Saturday, July 5, 2008

Clapping for Kepler! - Part II

I've been thinking of a way to explain Kepler's 2nd Law, and well, Newton provided two methods; in his Principle of Natural Philosophy, he outlined a geometrical method which involves highly accurate sketches.

Well too bad for you guys, I'm not about to go drawing things on my computer, because the angles to work out are simply too tedious! So kudos to Isaac Newton for working it out by hand.

But hey, he did come up with another easier method, which we all know now as Calculus - however, his work on his three laws of motion never once included the mechanics of calculus in his proofs.

So... how?

Like that lor.

No lah, of course not - let's just go through the main idea of using Calculus in proving Kepler's 2nd Law. Now, let us say we have an area A that is swept out by a planet moving around the Sun. If we can prove that the rate of change of area swept out (i.e. dA/dt) is zero, then we can then prove that the area swept out doesn't change with time!

Hurhur. Sounds easy.

Well, strictly speaking the area we should be considering should really be the area below, which shows the area swept out by Earth with the Sun at one focus:

But now we're considering the infinitesimal area that the planet sweeps out, and thus we are allowed to consider dA instead - that is, we consider the area swept out in a very small amount of time, such that the area swept out can be represented adequately by a triangle as such:

Well, I've removed the Earths for clarity's sake; and you may not yet be convinced as to why this small area can be a triangle - it is clearly a segment, is it not? Well, let me enlarge the diagram for you:

Are you now convinced that the segment is effectively reduced to a simple isoceles triangle when infinitesimally small areas are considered? It's as if the radius vector's length didn't change much! That's why we consider dA instead of A itself, and this allows us to work with nice approximations that hold. With this in mind, let us determine a formula for dA and I've enlarged the triangle OAB (yes, enlarged again!) for your viewing pleasure:

Notice that I've obliterated the Sun and the trajectory as well for clarity (Physicists tend to do that, haha!). Now,

1 comment:

telliott99 said...

Your post is truncated at the critical point. I wonder if you could explain further.

Also, you may be interested that Feynman's lectures at Cornell have been posted on the web (courtesy Bill Gates), and he covers the very same question. But I've never been able to understand this point in the talk (I have the book).