Abstract
Poincaré made the first attempt in 1896 on applying variational calculus to the three-body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions. Little progress had been made on resolving this difficulty until a recent breakthrough by Chenciner and Montgomery. Afterward, variational methods were successfully applied to the $N$-body problem to construct new classes of solutions. In order to avoid collisions, the problem is confined to symmetric path spaces and all new planar solutions were constructed under the assumption that some masses are equal. A question for the variational approach on planar problems naturally arises: Are minimizing methods useful only when some masses are identical?
This article addresses this question for the three-body problem. For various choices of masses, it is proved that there exist infinitely many solutions with a certain topological type, called retrograde orbits, that minimize the action functional on certain path spaces. Cases covered in our work include triple stars in retrograde motions, double stars with one outer planet, and some double stars with one planet orbiting around one primary mass. Our results largely complement the classical results by the Poincaré continuation method and Conley’s geometric approach.
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