An Approach to the Four Body Problem

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

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Section VII - Setting up for Angular Momentum Issues

Conservation of Angular Momentum :

• The { Conservation of Angular Momentum } arises from the fact that :
  • the attraction / replusive force between any 2 of the 4 { Point Mass's } are always equal and opposite .

  

  The { lines-of-action } for each pair of { equal and opposite forces } are always co-linear ,

  • so that their net [ moment ] about the { Center-Of-Mass } , [ Point P ] , is always [ zero ] .

• Thus , { angular momentum } is always conserved in such a { Force System } .

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The { Position Vectors } revisited :

• Let us now bring back , from { Section I } , the { Position Vectors } based on the { Translating Reference Frame } with the { Center-Of-Mass } as its [ Point-Of-Origin ] ,

  

  • the diagram on-the-right is then the { Mass-Position Vectors } diagram .

• The governing equation for the diagram on-the-left , above , is then :
  • 

  while the governing equations for the diagram on-the-right , above , is then :

  • 

• We also bring-in now the { Position-Vector Tetraherdon } and the { Mass-Position-Vector Tetrahedron } , as follows :

  

  We simply note here that :

  • [ Point P ] is always the { centroid } of the { Mass-Position-Vector Tetrahedron } on-the-right , and

  • [ Point P ] is not the { centroid } of the { Position-Vector Tetrahedron } on-the-left ,
    • unless the 4 { Point Mass's } are all [ equal-mass's ] .

  We then identify [ Point Q ] as the { centroid } of the { Position-Vector Tetrahedron } , as shown above , for our late use .

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Setting up the { Velocity Vectors } :

• Let us now bring-in the { Velocity Vectors } , as per the set of 2 diagrams below :

  

  And we note here that { Velocity Vectors } are simply the [ time-derivatives ] of corresponding { Position Vectors } .

• We then set-up , correspondingly , the { Velocity Vectors } and { Mass-Velocity Vectors } ,
  • using a { Reference Frame } identical to the { Translating Reference Frame } afore-mentioned , and again :
    • with [ Point P ] at the { Center-of-Mass } as the { Point-Of-Origin } .

  

• The governing equation for the diagram on-the-left , above , is then :
  • 

  while the governing equations for the diagram on-the-right , above , is then :

  • 

• We also bring-in now the { Velocity-Vector Tetraherdon } and the { Mass-Velocity-Vector Tetrahedron } , as follows :

  

  We simply note here that :

  • [ Point P ] is always the { centroid } of the { Mass-Velocity-Vector Tetrahedron } on-the-right , and

  • [ Point P ] is not the { centroid } of the { Velocity-Vector Tetrahedron } on-the-left ,
    • unless the 4 { Point Mass's } are all [ equal-mass's ] .

  We then identify [ Point R ] as the { centroid } of the { Velocity-Vector Tetrahedron } , as shown above , for our later use .

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The { Angular Momentum Vector } :

• The { Angular Momentum Vector } is simply equal to :
  • [ mass ] x { the cross-product of the [ Position Vector ] and the [ Velocity Vector ] } .

• Let us now denote the { angular momentum vector } for the 4 { Point Mass's } as [ ETA-1 ] , [ ETA-2 ] , [ ETA-3 ] and [ ETA-4 ] respectively :
  • and we have this set of 4 equations below , in { standard format } :
    • 

  • We then have this set of 4 equations below , in the { 1st alternate format } :
    • 

  • And we have this set of 4 equations below , in the { 2nd alternate format } :
    • 

• We simply note here that :
  • the { 1st alternate format } involves { vectors } from :
    • the { Mass-Position-Vector Tetrahedron } and the { Velocity-Vector Tetrahedron } ;

  • the { 2nd alternate format } involves { vectors } from :
    • the { Position-Vetcor Tetrahedron } and the { Mass-Velocity-Vector Tetrahedron } .

  And we shall be taking advantage of these 2 { alternate formats } in the next Section .

• Let us now also denote the { Total Angular Momentum Vector } for the { 4 Point-Mass's System } as [ ETA-T ] , i.e :
  • 

  And because of the { Conservation of Angular Momentum } , this { Total Angular Momentum Vector } will remain constant thru time , so that :

  • once the { initial conditions } are set , the { magnitude } and { direction } of this { Total Angular Momentum Vector } are also fully determined .

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go to the next section : Section VIII - Cross-Product Considerations

go to the last section : Section VI - Brief Summary on the Tetrahedron

return to The Four Body Problem HomePage

original dated 2007-12-16