Static and Dynamic Lever
Static lever:
Archimedes' law of lever:
|F[1]||D[1]|=|F[2]||D[2]|
for some scalars a and b:
|F[1]|/|F[2]|=|D[2]|/|D[1]|=-b/a
+-
| a|F[1]|+b|F[2]|=0
<
| b|D[1]|+a|D[2]|=0
+-
Matrix form of Static lever:
+- -+ +- -+ +- -+
| F | | F[1] F[2] | | a |
| | = | | x | |
| D | | D[2] D[1] | | b |
+- -+ +- -+ +- -+
Dynamic lever
extremes: |F[1,1]||D[1,1]|=|F[1,2]||D[1,2]|
new values: |F[2,1]||D[2,1]|=|F[2,2]||D[2,2]|
Neutral interaction (equatorial trajectories)
+- -+ +- -+ +- -+
| F[2,1] F[2,2] | | cos(a) sin(-a) | | F[1,1] F[1,2] |
| | = | | x | |
| D[2,2] D[2,1] | | sin(a) cos(-a) | | D[1,2] D[1,1] |
+- -+ +- -+ +- -+
Attraction towards infinity (the flower below)
+- -+ +- -+
| cos(a) sin(a) | | F[1,1] F[1,2] |
| | x | |
+- -+ | sin(a) cos(a) | | D[1,2] D[1,1] |
| F[2,1] F[2,2] | +- -+ +- -+
| | = --------------------------------------
| D[2,2] D[2,1] | cos(2a)
+- -+
Attraction towards zero (the flower above)
+- -+ +- -+ +- -+
| F[2,1] F[2,2] | | cos(a) sin(a) | | F[1,1] F[1,2] |
| | = cos(2a)| | x | |
| D[2,2] D[2,1] | | sin(a) cos(a) | | D[1,2] D[1,1] |
+- -+ +- -+ +- -+
.
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