Newton's derivation of Kepler's laws

1. Dec 1, 2012

inkyvoyd

Introduction
This is not a homework or coursework question (if it were it would be of the project type), and I am looking for hints not spoilers.
Hi,
I recently passed by kepler's laws again in a science class (this time earth science), and am concurrently taking calculus in my math class.

I realized that my current knowledge of calculus should let me be able to re-find kepler's laws (or show equivalence to newton's laws) - for kepler's first law, I should be able to prove (with newton's second law) that the elliptical orbit described is consistent. For his second law, I should be able to use polar integration to complete the consistency proof. As for the third, I haven't had any ideas, but my problems are really with where to start.

1. The problem statement, all variables and given/known data
I contacted my math teacher with this question, and we had a short discussion, with my teacher suggesting I get data of planetary locations over time. I searched for these, with no avail (I am not looking for conclusions - which are all I could seemingly find). I'm trying to understand kepler's first law and how it relates to newton's laws - but I do realize some problems. Since initial velocity (and position) must be known in order to determine the elliptical path, one must have these accounted for - and I have no idea how to do that.

I need hints, and if possible, data.
I do not want a result, or work and a result, because I want to in a sense "repioneer" this - the thinking that is involved with creating an idea previously unknown to one differs from that of learning about an idea.

tl;dr:I am trying to reconstruct Newton's derivation of Kepler's laws, and am stuck on how to translate newton's laws into kepler's first law. I need hints, and if possible, data.

2. Relevant equations
newton's second law: f=ma
newton's gravitational law:f=g(m_1m_2)/(r^2)
kepler's 3 laws

3. The attempt at a solution
I'm not sure where to start.
I know that once I get an idea I will break motion into x and y axis (put in parametric form), and try to convert to polar form as well. I am guessing I will encounter simple differential equations.

2. Dec 2, 2012

haruspex

The main problem relating Kepler's first law (elliptical orbits) to Newton's equations is that Kepler's law says nothing about time.
You could try:
- obtain a differential equation r, theta, t from Newton's laws;
- assume the relationship between r and theta implied by Kepler I, and on the basis of that obtain expressions for r and its time derivatives in terms of theta and its time derivatives;
- substitute in your ODE to eliminate references to r and show that appropriate assignments of the constants satisfy the equation.

3. Dec 2, 2012

inkyvoyd

I see - but since Kepler's second and third laws are about orbital time and velocity, I'm guessing I would have to incorporate all three simultaneously?

4. Dec 2, 2012

johns123

F = GMm/r^2 = mv^2/r ...... T = 2 x Pi x r / v and therefor v = 2 x Pi x r / T

sub for v in mv^2/r ( centrepital force ) and simplify to get T^2/r^3 = Constant

which holds for all the planets around the sun. That is Kepler's 3rd law.