The recent discovery of thirteen new and distinct three-body periodic planar orbits suggests that many more such orbits remain undiscovered. Searches in two-dimensional subspaces of the full four-dimensional space of initial conditions require computing resources that are available to many students, and the required level of computational and numerical expertise is also at the advanced undergraduate level. We discuss the methods for solving the planar three-body equations of motion, as well as some basic strategies and tactics for searches of periodic orbits. Our discussion should allow interested undergraduates to start their own searches. Users can submit new three-body orbits to a wiki-based website.
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We assume, without loss of generality, that the trajectory initially lies above the equator (see Fig. 11) and follows one of the upper directed semi-circles described by one of the caligraphic letters . At each equator crossing the subset of caligraphic letters that describe the trajectory changes from the “upper” subset to the “lower” subset and vice versa. We need to count the number of equator-crossings, starting from the chosen beginning of the orbit and use the parity of this number to determine which of these two subsets is the correct descriptor at any given time.
46.
Segment 2 does not correspond to the choice of the initial (Euler) configuration made in Sec. III E. Rather the latter corresponds to segment 3. Renumbering of particles is straightforward, but it would lead to unnecessary complications in the definition/application of the Jacobi vectors.
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