Conservation Laws

Now gravity is a conservative force. Hence, the gravitational force (285) can be written (see Sect. 3.5)

$${\bf f} = - \nabla U,$$ (288)

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

$$U({\bf r}) = - \frac{G\,M\,m}{r}.$$ (289)

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

$${\cal E} = \frac{v^2}{2} - \frac{G\,M}{r}$$ (290)

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 Sect. 3.6). In other words,

$${\bf h} = {\bf r}\times {\bf v},$$ (291)

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

$${\bf h}\cdot{\bf r} = 0.$$ (292)

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 27: Plane polar coordinates.