An Approach to the Three Body Problem

by Frank C. Fung ( 297 pages ) - 1st published in September / 1996

THE PROBLEM :

The Three Body Problem is simply trying to predict the positions of three (3) mass's moving in accordance to the Laws of Gravitation.

However, the size of the mass's may vary and the initial conditions may also be quite different.

At first glance, this would involve a { 2nd order differential equation } involving nine (9) variables.


SUMMARY OF THE APPROACH : ( this version date November, 2003 )

FIRST, we set up the [ Mass Mapping Matrix ] :

We set up the Position Vectors:

{ R1 } = { X1 , Y1 , Z1 } ; { R2 } = { X2 , Y2 , Z2 } ; { R3 } = { X3 , Y3 , Z3 }

And we set up the { MR } Vectors :

{ MR1 } = { M1*X1 , M1*Y1 , M1*Z1 }

{ MR2 } = { M2*X2 , M2*Y2 , M2*Z2 }

{ MR3 } = { M3*X3 , M3*Y3 , M3*Z3 }

We note that the positions of the mass's are defined by the { Position Vectors } but the forces are related to the 2nd derivatives of the { M*R Vectors }.

We then set up the [ Standardized Mass Ratios ] via the quantity [ MT ], defined by the relation:

[ MT^2 ] = [ M1^2 ] + [ M2^2 ] + [ M3^2 ]

And we set up :

[ m1 ] = [ M1 / MT ] ; [ m2 ] = [ M2 / MT ] ; [ m3 ] = [ M3 / MT ]

so that :

{ m1^2 } + { m2^2 } + { m3^2 } = 1

We then set up the 9-dimension { RT } & { MRT } Vectors, as follows :

{ RT } = { X1 , Y1 , Z1 , X2 , Y2 , Z2 , X3 , Y3 , Z3 }

{ MRT } = { m1*X1 , m1*Y1 , m1*Z1 , m2*X2 , m2*Y2 , m2*Z2 , m3*X3 , m3*Y3 , m3*Z3 }

And we adjust the Gravitation Constant accordingly to account for the new mass-unit for [ MT ] , [ m1 ] , [ m2 ] & [ m3 ].

Once we've picked a set of values for { m1 , m2 , m3 } , the { Mapping Matrix/function } for the conversion between the { R } & { M*R } Vectors became fixed and we can readily covert between { R } & { M*R }.

Thus, the [ Position / Force ] relations can be readily assessed.


SECONDLY , we noticed that the three (3) mass's forms a triangle.

If we let this triangle vary in size while keeping the shape of the triangle constant, this is a problem we can readily resolve using analysis for Gyroscopic motions/an un-even gyroscope.

We were able to identify that, once we set up the 'Principal Directions' for the [ Three-Mass's System ] :

The ratio of the Moments-of-Inertia [ { I-XX } vs. { I-YY } ] is controlling on the motions of the [ Three-Mass's System ].

And this ratio [ { I-XX } vs. { I-YY } ] is totally dependent on the shape of the triangle.

SINCE the Shape of the Triangle has only two (2) 'Degrees-Of-Freedom', the problem is reduced to :

A { second-order differntial equation } involving only two (2) variables.

Hopefully, with today's computer technologies, further insights may be gained here.

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