An Approach to the Four Body Problem

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

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Section I - Setting up the Reference Frames

A Statement Of the Problem :

• In the { Four Body Problem } , we simply have four (4) { Point-Mass's } moving in accordance to Newton's Law of Gravitation .
  • The problem here is to predict the positions of the four (4) { Point-Mass's } at any time , given an initial set of conditions .

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Settting up the Reference Frames :

• Let us first set-up a { Global Reference Frame } that is stationary and constant thru time ,
  • so that positions of the { 4 mass's } are identified by the { Position Vectors } [ R1 ] , [ R2 ] , [ R3 ] & [ R4 ] respectively .

  

• We simply note here that the { Conservation of Linear Momentum } is always observed under such a { Force System } .

  And we now identify the { Center-Of-Mass } , which is always moving at a { constant velocity } under such conditions ,

  • as per the diagram on-the-right , above .

• We then set up a { Translating Reference Frame } at the { Center-Of-Mass } , so that :
  • the positions of the { 4 mass's } are now identified by a new set of { position vectors } ,
    • namely : [ r1] , [ r2] , [ r3 ] & [ r4 ] respectively , as per the diagram on-the-right , below .

  

• We simply note here that the equation for the { Center-Of-Mass } :
  • 

    is always applicable .

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The { Mass-Position Vectors } :

• Let us now set up 4 { Mass-Position Vectors } , all referenced from the { Translating Reference Frame } at the { Center-Of-Mass } ,
  • namely , [ M1*r1 ] , [ M2*r2 ] , [ M3*r3 ] , & [ M4*r4 ] , as per the diagram on-the-right , below .

  

  The applicable equation here is then :

  • 

• We then present the { Position-Vector Tetrahedron } and the { Mass-Position-Vector Tetrahedron } , as shown below :

  

  • And because the { centroid } of the { Mass-Position-Vector Tetrahedron } is always positioned at the { Center-Of-Mass } ,
    • and the motion of the { Center-Of-Mass } is always predictable ,

    we shall be using this { Mass-Poistion-Vector Tetrahedron } as the primary { referencing tetrahedron } ,

    • and the { Position-Vector Tetrahedron } will be treated as a { derived tetrahedron } ,

    for our further analysis purposes .

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go to the next section : Section II - The 12 Degrees-Of-Freedom for the Tetrahedron

return to The Four Body Problem HomePage

original dated 2007-12-16