Special Session 38: Recent advances in the n-body problem

On behavior of solutions near collision singularities

Kuo-Chang Chen National Tsing Hua University Taiwan Co-Author(s):

Abstract: Near an isolated collision cluster, classical estimates by Sundman (1913) and Sperling (1970) are well-known to the celestial mechanics community. In this talk we will show a simpler approach based on some ODE techniques, and show some connections with elliptic partial differential equations.

Melnikov Method for Non-Conservative Perturbations of the Restricted Three-Body Problem

Marian Gidea Yeshiva University USA Co-Author(s):    Rafael de la Llave; Maxwell Musser

Abstract: We study the effect of small, time-dependent, non-conservative perturbations on homoclinic orbits to a normally hyperbolic invariant manifold in the planar circular restricted three-body problem. The homoclinic orbits can be described via the scattering map, which gives the future asymptotics of an orbit as a function of the past asymptotics. We add a time-dependent, non-conservative perturbation, and provide explicit formulas, in terms of convergent integrals, for the perturbed scattering map. The motivation of this work comes from astrodynamics. Low-energy space missions are often designed to follow hyperbolic invariant manifolds at constant energy. Applying an orbital maneuver amounts to adding a time-dependent non-conservative perturbation. We are interested in quantifying the effect of the maneuver on the orbit.

Braids, metallic ratios and periodic solutions of the 2n-body problem

Yuika Kajihara Kyoto University Japan Co-Author(s):    Eiko Kin and Mitsuru Shibayama

Abstract: Periodic solutions of the planar N-body problem determine braids through the trajectory of N bodies. Braid types can be used to classify periodic solutions. According to the Nielsen-Thurston classification of surface automorphisms, braids fall into three types: periodic, reducible and pseudo-Anosov. To a braid of pseudo-Anosov type, there is an associated stretch factor greater than 1, and this is a conjugacy invariant of braids. In 2006, Shibayama discovered a family of multiple choreographic solutions of the planar 2n-body problem. We prove that braids obtained from the solutions in the family are of pseudo-Anosov type, and their stretch factors are expressed in metallic ratios.

Existence of transit orbits in the planar restricted 3-body problem via variational methods

Taiga Kurokawa Kyoto University Japan Co-Author(s):    Mitsuru Shibayama

Abstract: We study the planar restricted 3-body problem (PR3BP). Although many numerical studies suggest the existence of transit orbits, few mathematical results have demonstrated their existence. For the case of two bodies in circular motion (PCR3BP), Moeckel (2005) provided a sufficient condition for their existence by minimizing Maupertuis` functional. However, in the case of elliptic motion (PER3BP), this variational structure does not apply because the system is non-autonomous, and no variational results had been known. We provide a different sufficient condition for PCR3BP by minimizing Lagrange`s functional without fixing time. Furthermore, we found that this variational structure is also applicable to non-autonomous systems, allowing us to establish a sufficient condition for PER3BP. We also numerically confirm that these sufficient conditions hold in specific cases where the two bodies have equal mass. In this talk, we will present these results.

Existence and nonexistence of first integrals near integral curves with finite time

Shoya Motonaga Ritsumeikan University Japan Co-Author(s):

Abstract: We consider existence and nonexistence of first integrals near integral curves with finite time for autonomous dynamical systems. We characterize how many first integrals can exist near periodic orbits. This characterization is an improvement of Poincar\`{e}`s classical criterion using the variational equation near periodic orbits. Moreover, we show that there is a flow-box coordinate for non-periodic integral curves with finite time.

Distance estimates for action-minimizing solutions of the n-body problem

Bo-Yu Pan Department of Applied Mathematics, National Chung Hsing University, Taiwan Taiwan Co-Author(s):    Kuo-Chang Chen

Abstract: In this talk, we estimate mutual distances of action minimizing solutions for the n-body problem. We will present some quantitative estimates for these solutions, including their action values and bounds for their mutual distances. These estimates will facilitate numerical explorations to locate and search for new orbits effectively.

