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)

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

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

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

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

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

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