Kepler's Second Law of Planetary Motion:

A line connecting a planet and the sun sweeps out equals amounts of area in equal amounts of time.

Begin by assuming the same situation as is assumed in the derivation of the first law.  The Lagrangian for the system would be

  .

As seen a diagram of the motion of the system,

  ,

which after being plugged into the expression for the Lagrangian gives

  .

One equation of motion resulting from this through the Euler-Lagrange equation is

  ,

which means that

and therefore

where l turns out to again be the angular momentum of the system.  Looking again at the diagram, and dividing the area swept out by the radius through the angle dq by the time interval dt this occurs in gives

  .

Plugging this result into the previous equation shows that

and therefore

  .

Since   is constant, this shows that the rate at which area is swept out by the line connecting the orbiting body to the central one is constant.  Another way of stating this is that the amount of area swept out in a given time is constant, which is

Background image courtesy of NASA's Planetary Photojournal.