Special Session 43: Hamiltonian Dynamics and Celestial Mechanics

Weak Compactness Criterion in $W^{k,1}$ with an Existence Theorem of Minimizers

Cheng Chen Sichuan University Peoples Rep of China Co-Author(s):    Mattie Ji, Yan Tang and Shiqing Zhang

Abstract: Nelson Dunford and Billy James Pettis [{\em Trans. Amer. Math. Soc.}, 47 (1940), pp. 323--392] proved that relatively weakly compact subsets of $ L^1 $ coincide with equi-integrable families. We expand it to the case of $ W^{k,1} $ - the non-reflexive Sobolev spaces - by a tailor-made isometric operator. Herein we extend an existence theorem of minimizers from reflexive Sobolev spaces to non-reflexive ones.

On finiteness of central configurations by symbolic computations

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

Abstract: Self-similar solutions for the n-body problem, whose configurations are known as central configurations, are of special importance in celestial mechanics. Its finiteness is a long standing open problem. In this talk we will briefly outline some breakthroughs in the past two decades, in particular Hampton-Moeckel`s work for the case n=4 by (Invent. Math. 2006), and Albouy-Kaloshin`s work for the case n=5 (Ann. Math. 2012). In this talk we will report our recent progress for the case n=6 by symbolic computations.

Exploration of billiards with Keplerian potential

Anna Maria Cherubini University of Salento Italy Co-Author(s):    Vivina L. Barutello and Irene De Blasi

Abstract: We study a class of elliptic billiards with a Keplerian potential inside, considering two cases: a reflective one, where the particle reflects elastically on the boundary, and a refractive one where the particle can cross the billiard`s boundary entering a region with a harmonic potential. In the latter case, the dynamics is given by concatenations of inner and outer arcs, connected by a refraction law. In recent papers these billiards have been extensively studied in order to identify which conditions give rise to either regular or chaotic dynamics. We complete the study by analysing the non focused reflective case. We then analyse the focused and non focused refractive case, where no results on integrability are known except from the centred circular case, by providing an extensive numerical analysis. We present also a theoretical result regarding the linear stability of homotetic equilibrium orbits in the reflective case for general ellipses, highlighting the possible presence of bifurcations even in the integrable case.

An efficient approach to the design of low-energy transfers in n-body systems

Elena Fantino Khalifa University of Science and Technology United Arab Emirates Co-Author(s):

Abstract: The invariant structures of the circular restricted three-body problem (CR3BP) have been used successfully to design transfers in the Earth-Moon-Sun system and tours of giant planet moons. At the same time, space mission concept studies often seek the computational efficiency and the geometrical insight offered by the two-body problem. Merging the two approaches leads to a design technique, known as the patched three-body/two-body model, that facilitates the design of low-energy transfers in $n$-body systems. Trajectories associated with hyperbolic invariant manifolds of orbits around the L$_1$ and L$_2$ libration points are propagated in the CR3BP until they reach the surface of an {\it ad hoc} sphere of influence where the state vectors are transformed to osculating orbital elements relative to the larger primary. In this way, computing transfers between CR3BPs with a common primary turns into the search for intersections between confocal elliptical orbits. The contribution analyses this methodology and its benefits, and presents applications to the design of a tour of the Saturn system and rendezvous missions to Near Earth Objects.

Geometric properties of normally hyperbolic invariant manifolds for conformally symplectic systems

Marian Gidea Yeshiva University USA Co-Author(s):    Rafael de la Llave; Tere M-Sera

Abstract: Normally hyperbolic invariant manifolds (NHIMs) are ubiquitous in Hamiltonian systems, including models from celestial mechanics. However, real-life systems are often subject to dissipative forces. Examples from celestial mechanics include tidal forces, Stokes drag, Poynting-Robertson effect, Yarkowski/YORP effects, atmospheric drag. Adding a dissipation to a Hamiltonian system is a singular perturbation that radically changes its long term behavior. In this work, we study geometric properties of NHIMs for conformally symplectic systems, which model mechanical systems with friction proportional to velocity. We show that certain conditions among rates and the conformal factor are equivalent to the NHIM being symplectic. Specifically, we show that the hyperbolicity rates for symplectic NHIMs satisfy pairing rules similar to those for Lyapunov exponents and eigenvalues of periodic orbits.

A generalized mountain pass lemma with a closed subset for locally Lipschitz functionals

Fengying Li Southwestern University of Finance and Economics Peoples Rep of China Co-Author(s):    Fengying Li, Bingyu Li, Shiqing Zhang

Abstract: The classical Mountain Pass Lemma of Ambrosetti-Rabinowitz has been studied, extended and modified in several directions. Notable examples would certainly include the generalization to locally Lipschitz functionals by K. C. Chang, analyzing the structure of the critical set in the mountain pass theorem in the works of Hofer, Pucci-Serrin and Tian, and the extension by Ghoussoub-Preiss to closed subsets in a Banach space with recent variations. In this paper, we utilize the generalized gradient of Clarke and Ekeland`s variatonal principle to generalize the Ghoussoub-Preiss`s Theorem in the setting of locally Lipschitz functionals. We give an application to periodic solutions of Hamiltonian systems.

