March 3, 2010
A strobed animation of the orbit of a slash and dot through space and time.
WOOSTER, Ohio - Isaac Newton mastered the orderly two-body problem (the motion of two spherical objects gravitationally interacting in space, such as the moon and the earth), but he struggled with the unpredictability and chaos of the three-body problem (the introduction of a third spherical object, such as the sun). And he may have missed a fascinating problem lurking in between.
John Lindner, professor of physics at The College of Wooster, and Wooster students Frank King and Amanda Logue (along with summer student Jacob Lynn) have devised the slashdot-body problem, or "/." for short, which involves the gravitational interaction of a line and a point. "The slashdot-body problem exhibits some of the richness of the three-body problem with only two bodies," says Lindner. "It demonstrates the interplay between rotation (spin) and revolution (orbit) in the interaction of physical bodies like asteroids and moonlets, which is absent in the corresponding two- and three-body problems. The spin of the slash can be a source or sink of energy
and angular momentum that can unbind the system in cases where the corresponding point masses would remain bound." Their findings will be published in the March edition of the journal Physical Review, the flagship publication of the American Physical Society.
An example of the slashdot-body problem is the docking of a space shuttle to the International
Space Station. "The space station can be approximated by the slash, and the shuttle can be approximated by the dot," explains Lindner. "You can't treat both of them as points, as points have no shape, and shape is important in this situation."
Another example is the dynamics of asteroid-moonlet pairs, where one object is spherical and the other is often elongated. "If you have two point bodies, they can orbit each other," says Lindner. "If you have two different shapes, like a slash and dot, they can both orbit and spin." Thus the
interplay of rotation and revolution.
It's not known whether Newton considered the slashdot-body problem, but Lindner is certain that he would have been interested in its solution. In fact, Lindner and his students used Newton's laws of motion and gravity as well as a major resource that Newton did not have - computers - to calculate the slashdot force and torque and discover some of its complicated and chaotic trajectories.
"The slashdot-body problem is partially solvable with pencil and paper," says Lindner. "But without computers, Newton would not have been able to compute most of the trajectories." The many complex trajectories also provide an unintended byproduct - visually captivating images of the graceful pirouettes of the slash and the dot. Of course the greatest benefit is the base of knowledge provided by the research, which will help engineers and physicists better understand irregular objects in space. "To our knowledge, we have pioneered this problem," says Lindner. "And we hope it becomes a canonical example that is incorporated into future textbooks."
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