I was trying deduce that if we suppose that the planet's orbits are circular and de Kepler's secong law is true:
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time
then, the linear velocity of the planets are constant.
That is my reasoning:
Let $\gamma (t)=r(\cos\theta(t),\sin\theta(t))$ be a orbit. We have to prove that: $$\Big|\frac{d\gamma}{dt}\Big|=\text{ constant }$$ or equivalently $\frac{d\theta}{dt}=\text{ constant}$.
The Kepler's second law is equivalent to: $$\int_0^r\int_{\theta(t_1)}^{\theta(t_2)} \rho d\theta d\rho=\int_0^r\int_{\theta(t'_1)}^{\theta(t'_2)}\rho d\theta d\rho \qquad\text{ forall } \ t_2-t_1=t'_2-t'_1$$ that implies: $$\theta(t_2)-\theta(t_1)=\theta(t'_2)-\theta(t'_1) \qquad\text{ forall } \ t_2-t_1=t'_2-t'_1$$
Any help for get $|\gamma'|=\text{ constant}$ by this way?
If is imposible by this reasoning, how could be this proof in terms of angular momentum conservation?