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Conservation Laws

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 quantity--see 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 quantity--see 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 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 - plane.

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