Kepler's Second Law Revisited 


Consider a small wedge of the orbit traced out in time dt:

So the area of the wedge is

 

And the rate at which area is swept out on the orbit is


 
 
Now, remember(?)  the  definition of Angular Momentum: 

 
 

Inserting this  previous equation , we get



 
"Equal areas in equal times" means the rate at which area is swept out on the orbit (dA/dt) is constant.
 
So Kepler's Second Law Revised:

 
The rate at which a planet sweeps out area on its orbit is equal to one-half its angular momentum divided by its mass (the specific angular momentum). Angular momentum is conserved.