Improbability results for non-collision orbits of Xia type
Volume 601 / 2025
Abstract
Non-collision singularities of the $n$-body problem are initial conditions without global solution, that however do not lead to collision in the limit. The question whether the set of non-collision singularities in the $n$-body problem is improbable, is open and the first in Barry Simon’s list of fifteen problems in mathematical physics from the year 1984. By now this question is only answered affirmatively in the case of $n=4$ bodies. We cannot answer the full question for five or more particles, but can give two different kinds of partial answers proving that some suitable subsets of the set of non-collision singularities are improbable.
One case is similar to the case of a non-collision singularity with four particles, but there might be more particles that do not come close to the diverging subsystem close to the escape time and have at most binary collisions at the escape time.
The second improbability result is for orbits similar to the first examples of non-collision singular orbits, constructed by Xia in 1992. They consist of five particles with two outer binaries and a messenger commuting between the binaries. Under a suitable mass restriction we can prove the improbability of these orbits.
The tool for proving these improbability results is the so-called Poincaré surface method developed by Fleischer (2019). The argument is based on a fine analysis of the orbits and deriving quantitative estimates at suitable reference times during the passages. The results are some application and extension of results proved in the author’s 2023 dissertation.
