If I have velocity at perihelion/apphelion, distance away from sun at perihelion/apphelion, and orbital period. How can I find the angular velocity function for earth and subsequently all the other planets in the solar system?

While reading Chapter 1 of an astrodynamics textbook, I came across the statement: $$\mathbf{v}\cdot \mathbf{{\dot{v}}}=v{\dot{v}}$$ In other words, the dot product of velocity and the time-rate-of-change of velocity is simply equal to the product of the magnitudes of the velocity and acceleration vectors. I think I see how this works mathematically: $$\frac{1}{2}\frac{d}{dt}\left ( \mathbf{v}\cdot \mathbf{v} […]

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