Well, it’s already past noon as I begin to draft this post. Maybe I’ll keep it short so I can get going on the mathematics and technical blogging I should be doing.
In many ways last weekend was very frustrating, and things didn’t get better when I grabbed some time during the week.
Color. I’m still trying to figure out how to design residuals which, when added to a sometimes negative fundamental spectrum, will give me a spectrum that is everywhere positive (actually, of course, everywhere nonnegative). Nothing I tried last week worked — but I do have another idea. That’s encouraging.
In addition, I really should have a go at working out the nonlinear characteristics of my computer screen.
Orbits. I know exactly which example I want to work out for myself — but I just haven’t gotten to it. The key to extra-planetary orbits is called “the patched conic approximation”. To fully describe an orbit, for example, from the earth to Mars, we patch together three orbits: a hyperbolic orbit with respect to the Earth, an elliptic orbit with respect to the sun, and then a hyperbolic orbit with respect to Mars. The boundaries between those orbits are at the “spheres of influence”: we solve a two body problem for a spacecraft in orbit around the Earth, until we reach a distance call the Earth’s sphere of influence — and then we solve a two body problem for a spacecraft in orbit around the sun. When we reach the sphere of influence of Mars, then we solve a two body problem for a spacecraft and Mars.
We would also want to specify the transfer from a bound earth orbit — what we would call a circular parking orbit — to the hyperbolic escape orbit; and similarly we might also want to specify the transfer from the inbound hyperbolic orbit around Mars to a bound orbit.
If we were using Mars for gravitational assist, instead of trying to orbit it, then we stay on the Mars hyperbolic orbit — come out the other side, and see what new elliptical orbit we have with respect to the Sun.
Anyway, the BMW book (Bate, Mueller, & White; see biblio) works out an example from the earth to the moon. Rather than using components of vectors — my preference in principle — they use magnitude and direction. For all I know, that may lead to easier solutions. It certainly leads to clear diagrams. I just haven’t brought myself to follow them through their calculations.
Now might be a good time to comment that the BMW book actually provided a great deal of geometric insight into orbits. Maybe I should go put this comment with the appropriate “books added” post….
I should also emphasize that the patched conic approximation is a first calculation. If you really want to travel through the solar system, you darned well better be monitoring your position and velocity as you go. It is not, in fact, a two body problem! But it is still rather nice to discover that we can get a good initial orbit by solving a sequence of two body problems.
Logic. I had trouble composing a post last weekend, so I took my own advice and didn’t push it: struggling with a post suggests that something is bothering me.
It seems to be coming along okay now, so maybe the one realization I had was the only thing I needed. I haven’t resolved it yet, but at least I recognize it: on the one hand, I can use truth tables to prove anything; on the other hand, when people lay out formal logic systems, they apparently need to assume at least one simple rule (called “modus ponens”, namely that from P imples Q, and P, we may conclude Q — how’s that for absolutely essential?).
It bothers me a little that just from the definitions, however, I can apparently derive modus ponens. I’m not going to worry about it, though, and my first post will show you how to use truth tables. (Instead, I will keep the question in the back of my mind, and someday something might trigger an understanding of why truth tables and formal systems seem to be different.)
Time series. I made it through one of the books on the little desk (described at the end of this post)…. It was there because I hoped that a second pass through the book would be more productive than the first pass had been — but the simple fact is that I cannot for the life of me figure out from his theory how to compute anything. (Okay, there was one computation I could follow — but I already knew how to do it!)
In the evenings, I’ve been looking through other books. The bad news is that time series analysis covers a lot of ground, and I am, quite frankly, intimidated by its breadth. The good news, however, is that my kid really likes the idea of doing simulations.
After all, it’s not difficult to generate artificial time series with specified structure — and then you get to go see how effective different techniques are at extracting that structure. That’s just the sort of thing my kid would be thrilled to do.
I guess I’ll close with the comment that my kid is doing his job. On the one hand, he is still finding things that look like fun right away, right now. On the other hand, interesting books keep passing my grown-up eyes. Late this week he saw the book on exotic smoothness in physics (there are an infinite number of essentially different ways to do calculus in R^4), and so I’m looking at differential topology again.
OK, to work.