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Conservation laws

As we have already seen, gravity is a conservative force. Hence, the gravitational force in Equation (4.1) can be written (see Section 2.4)

$\displaystyle {\bf f} = - \nabla U,$ (4.3)

where the potential energy, $ U({\bf r})$ , of our planet in the Sun's gravitational field takes the form

$\displaystyle U({\bf r}) = - \frac{G\,M\,m}{r}.$ (4.4)

(See Section 3.5.) It follows that the total energy of our planet is a conserved quantity. (See Section 2.4.) In other words,

$\displaystyle {\cal E} = \frac{\varv^{\,2}}{2} - \frac{G\,M}{r}$ (4.5)

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 2.5.) In other words,

$\displaystyle \index{orbital parameter!angular momentum} {\bf h} = {\bf r}\times {\bf v},$ (4.6)

which is actually the planet's angular momentum per unit mass, is constant in time. Assuming that $ \vert{\bf h}\vert>0$ , and taking the scalar product of the preceding equation with $ {\bf r}$ , we obtain

$\displaystyle {\bf h}\cdot{\bf r} = 0.$ (4.7)

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

Figure 4.1: Plane polar coordinates.
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next up previous
Next: Plane polar coordinates Up: Keplerian orbits Previous: Kepler's laws
Richard Fitzpatrick 2016-03-31