Question

Prove that angular momentum of a particle is equal to twice how does it lead to kepler's l

Answer 1

The Kepler's second law states that the radius vector from the Sun to the planet sweeps equal areas in equal times. In another words, the rate of change

dA

dt

is constant. Consider the figure below, enter image description here

The are element is dA=

1

2

r2dθ so in the time interval dt we have

dθ

dt

=

2

r2

dA

dt

,

On the other hand the angular momentum magnitude (with respect to O) is L=mr2

˙

θ

. Thus,

L=2m

dA

dt

,

which is constant.

However this does not prove that the vector

→

L

is constant. To prove that the vector does not change its direction one has to assumeeither the first Keppler's law (which implies the orbit lies in a plane) or that the force is central (which automatically implies in the angular momentum conservation).