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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.
![\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter05/fig5.02.eps}}
\end{figure}](img682.png) |
We can write the position vector of our planet as
![\begin{displaymath}
{\bf r} = r\,{\bf e}_r.
\end{displaymath}](img683.png) |
(220) |
Thus, the planet's velocity becomes
![\begin{displaymath}
{\bf v} = \frac{d{\bf r}}{dt} = \dot{r}\,{\bf e}_r + r\,\dot{\bf e}_r,
\end{displaymath}](img684.png) |
(221) |
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,
from differentiating Equation (218) with respect to time,
![\begin{displaymath}
\dot{\bf e}_r = \dot{\theta}\,(-\sin\theta,\,\cos\theta) = \dot{\theta}\,\,{\bf e}_\theta.
\end{displaymath}](img687.png) |
(222) |
Thus,
![\begin{displaymath}
{\bf v} = \dot{r}\,\,{\bf e}_r + r\,\dot{\theta}\,\,{\bf e}_\theta.
\end{displaymath}](img688.png) |
(223) |
Now, the planet's acceleration is written
![\begin{displaymath}
{\bf a} = \frac{d{\bf v}}{dt} = \frac{d^2{\bf r}}{dt^2}= \dd...
...eta})\,{\bf e}_\theta + r\,\dot{\theta}\,\,\dot{\bf e}_\theta.
\end{displaymath}](img689.png) |
(224) |
Again,
has a non-zero time-derivative because its
direction changes as the planet moves around.
Differentiation of Equation (219) with respect to time yields
![\begin{displaymath}
\dot{\bf e}_\theta = \dot{\theta}\,(-\cos\theta,\,-\sin\theta) = - \dot{\theta}\,{\bf e}_r.
\end{displaymath}](img690.png) |
(225) |
Hence,
![\begin{displaymath}
{\bf a} = (\ddot{r}-r\,\dot{\theta}^{\,2})\,{\bf e}_r + (r\,\ddot{\theta} + 2\,\dot{r}\,\dot{\theta})\,{\bf e}_\theta.
\end{displaymath}](img691.png) |
(226) |
It follows that the equation of motion of our planet, (212), can be written
![\begin{displaymath}
{\bf a} = (\ddot{r}-r\,\dot{\theta}^{\,2})\,{\bf e}_r + (r\,...
...\dot{\theta})\,{\bf e}_\theta = - \frac{G\,M}{r^2}\,{\bf e}_r.
\end{displaymath}](img692.png) |
(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,
![\begin{displaymath}
\ddot{r}-r\,\dot{\theta}^{\,2} = - \frac{G\,M}{r^2},
\end{displaymath}](img693.png) |
(228) |
and a tangential equation of motion,
![\begin{displaymath}
r\,\ddot{\theta} + 2\,\dot{r}\,\dot{\theta} = 0.
\end{displaymath}](img694.png) |
(229) |
Next: Conic Sections
Up: Planetary Motion
Previous: Conservation Laws
Richard Fitzpatrick
2011-03-31