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Conservation Laws

Now gravity is a conservative force. Hence, the gravitational force (210) can be written (see Section 2.5)
\begin{displaymath}
{\bf f} = - \nabla U,
\end{displaymath} (213)

where the potential energy, $U({\bf r})$, of our planet in the Sun's gravitational field takes the form
\begin{displaymath}
U({\bf r}) = - \frac{G\,M\,m}{r}.
\end{displaymath} (214)

It follows that the total energy of our planet is a conserved quantity--see Section 2.5. In other words,
\begin{displaymath}
{\cal E} = \frac{v^2}{2} - \frac{G\,M}{r}
\end{displaymath} (215)

is constant in time. Here, ${\cal E}$ is actually the planet's total energy per unit mass, and ${\bf v} = d{\bf r}/dt$.

Gravity is also a central force. Hence, the angular momentum of our planet is a conserved quantity--see Section 2.6. In other words,

\begin{displaymath}
{\bf h} = {\bf r}\times {\bf v},
\end{displaymath} (216)

which is actually the planet's angular momentum per unit mass, is constant in time. Taking the scalar product of the above equation with ${\bf r}$, we obtain
\begin{displaymath}
{\bf h}\cdot{\bf r} = 0.
\end{displaymath} (217)

This is the equation of a plane which passes through the origin, and whose normal is parallel to ${\bf h}$. Since ${\bf h}$ is a constant vector, it always points in the same direction. We, therefore, conclude that the motion of our planet is two-dimensional in nature: i.e., it is confined to some fixed plane which passes through the origin. Without loss of generality, we can let this plane coincide with the $x$-$y$ plane.

Figure 13: Polar coordinates.
\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter05/fig5.01.eps}}
\end{figure}


next up previous
Next: Polar Coordinates Up: Planetary Motion Previous: Newtonian Gravity
Richard Fitzpatrick 2011-03-31