## The Balance of Forces in a Planar Orbit

This is given by the fundamental kinematic equation:

F = m (d2r / dt2 – r omega squared)

This can be written as:

m d2r / dt2 = F + m r omega squared

Here m r omega squared is the outward or centrifugal force and F is the net force of attraction. In a Newtonian orbit (a conical section):

F = – mMG / r squared

which is the attraction between m and M, the inverse square law of universal gravitation. In a hyperbolic spiral orbit (Coats orbit):

F = – m r omega squared

so dtr/dt2 = dv / dt = 0, and v is constant as r goes to infinity as observed in the velocity curve of a whirlpool galaxy. There has always been confusion in the textbooks about the centrifugal force. It is in fact the force generated by the rotation of the plane polar axes – it is due to the spin connection of Cartan geometry. In pure Newtonian dynamics there is only the force of attraction, so m would fall into M. The rotation of the axes also generates a Coriolis force, but this vanishes in all planar orbits. The rotation of the axes also generates the orbital velocity omega x r. The angular velocity omega is a spin connection.