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Keplers 2nd law says that a planet revolving around the Sun at the focus of the elliptical orbit, sweeps the same areas in the same period of time.  We want to prove that.

The area ΔA swept (in time Δt) by a planet of mass m moving with instantaneous velocity v at a instantaneous distance r from the Sun, 
              ΔA = 1/2 (r Χ v Δt )          as  the Δs = arc length = v Δt
  the quantities in bold are vectors.
              ΔA/Δt = 1/2 (r Χ v)  = 1/2 (r Χ p/m) =  L / 2m
 L is the angular momentum vector of the planet.

As there is not external force on the planet and Sun system, the total angular momentum is a constant.  Hence L is constant.  That means ΔA/Δt is constant.

  Hence Area is directly proportional to time or Area swept is constant for a constant duration of time.

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