Proving that Angular Moment is Conserved - Kepler's 2nd Law

[BOAS]

Homework Statement

Kepler’s 2nd law of planetary motion says that the radius vector drawn from the Sun to any planet sweeps out equal areas in equal time intervals. By considering the small area swept out by the radius vector in time dt, show that $\frac{dA}{dt} = \frac{L}{2m}$ , where L is angular momentum. Then show that the angular momentum is conserved for this situation, thus proving Kepler’s 2nd law.

Homework Equations

$\vec L = \vec r \times m \vec v$

The Attempt at a Solution

This might be a little hard to follow without a diagram, but i'll do my best to label everything clearly.

The area of a sector is given by $A = \frac{1}{2} r^{2} \theta$, so the area swept out by the planet in time dt is given by $\frac{dA}{dt} = \frac{1}{2} r^{2} \frac{d \theta}{dt}$.

The velocity of the planet perpendicular to the radius vector is $v_{p} = v \sin \phi$ where $\phi$ is the angle between the radius and velocity vectors.

$\frac{dA}{dt} = \frac{1}{2} r v \sin \phi$

From the expression for the magnitude of the cross product $rv \sin \phi = |\vec r \times \vec v|$
Therefore, we can say $\frac{dA}{dt} = \frac{L}{2m}$. (after multiplying by a factor of 1/m).

I am confused by what it means to prove that angular momentum is conserved in this situation. I know how to prove that angular momentum is conserved for two bodies each exerting a force on one another, but I don't see the link to Kepler's second law.

Should I be understanding the 'situation' simply to mean a planet orbiting the sun, or can I use keplers second law to prove conservation of angular momentum?

This is a first year classical mechanics question, if that in any way affects how you'd respond.

thanks!

 

[mfb]

The area of a sector is given by $A = \frac{1}{2} r^{2} \theta$, so the area swept out by the planet in time dt is given by $\frac{dA}{dt} = \frac{1}{2} r^{2} \frac{d \theta}{dt}$.

What about changes in r?

I am confused by what it means to prove that angular momentum is conserved in this situation. I know how to prove that angular momentum is conserved for two bodies each exerting a force on one another, but I don't see the link to Kepler's second law.

Kepler's law follows from conservation of angular momentum and the equality you just derived.

Should I be understanding the 'situation' simply to mean a planet orbiting the sun, or can I use keplers second law to prove conservation of angular momentum?

That would be a circular argument.