Central Force Problem
Central Force Problem
Central Force Problem
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Qualitative Behavior<br />
• Integrating the radial motion<br />
isn’t always easy<br />
• More often impossible…<br />
• You can still tell general behavior by looking at<br />
E<br />
V′ () r ≡ V()<br />
r +<br />
2<br />
l<br />
2mr<br />
2<br />
• Energy E is conserved, and E – V’ must be positive<br />
2<br />
mr<br />
= + V′<br />
() r<br />
2<br />
mr<br />
2<br />
• Plot V’(r) and see how it intersects with E<br />
2<br />
2<br />
2 ⎛<br />
l<br />
r<br />
= ⎜E−V()<br />
r −<br />
m⎝<br />
2mr<br />
Quasi potential including<br />
the centrifugal force<br />
= E− V′<br />
() r > 0<br />
E<br />
2<br />
> V′<br />
() r<br />
⎞<br />
⎟<br />
⎠
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