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Now gravity is a conservative force. Hence, the gravitational force (210) can be written (see Section 2.5)

(213) 
where the potential energy, , of our planet in the Sun's gravitational field takes the form

(214) 
It follows that the total energy of our planet is a conserved quantitysee Section 2.5. In other words,

(215) 
is constant in time. Here, is actually the planet's total energy per unit
mass, and
.
Gravity is also a central force. Hence, the angular momentum
of our planet is a conserved quantitysee Section 2.6. In other
words,

(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 , we
obtain

(217) 
This is the equation of a plane which passes through the origin, and
whose normal is parallel to . Since is a constant vector,
it always points in the same direction. We, therefore, conclude that
the motion of our planet is twodimensional 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  plane.
Figure 13:
Polar coordinates.

Next: Polar Coordinates
Up: Planetary Motion
Previous: Newtonian Gravity
Richard Fitzpatrick
20110331