American Scientist

To the Editors:

The column by Brian Hayes ("The 100-billion-body problem," Computing Science, March–April) omits the issue of sensitivity to initial conditions (chaos). For example, is the small-N problem likely to be chaotic? If so, could large-N systems somehow "average-over" initial conditions to eliminate chaos, at least for coarse-grained solutions? The article provides a good intuitive feel for the galaxy model, at least in comparison to many other complex systems. As such, could it perhaps serve a more general role in facilitating basic insight into other complex systems—global weather, for example? Could the galaxy model address questions of chaos occurring at multiple spatial scales of observation?

Paul L. Nunez
Emeritus Professor
Tulane University
New Orleans, LA

To The Editors:

I have enjoyed reading Brian Hayes’s Computing Science column, not least for the interesting questions he suggests. In the March–April issue in "The 100-billion body problem," Hayes gives a lucid description of problems in astrophysics as researchers attempt "marching solutions" of stars modeled as point masses subject to Newtonian gravitation with its central forces and also to Newton’s law of conservation of momentum, F = Ma. I would like to offer some observations on a few aspects of the mechanics that might be useful.

I was surprised to find no mention of the great corollary to the central force hypotheses, namely the conservation of angular momentum. In principle, some problems in classical mechanics may not be solved without invoking this law, but in any case conservation of angular momentum most often elucidates the outcomes of direct calculation of central forces acting on point or aggregated masses modeled as such: rigid bars, the pendulum bob, and so on. Motion of the simple pendulum suspended in a gravity field becomes a moderately hard problem if one does not invoke angular momentum. Hayes mentions the inaccuracies introduced in computer calculations by choosing the finite value of Δt: The masses travel to their next positions at constant acceleration, rather than traveling under the strict Newtonian law of gravitation. In doing so the entire collection of particles in the model violates the conservation of angular momentum for the entire set of particles. Perhaps conservation of angular momentum might offer a less awkward way of correcting this inaccuracy so that larger time steps could be used in practice.

In one of physicist E. T. Jaynes’s papers on probability, he remarked that even if some genius were to find another integral of the motion in classical mechanics, it would not affect the results of statistics of motion in cases where we have measured only the energy (it seems self-evident now, but was not when he wrote it). Well, such a genius did happen by, Mikhail Gromov, who in the 1980s discovered an unexpected property of Hamiltonian systems and thus classical mechanics. In brief, Gromov’s theorem (the Gromov nonsqueezing theorem) tells us that it is impossible to deform the state space of a Hamiltonian system of particles to fit inside a cylinder in Hamiltonian (p, q) space unless the "radius" of the cylinder is at least as large as the "ball" in Hamiltonian space that contains the energy of the system of particles. Why does this come up? Every student of classical mechanics knows that the Liouville theorem requires that the sector of phase space occupied by the system of particles governed by Hamiltonian rules must deform without changing volume: This is an identity. But not all volume-preserving maps of that sector of phase space conform to Hamiltonian rules of dynamics. It is too much to go further on this subject here, but the computer scientists pursuing cosmic dynamics might find it helpful.

Charles A. Berg
Emeritus Professor
Northeastern University
Boston, MA

Mr. Hayes responds:

Conservation of angular momentum is indeed a crucial concept in celestial mechanics. However, in N-body simulations it is generally not a constraint imposed on the system but rather a property that should emerge from the dynamics of the particles. A failure to conserve angular momentum (or energy) acts as a diagnostic signal, indicating that something has gone wrong in the simulation.