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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.