Plane polar coordinates
Figure 4.1:
Plane polar coordinates.

We can determine the instantaneous position of our planet in the
 plane in terms of standard Cartesian coordinates, , ,
or plane polar coordinates, , , as illustrated in Figure 4.1. Here,
and
.
It is helpful to define two unit vectors,
and
, at the
instantaneous position of the planet. The first always points radially away from the origin,
whereas the second is normal to the first, in the direction of increasing . As is easily demonstrated, the Cartesian components of
and
are
respectively.
We can write the position vector of our planet as

(4.10) 
Thus, the planet's velocity becomes

(4.11) 
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,

(4.13) 
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 radial equation of motion,

(4.18) 
and a tangential equation of motion,

(4.19) 