Variational Persistence of Periodic and Quasi-Periodic Orbits in Symmetric $N$-Body Problems

Many mechanical systems, such as the Newtonian $n$-body system, are invariant under rotations. The rotational symmetry of the system gives rise to two commuting actions: the dynamical action and the action of the symmetry group. In this paper, we exploit this symmetry to construct families of periodic and quasi-periodic trajectories. In the strong-force setting, we extend Montgomery’s variational approach to free homotopy classes of curves that are periodic up to a prescribed nontrivial rotation. In the weak-force setting, we establish, under suitable hypotheses, global and local variational persistence results for action minimizers in loop spaces defined by general finite symmetry-group actions under sufficiently small rotations. As applications, we apply these results to the figure-eight orbit, under a numerically supported assumption of strict local minimality, to double choreographic loops with $q$-fold rotation symmetry, and to counter-rotating double choreographic loops in planar $(N+3)$-body problems.