1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Radial Equation for Two-Body Central Forces

  1. Oct 30, 2009 #1
    I'm getting two different radial equations depending on when I plug in the angular momentum piece. Here's the Lagrangian:

    [tex]L = \frac{1}{2} \mu (\dot{r}^2 + r^2 \dot{\phi}^2) - U(r)[/tex]

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

    [tex] \dot{\phi} = \frac{l}{\mu r^2} [/tex]

    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.

    [tex]L = \frac{1}{2} \mu \dot{r}^2 + \frac{l^2}{2 \mu r^2} - U(r)[/tex]

    [tex] \mu \ddot{r} = \frac{-l^2}{\mu r^3} - \frac{dU}{dr} [/tex]

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

    [tex] \mu \ddot{r} = \mu r \dot{\phi}^2 - \frac{dU}{dr} [/tex]

    [tex] \mu \ddot{r} = \frac{l^2}{\mu r^3} - \frac{dU}{dr} [/tex]

    These two radial equations have opposite signs for the "centrifugal term." Which is correct, and why is the other wrong?
  2. jcsd
  3. Nov 1, 2009 #2
    Remember that when you solved for [tex] \dot{\phi} [/tex], you held r constant. Similarly, [tex] \phi [/tex] should be held constant when you solve for the radial equation.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook