Significance of centrifugal potential

Question

While dealing with central forces (purely using newtonian mechanics) I've came across this result: $$U_\text{eff}(r)=\frac{l^2}{2\mu r^2}+ U(r) \, .$$ I'm not at all fluent with the lagrangian formalism often used to deduce such results. I just want to know why we are getting the centrifugal potential term $l^2/(2\mu r^2)$. What does it stands for? I suspect that the term has something to do with the motion that the planets go through.

Answer 1

This term is used to basically understand the motion in two dimensions $(r,\theta)$ in terms of an effective motion in just one dimension $(r)$. The extra term just encodes the effect of the angular aspect of the orbital motion.

You can write this because angular momentum $l$ is conserved, and you can basically trade angular velocity $\dot{\theta}$ for the radial distance $r$, using $$l=mr^2\dot{\theta}$$

This allows us to write the rotational kinetic energy as, $$K_R=\frac{1}{2}mr^2\dot{\theta}^2=\frac{l^2}{2mr^2}$$ which acts as an effective, extra, potential term.

Now, you can only concentrate on how $r$ changes as the planet moves around in orbit. E.g., How the distance between the sun and the earth changes.