Central Force Problem
Central Force Problem
Central Force Problem
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Energy Conservation<br />
1<br />
E = T + V = m r + r + V r = m r + l + V r =<br />
2 2 2 mr<br />
2<br />
2 2 2 2<br />
( θ ) ( ) <br />
( ) const<br />
2<br />
2<br />
2 ⎛<br />
l<br />
r<br />
= ⎜E−V()<br />
r −<br />
m⎝<br />
2mr<br />
• One can solve this (in principle) by<br />
t r dr<br />
t = ∫ dt = ∫<br />
= t()<br />
r<br />
0 r0<br />
2<br />
2 ⎛<br />
l ⎞<br />
⎜E−V()<br />
r −<br />
2 ⎟<br />
m⎝<br />
2mr<br />
⎠<br />
• Then invert t(r) r(t)<br />
• Then calculate θ(t) by integrating<br />
2<br />
⎞<br />
⎟<br />
⎠<br />
1 st order differential<br />
equation of r<br />
θ =<br />
l<br />
mr<br />
2<br />
NB: This never<br />
goes negative<br />
Done! (?)
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