One of our spins on the 3-body problem: What is the easiest direction to shoot a projectile out of the solar system, modeled as a planar circular restricted 3-body problem (1 infinitesimal mass + 2 finite masses)? We have created, most recently on a cluster of Mac OS X computers running under Xgrid, initial-conditions plots that dramatically illustrate the dynamical complexity of this problem; escape isn't so easy. For small (pseudo)energies, the projectile escapes rapidly; for large (pseudo)energies it is trapped; in between, its orbits are chaotic and arbitrarily complicated.
Through further computation and theoretical analysis, we hope to elucidate the structures of the escape set, especially its spiral, elliptical, and hyperbolic points. Many generalizations are possible, including incorporating Earth's spin, exploiting the third dimension. Most cool might be attempting to (arbitrarily?!) reduce the minimum escape speed by introducing and interacting with a second projectile.