S8. Cross-Product Condiserations

An Approach to the Four Body Problem

Section VIII - Cross-Product Considerations

Let us bring back the { Position-Vector Tetrahedron } and the { Velocity-Vector Tetrahedron } :

We note that we have previously identify :
- [ Point Q ] as the { centroid } of the { Position-Vector Tetrahedron } , and
- [ Point R ] as the { centroid } of the { Vecloity-Vetcor Tetrahedron } .
We then set up the [ RHO Vectors ] as per this set of 2 diagrams below :

We can then express the { Position Vectors } in terms of the { RHO Vectors } , as follows :
We then set up the [ MU Vectors ] as per this set of 2 diagrams below :

We can then express the { Velocity Vectors } in terms of the { MU Vectors } , as follows :

We then bring back this equation for the { Total Angular Momentum Vector } from previously :
We can then use the { 1st alternate format } to arrive at this equation below :
Expressing in terms of [ MU ] then yield this equation below :
But we note that we have this relation below , arising from the { Mass-Position-Vector Tetrahedron } :
And this gives rise to this relation below :
Thus , we can express the { Total Angular Momentum Vector } in this manner :
And we simply note here that the { MU Vectors } are { centroid-based vectors } for the { Velocity-Vector Tetrahedron } .

Again we bring back this equation for the { Total Angular Momentum Vector } from previously :
We can then use the { 2nd alternate format } to arrive at this equation below :
Expressing in terms of [ RHO ] then yield this equation below :
But we note that we have this relation below , arising from the { Mass-Velocity-Vector Tetrahedron } :
And this gives rise to the relation below :
Thus , we can express the { Total Angular Momentum Vector } in this manner :
And we simply note here that the { RHO Vectors } are { centroid-based vectors } for the { Poistion-Vector Tetrahedron } .

We then summarize our current procedure for the { Four Body Problem } as follows :
- First , we identify the { Mass-Position-Vector Tetrahedron } as our prime referencing { tetrahedron } .
- Next , we identify the { Mass-Velocity-Vector Tetrahedron } as its { time-derivative } .
- Next , we derive the { Position-Vector Tetrahedron } from the { Mass-Position-Vector Tetrahedron } .
- Next , we derive the { Velocity-Vector Tetrahedron } from the { Mass-Velocity-Vector Tetrahedron } .
- Next , we express the { Total Angular Momentum Vectors } as either :
  - the summation { Cross-Product } of { vectors } from the { Mass-Position-Vector Tetrahedron } and the { Velocity-Vector Tetrahedron } , or alternately ,
  - the summation { Cross-Product } of { vectors } from the { Position-Vector Tetrahedron } and the { Mass-Velocity-Vector Tetrahedron } .
And this last item is then a constraint placed on the { differential equations } to be derived .
There is a distinct advantage here , we are using { centroid-based } vectors here , namely :
- the { Mass-Position Vectors } , the { Mass-Velocity Vectors } , the { RHO Vectors } and the { MU Vectors } .
Thus , the technics we have developed for the { 6 DOF's } for the { tetrahedron } can be applied here .

There is a specific reason why we have taken this approach :
- In { An Approach to Matrix And Linear Algebra } back in 2004 , we were doing this :
  - We mapped a { 3-axis'ed Ellpisoid } onto a { 3-axis'ed Sphere } and visa-vera .
  - The relative orientation of the 2 set of { 3-axis } then determined :
    - the existance of { eigen-values } and { eigen-vectors } ;
    - the values of { eigen-values } and the magnitude and directions of the { eigen-vectors } ;
    - related properties such as whether we would encounter { multiple eigen-values } or an { eign-black-holes } , etc. .
- In here , we are looking at the summation { Cross-Product } of two (2) { 3-axis'ed Tetrahedra } :
  - with the { Total Angular Momentum Vector } being determined by the relative orientation of 2 { tetrahedra } .
May , just maybe , we can follow along the same line-of-thoughts to tackle the current problem .