Playtime!

OK, I spent *way* too much time playing this game last night: Orbitrunner. And because I'm the kind of guy that I am, I wanted to inflict it on you.

It's actually a very interesting bit of gaming, for as simple as seems at first glance. Here's the description from the site:

Control the Sun with your mouse. Use it to manipulate the planets' paths. The Sun's pull gets stronger as planets get closer. If the gravity is at a right angle to the direction of travel, an orbit can form. Make sure planets don't leave the screen or collide!

I'm sure that they have established some fairly basic approximations for your computer to manipulate, but it still addresses one of the classic problems of physics: how to calculate the orbital dynamics for two or more bodies in motion. Even if you restrict the interactions to one orbital plane, this is a surprisingly difficult problem for more than two bodies, and has been for centuries. From ScienceWorld:

The three-body problem considers three mutually interacting masses , , and . In the restricted three-body problem, is taken to be small enough so that it does not influence the motion of and , which are assumed to be in circular orbits about their center of mass. The orbits of three masses are further assumed to all lie in a common plane. If and are in elliptical instead of circular orbits, the problem is variously known as the "elliptic restricted problem" or "pseudorestricted problem" (Szebehely 1967, pp. 30 and 39).

The efforts of many famous mathematicians have been devoted to this difficult problem, including Euler and Lagrange (1772), Jacobi (1836), Hill (1878), Poincaré (1899), Levi-Civita (1905), and Birkhoff (1915). In 1772, Euler first introduced a synodic (rotating) coordinate system. Jacobi (1836) subsequently discovered an integral of motion in this coordinate system (which he independently discovered) that is now known as the Jacobi integral. Hill (1878) used this integral to show that the Earth-Moon distance remains bounded from above for all time (assuming his model for the Sun-Earth-Moon system is valid), and Brown (1896) gave the most precise lunar theory of his time.

And Wikipedia has a very good entry (beyond my math level) about the broader n-body problem:

General considerations: solving the n-body problem

In the physical literature about the n-body problem (n ? 3), sometimes reference is made to the impossibility of solving the n-body problem. However one has to be careful here, as this applies to the method of first integrals (compare the theorems by Abel and Galois about the impossibility of solving algebraic equations of degree five or higher by means of formulas only involving roots).

The n-body problem contains 6n variables, since each point particle is represented by three space (displacement) and three velocity components. First integrals (for ordinary differential equations) are functions that remain constant along any given solution of the system, the constant depending on the solution. In other words, integrals provide relations between the variables of the system, so each scalar integral would normally allow the reduction of the system's dimension by one unit. Of course, this reduction can take place only if the integral is an algebraic function not very complicated with respect to its variables. If the integral is transcendent the reduction cannot be performed.

Well, have fun with it. And be amused about that all that phenomenal computing power at your fingertips making a simple little game. Such is the future.

Jim Downey

(Via MeFi. Cross posted to my blog.)