Longterm inspection of orbits of a highly inclined triples system: a hierarchy exchange process including the ZKL mechanism

Masaya Saito University of Nagasaki, Siebold Japan Co-Author(s):    Masaya M Saito, Kiyotaka Tanikawa

Abstract: The gravitationally interacting three bodies are called hierarchical when their orbits are well approximated by double Kepler orbits which do not intersect each other. The presence of the mean motion resonances (MMRs), which have overlapped part, contributes the chaotic nature of the system. In highly inclined systems another process comes to the system, called the von Zeipel-Kozai-Lidov (ZKL) oscillation, that is, anti-correlated oscillation between the inner orbitfs eccentricity $e_1$ and mutual inclination $I$. Potentially, a configuration of higher $e_1$ and lower $I$ would lead a close approach and a member-change of hierarchy. In order to understand how systems lose their hierarchical stability under both the effects of MMRs and the ZKL mechanism, we will carry out a detailed inspection of a single orbit. Introducing three auxiliary quantities about the two outer bodies, which account the closeness, the alignment of two pericenters, and synchronous rotation around the central, we found via timecourse analysis that the ZKL mechanism and secular drift of the pericenters finally broke the hierarchy of the system.

Variational Construction of Orbits Realizing Symbolic Sequences in the Planar Sitnikov Problem

Mitsuru Shibayama Kyoto University Japan Co-Author(s):

Abstract: We study the limiting case of the Sitnikov problem. By using the variational method, we show the existence of various kinds of solutions in the planar Sitnikov problem. For a given symbolic sequence, we show the existence of orbits realizing it.

Some results of the enumeration problems for point vortex equilibria

Ya-Lun Tsai National Chung Hsing University Taiwan Co-Author(s):

Abstract: Point vortices in a plane form a dynamical system introduced by Helmholtz in 1858. A vortex equilibrium according to O`Neil is a solution where all vortices move with a common velocity, where the configuration formed by the vortices is stationary or translating depending on whether the velocity is zero or not. In this talk, we will present some results of the enumeration problems for $n$-vortex translating configurations and $n$-vortex stationary configurations. Especially, for $n\geq 4$, we will show there exist circulations yielding no translating configurations. Similarly, for $n\geq 5$, there are circulations yielding no stationary configurations. Then, such circulations satisfying the necessary conditions for vortex equilibria but yielding no vortex equilibria will be generalized.

Progress on four-body central configurations

Zhifu Xie The University of Southern Mississippi USA Co-Author(s):    Shanzhong Sun and Peng You

Abstract: Central configurations play crucial roles in comprehending the solution structure and dynamic behavior of the N-body problem. The renowned Chazy-Wintner-Smale conjecture states that, for any given positive masses, there exists only a finite number of central configurations. While the conjecture has been verified for the planar 4-body problem and partially for the planar 5-body problem, its generalization remains a formidable challenge for 21st-century mathematicians. The precise enumeration of central configurations for the 4-body problem remains an ongoing pursuit. Particularly the classifications of planar 4-body central configurations have attracted numerous researchers exploring various notions of symmetry within configuration and mass spaces. In this presentation, we will review the findings regarding central configurations for the four-body problem, as well as our recent contributions to this field.

Nonintegrability of dynamical systems near degenerate equilibria

Kazuyuki Yagasaki Kyoto University Japan Co-Author(s):

Abstract: In this talk, we prove that general three- or four-dimensional systems are real-analytically nonintegrable near degenerate equilibria in the Bogoyavlenskij sense under additional weak conditions when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues or two incommensurate pairs of purely imaginary eigenvalues at the equilibria. For this purpose, we reduce their integrability to that of the corresponding Poincar\{e}-Dulac normal forms and further to that of simple planar systems, and use a novel approach for proving the analytic nonintegrability of planar systems. Our result also implies that general three- and four-dimensional systems exhibiting fold-Hopf and double-Hopf codimension-two bifurcations, respectively, are real-analytically nonintegrable under the weak conditions. To demonstrate these results, we give two examples for the R\{o}ssler system and coupled van der Pol oscillators.