# American Institute of Mathematical Sciences

September  2008, 10(2&3, September): 569-595. doi: 10.3934/dcdsb.2008.10.569

## Making the moon reverse its orbit, or, stuttering in the planar three-body problem

 1 Department of Mathematics, University of North Carolina Asheville, CPO#2350 Asheville, NC 28804-8511, United States 2 Department of Mathematics, Penn State, State College, PA 16801, United States 3 Mathematics Department, University of California, Santa Cruz, CA 95064, United States

Received  October 2006 Revised  November 2007 Published  June 2008

We investigate the planar three-body problem in the range where one mass, say the ‘sun’ is very far from the other two, call them ‘earth’ and ‘moon’. We show that “stutters” : two consecutive eclipses in which the moon lies on the line between the earth and sun, occur for an open set of initial conditions. In these motions the moon reverses its sense of rotation about the earth. The mechanism is a kind of tidal torque (see the ‘key equation’). The motivation is to better understand the limits of variational methods. The methods of proof are classical estimates and bounds in this asymptotic regime.
Citation: Samuel R. Kaplan, Mark Levi, Richard Montgomery. Making the moon reverse its orbit, or, stuttering in the planar three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 569-595. doi: 10.3934/dcdsb.2008.10.569
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