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Old 2011-08-07, 06:51   #1
davieddy
 
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"Lucan"
Dec 2006
England

2·3·13·83 Posts
Default My(?) Three body theorem.

Three bodies are all distance a apart, the ith located at ri relative to the centre of mass. The sum of their (different) masses is M.

1) The acceleration of the ith body is -GMri/a3

\omega is a vector perpendicular to the plane in which
the bodies lie.

2) If they are given initial velocities \omega x ri,
they will follow conic section orbits, focus at the centre of mass, and remain equidistant from each other.

Proof (or disproof) left to the reader.
Alternatively, read my posts.

How could Euler (or even Newton) have failed to spot such a simple and
elegant result, and hence the existence the L4 and L5 Lagrangian points?

David

Last fiddled with by davieddy on 2011-08-07 at 07:09
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Old 2011-08-08, 03:14   #2
davieddy
 
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"Lucan"
Dec 2006
England

2×3×13×83 Posts
Default Simple yes. Trivial no.

Note that imparting the initial velocities ("Rigid Body" rotation) adds 0
to the momentum of the system.

Now acceleration = -GMri/a3 also
applies to 4 equidistant masses (regular tetrahedron).

If they were released from rest they would collide simultaneously
at the Centre of Mass: When?

Could we give them initial velocities to maintain equal distance
from each other while doing something more entertaining?

David

Last fiddled with by davieddy on 2011-08-08 at 03:17
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