# Registered Data

## [00475] Variational methods and periodic solutions in the n-body problem

**Session Date & Time**:- 00475 (1/3) : 1C (Aug.21, 13:20-15:00)
- 00475 (2/3) : 1D (Aug.21, 15:30-17:10)
- 00475 (3/3) : 1E (Aug.21, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: The n-body dynamics have been studied by many prominent mathematicians and physicists for centuries. With the developments of mathematical and computational tools, there has been exciting progress during the past two decades. This progress includes variational approaches, stability, chaotic phenomenon, integrability, central configurations, solar system, space mission designs, and planetary formations, among many others. In this minisymposium, we aim to provide a forum for researchers to share the latest developments and exchange ideas.**Organizer(s)**: Mitsuru Shibayama**Classification**:__70F10__,__37D35__,__70M20__,__70G75__**Speakers Info**:**Mitsuru Shibayama**(Kyoto University)- Bo-Yu Pan (National Chung-Hsing University)
- Takuya Chikazawa (The University of Tokyo)
- Taiga Kurokawa (Kyoto University)
- Hiroshi Fukuda (Kitasato University)
- Naoki Hiraiwa (Kyushu University)
- Eiko Kin (Osaka University)
- Yuika Kajihara (Kyoto University)
- Guowei Yu (Nankai University)
- Kenta Oshima (Hiroshima Institute of Technology)
- Toshihiro Chujo (Tokyo Institute of Technology)
- Kuo-Chang Chen (National Tsing Hua University)

**Talks in Minisymposium**:**[03430] Braids and periodic solutions of the planar N-body problem****Author(s)**:**EIKO KIN**(Osaka University)- Yuika Kajihara (Kyoto University)
- Mitsuru Shibayama (Kyoto University)

**Abstract**: Periodic solutions of the planar N-body problem determine braids through the trajectory of N bodies. Braids fall into three types: periodic, reducible and pseudo-Anosov. The last type is significant for the study of dynamical systems. In this talk I discuss a family of braid types obtained from periodic solutions, simple choreographies of the chain types by Guowei Yu and multiple choreographic solutions of the planar 2n-body problem by Shibayama.

**[03812] Periodic and homo/heteroclinic solutions of the restricted three-body problem****Author(s)**:**Mitsuru Shibayama**(Kyoto University)

**Abstract**: The restricted three-body problem is an important research area that deals with significant issues in celestial mechanics, such as analyzing asteroid movement behavior and orbit design for space probes. We aim to show the existence of periodic and heteroclinic orbits in the planar circular R3BP. To find these orbits, we adopt a variational approach and symmetry.

**[03894] regularizable collinear periodic solutions in the n-body problem with arbitrary masses****Author(s)**:**Guowei Yu**(Nankai University)

**Abstract**: For n-body problem with arbitrary positive masses, we prove there are regularizable collinear periodic solutions for any ordering of the masses, going from a simultaneous binary collision to another in half of a period with half of the masses moving monotonically to the right and the other half monotonically to the left. When the masses satisfy certain equality condition, the solutions have extra symmetry. This also gives a new proof of the Schubart orbit, when n=3.

**[03973] Distance estimates for action-minimizing solutions of the n-body problem****Author(s)**:**Bo-Yu Pan**(National Chung-Hsing University)

**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 new orbits effectively.

**[04028] Transfer between Resonances via Lobe Dynamics in the Standard Map****Author(s)**:**Naoki Hiraiwa**(Kyushu University)- Isaia Nisoli (Universidade Federal do Rio de Janeiro)
- Yuzuru Sato (Hokkaido University)
- Mai Bando (Kyushu University)
- Shinji Hokamoto (Kyushu University)

**Abstract**: Lobe dynamics is a useful structure to reveal phase space transport of chaotic trajectories by stable and unstable manifolds of resonant orbits. Based on lobe dynamics, this study formulates the transfer problem between two quasi-periodic orbits to find the optimal lobe sequence. Especially, the lobe dynamics of resonant orbits of the standard map are extracted and used to solve the problem. Applications to spacecraft trajectory design are also discussed.

