Time-renormalization for the search of periodic solutions to the three-body problem

Abstract

Researchers Antoñana et al. developed a technique for global time-renormalization of the gravitational N-body problem. In their paper, it is speculated that it may be useful for finding periodic orbits, but they do not perform any experiments to test this hypothesis. Influenced by their work, the aim of this project is to find planar three-body choreogra- phies of different topologies. This project takes a lot of inspiration from Simó’s work done on N-body choreographies and the figure eight. In his paper, he proposes efficient methods for finding planar chore- ographies. The main driver of our work is that some of the problems Simó faced in his work could be lessened by making use of global time-regularization. Two completely different approaches were taken to tackle the problem. The first approach consists of finding choreographies by solving an optimization problem in the space of so- lutions to the differential equations of Newton’s law of gravitation. The second approach involves generating curves with the desired topology, and then using variational calculus to find solutions that satisfy Newton’s laws. With the first approach, we found thousands of choreographies of many different topolo- gies. We also managed to show that the second approach is viable, although the results were not anywhere close those of the first approach. Experiments showed that global time renormalization reduces the number of Fourier coefficients for curve representation. It was also experimentally verified that the integration of differential equations was much more accurate with time-renormalization when using a constant step-size. Two conclusions can be drawn from the results of the experiments. For one, both ap- proaches greatly benefit from global time-renormalization. Secondly, our first approach is more effective than the second in finding the most choreographies. However, the ability to control the topology of the solutions is limited with the first approach, and not with the second.