The Virial Middle

In celestial mechanics, “heaven” is the zero-energy surface — where bodies move in unbound trajectories, just barely escaping each other forever. “Hell” is total collision — all bodies converging to a single point. Montgomery asks: what lives between them?

The virial equation relates kinetic energy to potential energy along solutions of the N-body problem. The Jacobi-Maupertuis reformulation replaces time with arc length, transforming Newton’s equations into a geodesic problem on a curved space. Montgomery connects these two perspectives and poses questions about the territory they jointly illuminate.

The answers he provides are selective — this is a paper of questions more than resolutions. But the questions are structural. Between the escape boundary and the collision singularity, the dynamics of N gravitating bodies inhabit a landscape whose geometry the virial equation constrains and the Jacobi-Maupertuis metric measures. Some energy levels permit certain topological configurations; others exclude them. The virial equation says how much room there is; the Jacobi-Maupertuis metric says what shape the room has.

The title is precise. Heaven (escape) and hell (collision) are the energy extremes. The dynamics of actual solar systems, actual galaxies, actual clusters of interacting bodies — these live halfway between, in the regime where neither extreme dominates and both constrain. The physics that matters is the physics of the middle.