Geometry of the Twist Sim Math

Here is a drawing of the forces on the twist path that the simulator attempts to model.

Here is a drawing of the forces on the twist path that the simulator attempts to model.

I created a picture that hopefully shows the geometry of the simulation math described in the previous post (see in particular the PPS update).  This picture attempts to show a generator twist path about point A in red, with the two force sources F(loop) and F(twist), which are delta 1/r^2 and 1/r^3 flux field generators respectively.  The destination point D path is shown in blue.  The parametric integral must be computed for every source point on each destination point–this will give a potential field.  When the entire set of curves lies on an equipotential path, one of many possible stable solutions has been found (it’s already easy to establish that any topologically unique closed loop solution will not degenerate because the 1/r^3 force will repel twist paths from crossing each other).  There probably is a good LaGrange method for finding stable solutions, but for now I will work iteratively and see if convergence for various linked or knotted loops can be achieved.

 

Agemoz

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