**[04108] Regularizing Fuel-Optimal, Multi-Impulse Trajectories with Second-Order Derivatives****Author(s)**:**Kenta Oshima**(Hiroshima Institute of Technology)

**Abstract**: The present work implements analytical second-order derivatives for a direct multiple shooting-based regularized method of minimizing the fuel expenditure for spacecraft trajectories. The high-order dynamical information, such as the state transition tensor, expresses the Hessian matrix of the Lagrange function in the nonlinear programming problem. The result is an efficient tool for robustly and accurately computing fuel-optimal, multi-impulse trajectories in the regularized framework of removing singularities associated with null thrust impulses.

**[04318] Periodic solutions bifurcated from the figure-eight choreography: non-planar eight and non-symmetric eight****Author(s)**:**Hiroshi Fukuda**(Kitasato University)- Toshiaki Fujiwara (Kitasato University)
- Hiroshi Ozaki (Tokai University)

**Abstract**: The figure-eight choreography of equal mass three bodies under the homogeneous potential $-1/r^\alpha$ and under the Lennard-Jones type potential $1/r^{12}-1/r^6$, bifurcates in power $\alpha$ and in period T, respectively, where $r$ is a distance between bodies. We found two interesting bifurcation solutions: a figure-eight choreography with an orbit having no spatial symmetry, and a non-planar figure-eight solution which is unfortunately not choreographic.

**[04549] Low-energy Transfer to the Earth-Moon Periodic Orbit: CubeSat Application****Author(s)**:**Takuya Chikazawa**(The University of Tokyo)

**Abstract**: This work investigates trajectories design for the ride-share spacecraft that begin with the Moon swing-by. In this type of trajectory design, mission designers need to consider uncertainties under limited propulsion capability. To aid trajectories design, established theory, such as manifold from periodic orbit in circular restricted three-body system, are often used. We demonstrate how to use such theory in actual mission design and operation phases.

**[04642] Floquet Mode-Based Transfer between Halo Orbits Using Solar Sails****Author(s)**:**Toshihiro Chujo**(Tokyo Institute of Technology)

**Abstract**: Transfer between halo orbits around the sun-Earth L2 using solar sails is discussed in the circular restricted three-body problem. The path planning and the corresponding tilting angle of the sail are determined based on the Floquet mode, such that the coefficient of the center manifold for transition to another family of the halo orbit is maximized while that of the unstable manifold is suppressed under a certain threshold.

**[04714] Existence of transit orbits in the restricted three-body problem****Author(s)**:**Taiga Kurokawa**(Kyoto University)- Mitsuru Shibayama (Kyoto University)

**Abstract**: We discuss the existence of transit orbits with fixed energy in the planar circular restricted three-body problem. In 2005, Moeckel provided sufficient conditions for the existence of transit orbits using the Maupertuis functional. In this talk, we give another sufficient condition using the Lagrange functional. This is joint work with Mitsuru Shibayama.

**[04797] Variational structures for infinite transition orbits of monotone twist maps****Author(s)**:**Yuika Kajihara**(Kyoto university)

**Abstract**: There is a lot of study on the dynamics of area-preserving maps, and Poincare and Birkhoff's works are well-known. In this talk, we define a special class of area-preserving maps called monotone twist maps to consider the variational structures of area-preserving maps. Variational structures determined from twist maps can be used for constructing characteristic trajectories of twist maps. Our goal is to define the variational structure such as giving infinite transition orbits through minimizing methods.

**[04970] Some progress on the N-center problem by variational methods****Author(s)**:**Kuo-Chang Chen**(National Tsing Hua University)

**Abstract**: Since Chenciner-Montgomery’s construction of the figure-8 orbit for the 3-body problem, in the past 20+ years variational methods have succeeded in discovering new solutions for the N-body and N-center problems, within certain symmetry of topological classes. I will briefly outline recent progress, main ideas and obstacles, and some ongoing research topics for the N-center problem. In particular, I will outline variational construction of satellite orbits for N=2, and recent joint works with Guowei Yu on periodic and chaotic orbits for N>2.