Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem. (English) Zbl 1364.70024
Summary: In this paper, we give a rigorous proof of the existence of infinitely many simple choreographic solutions in the classical Newtonian 4-body problem. These orbits are discovered by a variational method with structural prescribed boundary conditions (SPBC). This method provides an initial path that is obtained by minimizing the Lagrangian action functional over the SPBC. We prove that the initial path can be extended to a periodic or quasi-periodic solution. With computer assistance, a family of choreographic orbits of this type is shown to be linearly stable. Among the many linearly stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution. We also prove the existence of infinitely many double choreographic periodic solutions, infinitely many non-choreographic periodic solutions and uncountably many quasi-periodic solutions. Each type of periodic solutions has many stable solutions and possibly infinitely many stable solutions. Our results with SPBC largely complement the current results by minimizing the action on a loop space.
MSC:
| 70F10 | \(n\)-body problems |
| 70K42 | Equilibria and periodic trajectories for nonlinear problems in mechanics |
| 70K20 | Stability for nonlinear problems in mechanics |
| 70H05 | Hamilton’s equations |
Keywords:
variational method; choreographic periodic solutions; variational method with structural prescribed boundary conditions; stability; central configurations; \(n\)-body problemReferences:
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