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The gravitational force fields are both scalar and vector. The forces are the first derivative of the energy.
The derivative is a Quaternion derivative X=[d/dr,Del].
The total energy is W = -mGM/r + cP and there are five forces,
F = XW= [d/dr,Del][-mGM/r, cP] = [vp/r -cDel.P, cdP/dr - Del mGM/r + cDelxP]
The scalar forces are :
vp/r centripetal scalar force
-cDel.P= -cp/r cos(PR) centrifugal scalar force
the vector forces are:
cdP/dr = -cp/r R/r tangent force
Del -mGM/r = vp/r R/r gradient force
cDelxP = RxP cp/r sin(PR) Curl force.
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