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 non-zero 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) |