My(?) Three body theorem.

davieddy · 2011-08-07, 06:51

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

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

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

2) If they are given initial velocities $\omega$ x r_i,
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

davieddy · 2011-08-08, 03:14

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

Now acceleration = -GMr_i/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

2011-08-07, 06:51	#1
davieddy "Lucan" Dec 2006 England 2·3·13·83 Posts	My(?) Three body theorem. Three bodies are all distance a apart, the ith located at r_i relative to the centre of mass. The sum of their (different) masses is M. 1) The acceleration of the ith body is -GMr_i/a³ $\omega$ is a vector perpendicular to the plane in which the bodies lie. 2) If they are given initial velocities $\omega$ x r_i, 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

2011-08-08, 03:14	#2
davieddy "Lucan" Dec 2006 England 6474₁₀ Posts	Simple yes. Trivial no. Note that imparting the initial velocities ("Rigid Body" rotation) adds 0 to the momentum of the system. Now acceleration = -GMr_i/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

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Four chador theorem	axn	Soap Box	19	2009-07-01 05:24
Body Language	Orgasmic Troll	Lounge	2	2005-11-29 16:52
Lagrange's Theorem	Numbers	Math	2	2005-09-07 15:22
failing all prime tests, can any body help this nooby	madbullet	Hardware	5	2005-06-12 03:10
any body play with the soft fft crossover yes?	crash893	Software	9	2002-09-18 20:45