Celestial Mechanics

By Helmut Hofer and Derek Bermel 

This image (produced with a Java applet by Alec Jacobson at http://alecjacobåson.com/programs/three-body-chaos) shows color­ful trackings of the paths of satellites as they evolve from a simple single orbit to a complex multicolored tangle of orbits.

I can’t understand why people are frightened of new ideas. I’m frightened of the old ones.—John Cage

Helmut Hofer, Professor in the School of Mathematics, writes:

Last September, the School of Mathematics launched its yearlong program with my Member seminar talk “First Steps in Symplectic Dynamics.” About two years earlier, it had become clear that certain important problems in dynamical systems could be solved with ideas coming from a different field, the field of symplectic geometry. The goal was then to bring researchers from the fields of dynamical systems and symplectic geometry together in a program aimed at the development of a common core and ideally leading to a new field—symplectic dynamics.

Not long before, in my 2010 inaugural public lecture at IAS, “From Celestial Mechanics to a Geometry Based on the Concept of Area,” I had described the historical background and some of the interesting mathematical problems belonging to this anticipated field of symplectic dynamics. The lecture began with a computer program showing chaos in the restricted three-body problem. This problem describes the movement of a satellite under the gravity of two big bodies, say the earth and the moon, in a rotating coordinates system in which the earth and the moon stay at fixed positions. The chaos in the system is illustrated by putting about ten satellites initially at almost the same position with almost the same velocity.

When the system starts evolving, the program shows colorful trackings of the paths of the satellites as they evolve from a simple single orbit to a complex multicolored tangle of orbits, once the orbits of the different satellites start separating.

By Scott Tremaine 

Scott Tremaine explores the stability of our solar system, one of the oldest problems in theoretical physics, dating back to Isaac Newton.

The stability of the solar system is one of the oldest problems in theoretical physics, dating back to Isaac Newton. After Newton discovered his famous laws of motion and gravity, he used these to determine the motion of a single planet around the Sun and showed that the planet followed an ellipse with the Sun at one focus. However, the actual solar system contains eight planets, six of which were known to Newton, and each planet exerts small, periodically varying, gravitational forces on all the others.

The puzzle posed by Newton is whether the net effect of these periodic forces on the planetary orbits averages to zero over long times, so that the planets continue to follow orbits similar to the ones they have today, or whether these small mutual interactions gradually degrade the regular arrangement of the orbits in the solar system, leading eventual ly to a collision between two planets, the ejection of a planet to interstellar space, or perhaps the incineration of a planet by the Sun. The interplanetary gravitational interactions are very small—the force on Earth from Jupiter, the largest planet, is only about ten parts per million of the force from the Sun—but the time available for their effects to accumulate is even longer: over four billion years since the solar system was formed, and almost eight billion years until the death of the Sun.