Kepler's law and my problem

Question

I was doing Kepler's first law (basically that is the only law in which I have a problem) other two laws are easy to understand). During the explanation of the first law my instructor said that we need to establish the fact that angular momentum does not change per unit time i.e. stays constant.

Problem 1.) Instructor says: "First of all, it can be shown that the angular momentum $L$ is constant. Since in the first term, the centripetal force $F$ is collinear with $r$, and in the second term $v$ is obviously collinear with $v$, both terms are zero, which implies that $L$ is constant and that the orbit lies in a plane!" What does this mean (specifically what does "lies in a plane mean"?)

Problem 2.) I learned Kepler's second law which was about equal areas and we PROVED that law of areas mathematically and also in his third law we PROVED, mathematically, that the period squared is equal to semi-major axis cubed, but in his first law which is:"All planets move about the Sun in elliptical orbits, having the Sun as one of the foci" we did not PROVE this law. We just said that the orbit lies in a plane, not that it is elliptical which is actually the law. Should not we be proving that the orbit is elliptical with Sun as on of the foci? Or am I missing something?

Problem 3.) Here is a screenshot from the website:

It says dr/dt=v (vector) and it is in the same direction as p (vector), but the change in r (vector) with respect to time is not v, it is the one component of v which is directed inwards. Obviously v is collinear with v but it is not "v" which is causing change in r, it is the component of v which causes the change in r and it is directed inwards, not collinear.

Answer 1

Solution 1) "Lies in a plane" means that the orbit forms a 2-dimensional shape. It means that you could define some x, y, and z axes and the paths for both the Sun and the planet would always have a z coordinate of $0$.

Solution 2) It's true that saying the orbit lies in a plane is not the same thing as proving Kepler's Law of Orbits mathematically. However, if you go back to the website your screenshot comes from, you'll find that they continue on to show the complete and rigorous derivation for the orbital path and show how that is a conic section, which, for a gravitationally bound orbit, is the definition of an ellipse.

Solution 3) it is important that we distinguish between $\vec r$ and $r$. $r$ is the magnitude of the radius. You are correct in saying that $\frac{dr}{dt}$ is the component of $\vec v$ that is directed along the radius, because it corresponds to a change in distance from the center. However, $\vec r$ is the position vector. It represents the coordinates of the orbiting body in the orbital plane. As a vector, it would be written as $\vec r=(r,\theta)$; that is, it has components that represent the distance from the origin, $r$, and the angle from some pre-defined line of reference, $\theta$. So you can see that $\frac{d\vec r}{dt}=\frac{dr}{dt}\hat r+r\frac{d\theta}{dt}\hat\theta$ (the hat notation indicates a direction unit vector). What was used in the equation you showed was $\frac{d\vec r}{dt}$, which doesn't necessarily have to point in the radial direction because the position vector might only be changing in $\theta$. So you see that $\frac{d\vec r}{dt}=\vec v$, which is a definition.

Answer 2

I am only going to answer the first part of your question as the second part, that is why planets move in elliptical (more generally in a conic section) orbit is just solving the differential equation for Newton's Law and if you are not an undergraduate student, I doubt that it is relevant for you.

First of all note that the definition of $\vec v$ is in fact $\dot{\vec r}$. I don't know how else you would define it. You said the change of $\vec r$ is not $\vec v$ but "it is the one component of $v$ which is directed inwards". Is it possible that you are confusing the change of $\vec v$, which is the acceleration and is directed inwards due to Newton's law, with the change of $\vec r$, which is just $\vec v$ ?

Let's take an example. Assume that the orbit is a circle with radius $r$ and the planet is in the position $(r,0)$ now the radius has to stay constant, so the planet is going inwards as you guessed but it also has to go sideways because otherwise it would be at a point $(r-\Delta x, 0)$, which would not be on the circle, so the change in $\vec r$ is not just downwards but also sideways too!

I don't know if this helps but if you can provide your definition of $\vec v$ and why you think think that only the radial part of $\vec v$ is changing the position, I can help more.