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The formula to calculate force F is the first derivative of energy, XW.

Energy is a quaternion quantity W = Scalar S + Vector V, the Force calculation is:

Force F = [d/dr, Del] [S, V] = [dS/dr - Del.V, dV/dr + Del S + DelxV]

There are five forces:

Centripetal force dS/dr, a scalar

Centrifugal force Del.V, a scalar

Tangent Force dV/dr a vector

Gradiant/Normal force Del S

Curl/Circular Force DelxV, a vector

when the Force is zero the Formula is the Invariant Condition for Energy:

0 = [dS/dr - Del.V, dV/dr + Del S]

this reduces to

0 = dS/dr - Del.V and

0 = dV/dr + Del S

The DelxV term is zero at Invariant Condition. DelxV = 0, indicating that

dV/dr is equal and Opposite to Del S, ( Newton's Action-Reaction Law.)

An example of force calculation is W = -mGM/r + cmV for Gravity,

S = -mGM/r and V = cmV = cP, P is the Momentum of m and cP is the "Dark Energy".

F = [mGM/r2 - cDel.P, cdP/dr + mGM/r3 R + cDelxP]

F = [mv2/r -cmv/r cos(PR), mdV/dt + mw2R + cmv/r sin(PR) H]

The Force is a vector when the scalar part is zero, or mv2/r = cmv/r cos(PR) or v/c = cos(PR), this is the redshift !

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