effective potential

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A particle under the influence of a central force f(r)\, is described by the Lagrangian

L = \frac{1}{2}m \dot{r}^2 + \frac{1}{2}m r^2 \dot\phi^2 - V(r)\,,

where

f(r) = -\frac{\partial V}{\partial r}\,.

The angular momentum

l = m r^2 \dot\phi\,

is conserved, and the Euler-Lagrange equation for r\, is

m \ddot{r} - m r \dot\phi^2 + \frac{\partial V}{\partial r} = 0\,,

which can be written as

m \ddot{r}\, = \frac{l^2}{m r^3} - \frac{\partial V}{\partial r}\,,
= -\frac{\partial}{\partial r} \left( \frac{l^2}{2m r^2} \right) - \frac{\partial V}{\partial r}\,,
= -\frac{\partial}{\partial r} \left( V(r) + \frac{l^2}{2m r^2}\right)\,,

and so the radial motion of the particle can be described as that of a particle in one dimension under the influence of an effective potential, given by

V_{eff}(r) = V(r) + \frac{l^2}{2m r^2}\,.


back to reduction from two-body to one-body problem
back to Central force motion
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