Plane polar coordinates

respectively.

We can write the position vector of our planet as

(4.10) |

Thus, the planet's velocity becomes

where is shorthand for . Note that has a nonzero time derivative (unlike a Cartesian unit vector) because its direction changes as the planet moves around. As is easily demonstrated, by differentiating Equation (4.8) with respect to time,

(4.12) |

Thus,

The planet's acceleration is written

(4.14) |

Again, has a nonzero time derivative because its direction changes as the planet moves around. Differentiation of Equation (4.9) with respect to time yields

(4.15) |

Hence,

(4.16) |

It follows that the equation of motion of our planet, Equation (4.2), can be written

(4.17) |

Because and are mutually orthogonal, we can separately equate the coefficients of both, in the preceding equation, to give a

and a