An Approach to the Four Body Problem

by Frank C. Fung - 1st published in December, 2007.

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Epilog II - On the { [ Zero ] Angular Momentum Vector }

Brief Introduction :

• In this 2nd Epilog , we shall be looking at several interesting properties associated with a { [ Zero ] Angular Momentum Vector } ,
  • but our conecntration here will be primarily on the { [ zero ] Cross-Porduct } of particular { Tetrahedra } .

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An Example of a { [ Zero ] Cross-Product } :

• Let us now bring in { Tetrahedron A } and { Tetrahedron C } , whose { Cross-Product [ A ] x [ C ] } is the [ Zero ] vector .

  

  We now also present the { co-ordinates } for the 4 { Centroid-based Position Vectors } for { Tetrahedron A } and { Tetrahedron C } , for authentication purposes .

  Nodal Point	Tetrahedron A			Tetrahedron C			Cross Product
  	X	Y	Z	X	Y	Z	X	Y	Z
  Mass #1	+0.000	-180.000	-135.000	-165.789	-318.300	+264.400	-90,562.500	+22,381.579	-29,842.105
  Mass #2	+240.000	+0.000	-180.000	-165.789	+366.100	+31.867	+65,898.000	+22,194.105	+87,864.000
  Mass #3	-240.000	+180.000	-160.000	+194.211	+96.100	-328.133	-43,688.000	-109,825.684	-58,021.895
  Mass #4	+0.000	+0.000	+475.000	+137.368	-143.900	+31.867	+68,352.500	+65,250.000	+0.000
  Total	+0.000	+0.000	+0.000	+0.000	+0.000	+0.000	+0.000	+0.000	+0.000

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The { Rotation Symmetry } Property :

• Let us now bring back the 3 diagrams from above :

  

  

• Let us now set up , again , three (3) { 3-orthogonal-unit-vectors Reference Frames } , initially co-incidental to one-another , namely :
  • the { [ w1 ] - [ w2 ] - [ w3 ] Reference Frame } ,
  • the { [ g1 ] - [ g2 ] - [ g3 ] Reference Frame } ,
  • the { [ N1 ] - [ N2 ] - [ N3 ] Reference Frame } ;

    so that the { [ w3 ] , [ g3 ] , [ N3 ] unit-vectors } are initially in the same direction , but this being an arbitrary direction .

• Let us now :
  • mount { Tetrahedron A } firmly onto the { [ w1 ] - [ w2 ] - [ w3 ] Reference Frame } , and
  • mount { Tetrahedron C } firmly onto the { [ g1 ] - [ g2 ] - [ g3 ] Reference Frame } .

• Let us now rotate the said 3 { 3-orthogonal-unit-vectors Reference Frames } in unison , together with the 2 { Tetrahedra } mounted thereupon ,
  • with the { [ w3 ] / [ g3 ] / [ N3 ] direction } being the { axis-of-rotation } .

  Question , what will happen to the { HT Vector } , the { Cross-Product [ A ] X [ C ] } ?

  Very simply , it will remain constant at [ zero ] . And this is because :

  • we have rotated the 3 { Reference Frames } in unison , so that the { HT Vetcor } will always retain its constant [ zero ] value , "vectorially" speaking .

• And we simply state here that ,
  • when the { HT Vector } attains a [ zero ] value ,

    we can use an arbitrary { direction } as the { axis-of-rotation } , to obtian this { Rotational Symmetry } property in the { mapping relations } .

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On { Size } Considerations :

• Let us bring back { Tetrahedron A } and { Tetrahedron C } from above :

  

• Let us now do the following :
  • Vary the size of { Tetrahedron A } from [ zero ] to [ infinity ] , and
  • Vary the size of { Tetrahedron C } from [ zero ] to [ infinity ] .

  Question , what will happen to the { HT Vector } ?

  Again , the { HT Vector } will remain constant at [ zero ] . This is because :

  • the 4 { sub-Cross-Products } arising from the 4 pairs of { corresponding Position Vectors } will :
    • always retain their individual orientations , and
    • their sizes relative to one-another will also be maintained ;

    so that their { sum-total } , "vectorially" speaking , will always remain at [ zero ] .

• We therefore have an added flexibility here when the { HT Vector } becomes [ zero ] .

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Next Question :

• { Eigen-Values } and { Eigen-Vectors } any one ? HUH ?

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go to the next section : Epilog III - On { Mirror / Rotated / Counter Images }

go to the last section : Epilog I - An Elaboration on the Cross-Product [ T-on-T ]

return to The Four Body Problem HomePage

original dated 2008-1-07