Elementary Mechanics Using Matlab

Abstract

The Trojan asteroids, discovered almost a century ago, are directevidence for stability in the pure three-body problem. Two groups ofasteroids share Jupiter's orbit, preceding or trailing it by an angle of60 degrees. Because the Sun and Jupiter are by far the heaviest bodies,the restricted three body model is perfectly adequate to describe therelevant dynamics, and the perturbations due to the attraction of otherplanets do not significantly modify the orbit. For a hypotheticalLagrangian satellite bound to Earth and the Moon, we cannot discard theSun's influence a priori. The problem of stability in this case is veryhard; no analytic result is known, up to now. However, numericalanalysis can give us a plausible answer. In spring 1999, I made thisproblem a classroom activity for my physics undergraduate students atthe University of Parma. These students had an elementary background inclassical mechanics but no computational-physics training. The choice ofMatlab as a working environment was rather natural. With a littlesacrifice in speed compared to Fortran or C, Matlab let us build aworking program in a few days, including visualization and a friendlyuser interface. We easily found clear numerical evidence that theequilateral orbits L4 and L5 in the Earth-Moonsystem are unstable. I describe the program's setup, its Matlabimplementation and the results