmechanic
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Luessiaw
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    You cannot use a conservation law in the action before deriving the equations of motion, since $l$ itself depends on the initial conditions. – Hossein Sep 12 '22 at 13:38
  • @Hossein but in Eq(1) I used $p_\varphi＝0$, otherwise an extral item will go into Eq(2), and $p_\theta$ is not conserved. Is this process legal? – Luessiaw Sep 12 '22 at 13:45
  • The conserved momentum $p_\theta$ can be regarded as a constraint on the particle. So the system has only one free variable, which can be chosen to be $r$. – Luessiaw Sep 12 '22 at 13:49
  • For $(\varphi,p_{\varphi})$ the coordinate is constant and the momentum is zero. For $(\theta,p_{\theta})$ neither the coordinate $\theta$ and the momentum $p_\theta$ are not zero. In fact, for the special case of $p_{\theta}=0$ where the motion is actually along a line, you can substitute in the Lagrangian and you will get the correct equations of motion. – Hossein Sep 12 '22 at 13:57
  • @Hossein You are right, thank you. – Luessiaw Sep 12 '22 at 14:08
  • I asked a very similar question several years ago, see here and the very illuminating answer: https://physics.stackexchange.com/q/190602/ – psm Sep 12 '22 at 14:26
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    Related 1, Related 2. and links therein. – Mr. Feynman Sep 12 '22 at 15:30

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