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Force is the space derivative of Energy. Energy and spac e are four dimensional quaternions. So Energy is E=Er + Ev where Er is the real energy and Ev is the vector (3) energy.

Force F=( d/dr + Del) (er + Ev) = (dEr/dr - Del.Ev) + (dEv/dr + Del Er + DelxEv)

So here you see there are five forces.

The first two forces are real forces dEr/dr is the centripetal force and Del.Ev is the centrifugal force. For gravity Er = -mGM/r and Ev=mcV.

Thre remaining three forces are vectors. dEv/dr is the Tangential (Hypotenuse) force;
Del Er is the Normal Force ( for gravity muR/r^3); the last force DelxEv is the transverse force that causes rotation.


When energy is conserved, the force is zero or in equilibrium. Here the centripetal force is equal the centrifugal force and the sum of the three vector forces is zero.

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