I’d like to do a little bit of classical mechanics, but the particular thing I want to do is a little hefty for one post. We’ll split it in two. Today I’ll set the stage and tomorrow we’ll use it to solve an interesting problem.
The problem involves the orbit of a planet in a gravitational field, and fortunately the initial approach to the problem is not complicated. First we write everything in terms of energy. The total energy of a system is the sum of its kinetic and potential energy. Kinetic energy is the energy associated with motion, and potential energy is (in the case of orbits) the energy associated with its particular position in the gravitational field. We’ll write the kinetic energy in terms of polar coordinates, and then write that in terms of the angular momentum L. I’m skipping the details, since while interesting they’re not vital at the moment. If you’re curious they’re in any mechanics textbook (I first learned mechanics from Thornton and Marion).
Some explanation: r is the distance from the center to the oribiting body (say, from the sun to the earth), r-dot is the rate of change of r, L is the angular momentum (roughly, a measure of the mass of the earth and the speed of its orbit), and U is the gravitational potential energy. We can solve this for r-dot if we want:
Now why are we doing this? The orbital dynamics of stars and planets are well-known, why should we start from the very beginning? The reason I’m starting here is that this basic equation doesn’t care about the form of the potential energy. We know that in classical gravity it’s proportional to 1/r, but for all we know it might have been something else. And if it were, what might the universe have been like? Perhaps orbits wouldn’t be stable, or perhaps they wouldn’t have been possible in the first place. Maybe there wouldn’t have been any such thing as escape velocity, or maybe the relationship between distance and orbital speed would have been different.
While it’s an interesting philosophical bull-session question, there is some relevance. Looking at what laws produce what results can tell us something, even if we think it’s purely speculative. Small differences between the predictions of Newton’s 1/r potential and the actual orbit of Mercury helped spur the development of relativity, and today there’s some (admittedly dubious) speculation that the laws may need to be further modified to account for the lack of success of dark matter searches.
I don’t know how that will turn out, but I do know that mathematical physics has got the tools to make predictions from it. Counterfactual “what if” scenarios don’t always stay counterfactual forever.