Numerical search for three-body periodic free-fall orbits with central symmetry

A specialized high-precision numerical search for equal-mass collisionless three-body periodic free-fall orbits with central symmetry is conducted. The search is based on Newton’s method with initial approximations obtained by the grid-search method. Instead of solving the standard periodicity equation on the entire period a quarter-period equation that also characterizes the periodic orbits is solved. The number of the known orbits from the class is significantly enlarged. The linear stability of the orbits is also investigated. All of them are unstable. A discussion in relation to the efficiency of Newton’s method applied with grid-search initial approximations is held.