| Abstract: |
| In the past two decades, since the discovery of the figure-8 orbit by Chenciner and Montgomery,
the variational method has become one of the most popular tools for constructing new
solutions of the $N$-body problem and its extended problems. However, finding solutions to the restricted
three-body problem, in particular, the two primaries form a collision Kepler system, remains a
great difficulty. One of the major reasons is the essential differences between two-body collisions and
three-body collisions.
In this paper, we consider a similar three-body system with less difficulty, i.e. the restricted one-center-
two-body system, that is involving a massless particle and a collision Kepler system with
one body fixed. It is an intermediate system between the restricted three-body problem and the two-center
problem. By an in-depth analysis to the asymptotic behavior of the minimizer, and an argument
concerning critical and inflection points, we prove the Sundman-Sperling estimates near the three-body
collision for the minimizers.With these estimates, we provide a class of collision-free solutions
with prescribed boundary angles. Finally, under the extended collision Kepler system from Gordon,
we construct a family of periodic and quasi-periodic solutions. |
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