An Approach to the Four Body Problem

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

Top Of Page

Epilog I - An Elaboration on the Cross-Product [ T-on-T ]

Brief Introduction :

• In this 1st Epilog , we shall discuss in fuller details the { Cross-Product } of 2 { Tetrahedra } .

  And for our discussion below , we shall be using only :

  • the { 4 centroid-based Position Vectors } for each and every { Tetrahedron } , and
  • the { Cross-Products } arising therefrom .

• But let us first develop a system for defining the { relative orientations } of { Reference Frames } , to facilitate our discussions below .

go to Top Of Page

Relative Orientations of { Reference Frames } :

• Let us first set-up two (2) { 3-orthogonal-unit-vectors Reference Frames } , as follows :
  • the { [ i ] - [ j ] - [ k ] Reference Frame } ,
    • This is the { Global Refrence Frame } which is constant and invariant thru time .

  • the { [ e1] - [ e2 ] - [ e3 ] Reference Frame } ,
    • This { Reference Frame } is initially co-incidental to the { Global Reference Frame } , but is ready for later rotations and re-orientations .

  

• Let us now rotate the { [ e1 ] - [ e2 ] - [ e3 ] Reference Frame } thru an [ angle ALPHA ] , using [ e3 ] as the { axis-of-rotation } ,
  • to arrive at the { [ f1 ] - [ f2 ] - [ f3 ] Reference Frame } , as shown in the diagram on-the-left , below .

  

• Let us now rotate the { [ f1 ] - [ f2 ] - [ f3 ] Reference Frame } thru an [ angle BETA ] , using [ f2 ] as the { axis-of-rotation } ,
  • to arrive at the { [ g1 ] - [ g2 ] - [ g3 ] Reference Frame } , as shown in the diagram in-the-middle , above .

• Let us now rotate the { [ g1 ] - [ g2 ] - [ g3 ] Reference Frame } thru an [ angle PHI ] , using [ g3 ] as the { axis-of-rotation } ,
  • to arrive at the { [ h1 ] - [ h2 ] - [ h3 ] Reference Frame } , as shown in the diagram on-the-right , above .

• We then bring-in this set of 2 diagrams for a quick comparison .

  

• We then summarize the above procedure in this manner :
  • We first select the { [ e3 ] unit-vector } as our primary { unit-vector } , and
    • the { [ e1 ] unit-vector } and the { [ e2 ] unit-vector } are then 2 orthogonal { unit-vectors } residing in the plane orthogonal to the { [ e3 ] unit-vector } .

  • We then re-orient the { [ e3 ] unit-vector } via the angles [ angle ALPHA ] and [ angle BETA ] ,
    • and [ angle ALPHA ] and [ angle BETA ] then defines the new position of the { [ e3 ] unit-vector } , relative to the { Global Reference Frame } .

      ( [ ALPHA ] and [ BETA ] are then the { polar co-ordinates } of the new { [ e3 ] unit-vector } ) .

  • We then release-the-hold on [ angle PHI ] and let the { [ e1 ] and [ e2 ] unit-vectors } rotate freely in the plane orthogonal to the new { [ e3 ] unit-vector } .

go to Top Of Page

The { Cross-Product [ A ] X [ B ] } :

• For our discussion here-on-in , let us set up three (3) { 3-orthogonal-unit-vectors Reference Frames } , initially co-incidental to one-another , as follows :
  • the { [ u1 ] - [ u2 ] -[ u3 ] Reference Frame } ,
  • the { [ e1 ] - [ e2 ] -[ e3 ] Reference Frame } ,
  • the { [ i ] - [ j ] - [ k ] Reference Frame } , which is the { Global Reference Frame } , constant and invariant thru time .

  

  

• Let us now bring-in :
  • { Tetrahedron A } , which is firmly mounted onto the { [ u1 ] - [ u2 ] - [ u3 ] Reference Frame } ,
  • { Tetrahedron B } , which is firmly mounted onto the { [ e1 ] - [ e2 ] - [ e3 ] Reference Frame } .

  And the { Cross-Product [ A ] X [ B ] } is then the [ HT Vector ] , as shown in the diagram on-the-right , above .

• Let us now do the following :
  • Freeze { Tetrahedron A } and the { [ u1 ] - [ u2 ] - [ u3 ] Reference Frame } in its original position , and
  • rotate / re-orient { Tetrahedron B } and the { [ e1 ] - [ e2 ] - [ e3 ] Reference Frame } freely .

  In the 1st Stage , we shall :

  • let [ angle ALPHA ] vary from [ 0 degrees ] to [ 360 degrees ] , and let [ angle BETA ] vary from [ 0 degrees ] to [ 180 degrees ] ; and
  • freeze [ angle PHI ] at the value [ zero ] for the time-being .

    The [ HT Vector ] , being the { Cross-Proudct [ A ] X [ B ] } , will vary accordingly , both in terms of { size } and { position } .

  In the 2nd Stage , we shall un-freeze [ Angle PHI ] and let it vary from [ 0 degrees ] to [ 360 degrees ] ,

  • and again , the [ HT Vector } will vary accordingly .

• This then sets the stage / platform for further investigations into these { mapping relations } .

go to Top Of Page

The { Rotational Symmetry } Property :

• Let us now bring back { Tetrahedron A } and { Tetrahedron B } from above , for this special property .

  

  

• Let us now set up 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 as the [ HT Vector ] , as shown above .

• Let us now :
  • mount { Tetrahedron A } firmly onto the { [ w1 ] - [ w2 ] - [ w3 ] Reference Frame } , and
  • mount { Tetrahedron B } 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 [ B ] } ?

  Very simply , it will remain constant . And this is because :

  • we have rotated the 3 { Reference Frames } in unison , so that the { HT Vetcor } will retain its { size } and orientation in the { [ N3 ] unit-vector } direction .

• This special feature , on using the { [ HT Vector ] direction } as the { axis-of-rotation } , is then known as the { Rotational Symmetry } property in these { mapping relations } .

go to Top Of Page

On { Size } Considerations :

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

  

• Let us now do the following :
  • Double the size of { Tetrahedron A } by doubling the size of all 4 { Position Vectors } ,
  • Half the size of { Tetrahedron B } by halving the size of all 4 { Position Vectors } .

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

  Again , the { HT Vector } will remain constant . This is because :

  • the 4 { sub-Cross-Products } arising from the 4 pairs of { corresponding Position Vectors } will all retain their original individual values and orientations ,

    so that their { sum-total } , "vectorially" speaking , will always remain the same .

• This then "opens-the-door" for strong negotiations on { relative size } issues .

go to Top Of Page

go to the next section : Epilog II - On the { [ Zero ] Angular Momentum Vector }

go to the last section : Section IX - Concluding Remarks

return to The Four Body Problem HomePage

original dated 2008-1-07