Research Interests

Dancing, with the Stars.

Have you ever seen the planets do-si-do?

It's not something you're likely to observe looking up at the night sky — but for centuries, astronomers and physicists have been fascinated by the ways planets and stars move about one another. Familiar interactions are apparently at work our solar system: the earth orbits the sun along a roughly circular path, and likewise the moon orbits the earth. But will it continue this way forever? And is this the only way for three celestial bodies to sustainably coexist?

The question is at once theoretical and existential, and has been unsettled since it was first posed by Sir Isaac Newton. The question has since motivated insightful studies in mathematics and physics over the years, attracting the eye of such giants as Euler, Hamilton, Lagrange, Noether, and Poincaré. But while advanced mathematics has provided a great deal of new information on the problem, it has yet to be successful in delivering a complete solution.

What would a solution look like? Recent work has focused on what has come to be known as choreography: solutions where all three bodies share the same fixed path in space, tracing out the same dance step again and again as each endlessly chases the next.

Doing Newton One Better.

Newton's Second Law, the famous equation F = ma, describes how a force on an object gives rise to an acceleration. This physical law is the foundation of our understanding of how all (macroscopic) objects, from apples to galaxies, move under the influence of gravity.

But is Newton missing a bigger picture? Force and acceleration are local quantities that are measured separately at each point in space and time. The Second Law, in essence, dictates "which way" a body will move. What it does not dictate is "how" it will move over long time scales: over time, what overall shape will its motion take?

This global question is more challenging, and Newton himself was unable to answer it definitively (except in the simplest nontrivial case, that of only two bodies). Since Newton's time, mathematicians and physicists have developed and applied techniques of global analysis to make progress toward a complete understanding of the "N-body problem."

In recent years, techniques from the fields of Lagrangian mechanics and non-Euclidean geometry have been used to discover and prove the existence of a variety of solutions, with perhaps the most notorious being the figure-eight orbit of the 3-body problem proved by A. Chenciner and R. Montgomery in 2000.

The idea behind my research is this. Jacobi showed that the potential energy of a conservative system - which is responsible for all its motion - can be "built in" to the system as geometry instead. The bodies then move freely in this geometry along exactly the same paths they would have moved according to Newton's laws! This turns a dynamics problem into a geometry problem, and by studying the geometry we gain insight into the dynamics.

Uncovering Negative-Curvature Mechanics.

Up to reparameterization, the trajectories of a mechanical dynamical system on a smooth Riemannian manifold \( (M, ds^2) \) arise as geodesics of the Jacobi metric \( d\rho^2 = (h + U)ds^2 \), with \( h \) the (conserved) total energy and \( U = U(p) \) the potential function.

Meanwhile, the topological types of these geodesics are known in the case where the metric \( d\rho^2 \) has negative curvature: namely, in this case a unique geodesic representative exists in each homotopy class of \( M \). This strong topological classification has long been the "holy grail" in the study of \(N\)-body systems. My research seeks to explore the question: What circumstances (on \(M\) and/or on \(U\)) are sufficient for a nonpositively-curved Jacobi metric?

That affirmative examples exist to this question is suggested by the observation that \(N\)-body shape space (i.e., a configuration space on \(N\) particles) has negative Euler characteristic for \( N \geq 3 \). Thus the Gaussian curvature (or sectional curvature, or Euler class as appropriate) must be negative on at least a set of positive measure in \(M\). Whether and when it is negative, or at least nonpositive, on all of \(M\) provides the intrigue in this question.

From the point of view of the potential U, one answer has come from R. Montgomery (2006) for 3-body shape space \( M = S^2 \setminus \{3\} \), in which an inverse-square (strong force) potential was shown to lead to nonpositive Jacobi curvature. Here, questions include: Does this result hold for \(N > 3\)? How might the rigidity of hyperbolic metrics in real dimension ≥ 3 be reflected by Jacobi metrics? What is the nature of topological progress measures (such as rotation measure) on these manifolds?

From the point of view of the manifold \(M\), if \(M\) is a surface this question has the flavor of the Yamabe problem. Since \(d\rho^2\) is conformally related to \(ds^2\), the question becomes: What metrics \(ds^2\) are conformally related to some hyperbolic metric (for instance, of constant curvature -1)?