John Lindner Lands Another Cover Article in Science Journal

Research, co-authored with students, published in International Journal of Bifurcation and Chaos

WOOSTER, Ohio - John Lindner, professor of physics at The College of Wooster, has had another cover article published in a science journal. Lindner, whose "Invitation to Embarrassingly Parallel Computing" was recently featured as the cover article in the American Journal of Physics, has also had "Precession and Chaos in the Classical Two-Body Problem in a Spherical Universe" selected for the February issue of the International Journal of Bifurcation and Chaos.

Lindner, who conducted the research with four students over a five-year period, added a twist - or more precisely a curve - to the two-body problem, which was first solved by Isaac Newton. Newton found that solutions to the problem of two bodies interacting gravitationally were simple, elliptical orbits for bound particles (like Earth and Moon) or hyperbolic orbits for unbound particles (like a non-periodic comet). "By contrast, the corresponding three-body problem is now known to be infinitely complicated, including orbits that are chaotic, for which a tiny change in initial conditions can lead to a large change in final conditions," said Lindner.

"We generalized the classical two-body problem from flat space to spherical space and realized much of the complexity of the classical three-body problem with only two bodies," said Lindner. "We showed algebraically, by perturbation theory, that small, nearly circular orbits of identical particles in a spherical universe precess at rates proportional to the square root of their initial separations and inversely proportional to the square of the universe's radius. We showed computationally, by graphically displaying the outcomes of large open sets of initial conditions, that large orbits can exhibit extreme sensitivity to initial conditions, the signature of chaos. Although the spherical curvature causes nearby initially parallel orbits to converge, the compact space enables infinitely many close encounters, which is the mechanism of the chaos."

Lindner conducted the research successively with three rising sophomores and one senior. It began during the 2003 National Science Foundation-Research Experience for Undergraduates (NSF-REU) summer program with rising sophomore Katherine Olaksen, a physics major who graduated in 2006. It continued through the Senior Independent Study project of Nick Harmon, a physics and mathematics double major who graduated in 2004, and was followed that summer by rising sophomore Danny Shai, a physics and chemistry double major who graduated in 2007. The project was put on hold during Lindner's 2004-2005 sabbatical at Georgia Tech, but was completed during the summer 2006 NSF-REU program by rising sophomore physics major Martha Roseberry.

Roseberry created the cover image using a network of computers in Taylor Hall (assisted by rising sophomore math and physics double major Even Heidtmann, who operated the computer cluster). The colorful graphic, which resembles a psychedelic whirlpool affectionately known as "the vortex", demonstrates the dynamic complexity of the motion of just two attractive particles in a two-dimensional spherical universe (resembling ants crawling on the surface of a basketball). The colors code the particles' final separations as a function of initial separation and initial relative speed.

"We're thrilled to be able to say something new about such a classic problem," said Lindner. "It's a fascinating project that demonstrates how to sustain a novel research program by collaborating with a succession of talented undergraduates, without whom it would not have happened."