Intermediate Mechanics:Central Force Motion

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Revision as of 18:19, 4 November 2005

Second coolest equation ever (T&M 8.21):

$\frac{\partial u}{\partial \dot{\theta}} + u = -\frac{\mu r^2}{l^2}F(r)$

$u = \frac{1}{r}$

$\frac{\alpha}{r} = 1 + \epsilon \cos \theta$

$\epsilon = \sqrt{1 + \frac{2 E L^2}{m K^2}}$

@@ Line 4: / Line 4: @@
 <math>u = \frac{1}{r}</math>
+==Central Force Orbits==
+<math>
+\frac{\alpha}{r} = 1 + \epsilon \cos \theta
+</math>
+<math>
+\epsilon = \sqrt{1 + \frac{2 E L^2}{m K^2}}
+</math>
+{| border="1" cellpadding="5" cellspacing="0"
+|+'''Central Force Motion Orbits'''
+!<math>\epsilon</math> value
+!Orbit Type
+!Energy Value
+|-
+|<math>\epsilon = 0 </math>
+| Circle
+|
+|-
+|<math>0 < \epsilon < 1 </math>
+| Ellipse
+|<math>E < 0 </math>
+|-
+|<math>\epsilon = 1 </math>
+| Parabola
+|<math>E = 0 </math>
+|-
+|<math>\epsilon > 1 </math>
+| Hyperbola
+|<math>E > 0 </math>
+|}