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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) (4.3)

where the potential energy, , of our planet in the Sun's gravitational field takes the form (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, (4.5)

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 2.5.) In other words, (4.6)

which is actually the planet's angular momentum per unit mass, is constant in time. Assuming that , and taking the scalar product of the preceding equation with , we obtain (4.7)

This is the equation of a plane that passes through the origin, and whose normal is parallel to . Because 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 - plane.    Next: Plane polar coordinates Up: Keplerian orbits Previous: Kepler's laws
Richard Fitzpatrick 2016-03-31