Radial Equation for Two-Body Central Forces

[Proofrific]

I'm getting two different radial equations depending on when I plug in the angular momentum piece. Here's the Lagrangian:

$$L = \frac{1}{2} \mu (\dot{r}^2 + r^2 \dot{\phi}^2) - U(r)$$

The Euler-Lagrange equation for phi gives angular momentum (conserved), which can be solved for $$\dot{\phi}$$:

$$\dot{\phi} = \frac{l}{\mu r^2}$$

Now, let's find the radial equation (that is, the Euler-Lagrange equation for r):

Method 1: Substitute angular momentum piece into Lagrangian, then find the Euler-Lagrange equation for r.

$$L = \frac{1}{2} \mu \dot{r}^2 + \frac{l^2}{2 \mu r^2} - U(r)$$

$$\mu \ddot{r} = \frac{-l^2}{\mu r^3} - \frac{dU}{dr}$$

Method 2: Find the Euler-Lagrange equation for r, then substitute angular momentum piece into the radial equation.

$$\mu \ddot{r} = \mu r \dot{\phi}^2 - \frac{dU}{dr}$$

$$\mu \ddot{r} = \frac{l^2}{\mu r^3} - \frac{dU}{dr}$$

These two radial equations have opposite signs for the "centrifugal term." Which is correct, and why is the other wrong?

 

[badphysicist]

Remember that when you solved for $$\dot{\phi}$$, you held r constant. Similarly, $$\phi$$ should be held constant when you solve for the radial equation.