A continuum of periodic solutions to the four-body problem with various choices of masses

Abstract

In this paper, we apply the variational method with the Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for planar four-body problem with $m_1=m_3$ and $m_2=m_4$. A path $q(t)$ in $[0,T]$ satisfies SPBC if the boundaries $q(0)\in \mathbf{A}$ and $q(T)\in \mathbf{B}$, where $\mathbf{A}$ and $\mathbf{B}$ are two structural configuration spaces in $(\mathbf{R}^2)^4$ and they depend on a rotation angle $θ\in (0,2π)$ and the mass ratio $μ=\frac{m_2}{m_1}\in \mathbf{R}^+$. We show that there is a region $Ω\subseteq (0,2π)\times R^+$ such that there exists at least one local minimizer of the Lagrangian action functional on the path space satisfying SPBC $\{q(t)\in H^1([0,T],$ $(\mathbf{R}^2)^4)| $ $q(0)\in $ $\mathbf{A}, q(T)\in $ $\mathbf{B}\}$ for any $(θ,μ)\in Ω$. The corresponding minimizing path of the minimizer can be extended to a non-homographic periodic solution if $θ$ is commensurable with $π$ or a quasi-periodic solution if $θ$ is not commensurable with $π$. In the variational method with SPBC, we only impose constraints on boundary and we do not impose any symmetry constraint on solutions. Instead, we prove that our solutions extended from the initial minimizing pathes have the symmetries. The periodic solutions can be further classified as simple choreographic solutions, double choreographic solutions and non-choreographic solutions. Among the many stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution when $(θ,μ)=(\frac{4π}{5},1)$. Remarkably the unequal-mass variants of the stable star pentagon are just as stable as the basic equal mass choreography (See figure 1).