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Polar Coordinates
We can determine the instantaneous position of our planet in the
 plane in terms of standard Cartesian coordinates, (, ),
or polar coordinates, (, ), as illustrated in Figure 13. 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.
Figure 14:
An ellipse.

We can write the position vector of our planet as

(220) 
Thus, the planet's velocity becomes

(221) 
where is shorthand for . Note that
has a nonzero timederivative (unlike a Cartesian unit vector) because its
direction changes as the planet moves around. As is easily demonstrated,
from differentiating Equation (218) with respect to time,

(222) 
Thus,

(223) 
Now, the planet's acceleration is written

(224) 
Again,
has a nonzero timederivative because its
direction changes as the planet moves around.
Differentiation of Equation (219) with respect to time yields

(225) 
Hence,

(226) 
It follows that the equation of motion of our planet, (212), can be written

(227) 
Since and
are mutually orthogonal, we can separately equate the coefficients of both, in the above equation, to give
a radial equation of motion,

(228) 
and a tangential equation of motion,

(229) 
Next: Conic Sections
Up: Planetary Motion
Previous: Conservation Laws
Richard Fitzpatrick
20110331