On the uniqueness of the planar 5-body central configuration with a trapezoidal convex hull

Yangshanshan Liu Chern Institute of Mathematics at Nankai University Peoples Rep of China Co-Author(s):    Yangshanshan Liu&Shiqing Zhang

Abstract: In order to apply Morse`s critical point theory, we use mutual distances as coordinates to discuss a kind of central configuration of the planar Newtonian 5-body problem with a trapezoidal convex hull, i.e., four of the five bodies are located at the vertices of a trapezoid, and the fifth one is located on one of the parallel sides. We show that there is at most one central configuration of this geometrical shape for a given cyclic order of the five bodies along the convex hull. In addition, if the parallel side containing the three collinear bodies is strictly shorter than the other parallel side, the configuration must be symmetric, i.e., the trapezoid is isosceles, and the last body is at the midpoint of the shorter parallel side.

The symplectic geometry of the restricted three-body problem

Agustin Moreno Heidelberg University Germany Co-Author(s):

Abstract: In this talk, I will survey recent advances in the classical (circular, restricted) three-body problem, from the perspective of symplectic geometry, and discuss applications to trajectory design. Based on joint work with several authors: Otto van Koert, Urs Frauenfelder, Dan Scheeres, Cengiz Aydin, Dayung Koh, Gavin Brown.

Real-analytic nonintegrability of nearly integrable systems and Melnokov method

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

Abstract: We study necessary conditions for existence of real-analytic first integrals and real-analytic integrability for perturbations of integrable systems in the sense of Bogoyavlenskij including non-Hamiltonian ones. Moreover, we compare our results with classical results of Poincar\`{e} and Kozlov for systems written in action and angle coordinates and discuss their relationships with Melnikov methods for periodic perturbations of single-degree-of-freedom Hamiltonian systems. The latter discussion reveals that the perturbed systems can be real-analytically nonintgrable even if there exists no transverse homoclinic orbit to a periodic orbit. This is a joint work with Kazuyuki Yagasaki (Kyoto University).

Semi-analytical exploration of drift trajectories near $L_1$ in the Spatial RTBP

Pablo Roldan Universitat Politecnica de Catalunya Spain Co-Author(s):    Amadeu Delshams and Marian Gidea

Abstract: Consider the spatial restricted three-body problem, as a model for the motion of a spacecraft relative to the Sun-Earth system. We focus on the dynamics near the equilibrium point $L_1$, located between the Sun and the Earth. We show that we can make the spacecraft transition from an orbit that is nearly planar relative to the ecliptic, to an orbit that has large inclination, at zero energy cost. (In fact, the final orbit has the maximum inclination that can be obtained through the particular mechanism that we consider. Moreover, the transition can be made through any prescribed sequence of inclinations in between). We provide several explicit constructions of such orbits, and also develop an algorithm to design orbits that achieve the \emph{shortest transition time} for this particular mechanism. Our main new tool is the `Standard Scattering Map` (SSM), a series representation of the exact scattering map. The SSM can be used in many other situations, from Arnold diffusion problems to transport phenomena in applications.

Computational symplectic topology and the restricted three-body problem

Otto van Koert Seoul National University, Department of Mathematics Korea Co-Author(s):    Chankyu Joung

Abstract: In this talk we will give an overview of applications of methods from symplectic topology to Hamiltonian dynamical systems. In particular, we describe how information about periodic orbits and global surfaces of section can be obtained. After that we will outline how to apply validated numerics to get concrete results concerning the restricted three-body problem. This is joint work with Chankyu Joung.

Investigation of Bifurcations of Central Configurations

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

Abstract: The system of central configuration equations includes invariants such as scaling, rotation, and translation, which introduce trivial zero eigenvalues to the Jacobian matrix for a central configuration. To determine if a central configuration undergoes a bifurcation, it is crucial to identify if there is a non-trivial zero eigenvalue. In this talk, we will discuss methods to isolate these trivial zero eigenvalues, allowing us to compute the determinant of the remaining matrix. If this determinant is non-zero, it indicates that no bifurcation occurs.

Global dynamics of the N-body problem.

Jinxin Xue Tsinghua University Peoples Rep of China Co-Author(s):    Guan Huang, J. Gerver

Abstract: We explain our construction of noncollision singularities and superhyperbolic orbits and show that how these special orbits play an role in understanding the generic global dynamics of the N-body problem.

Nonintegrability of the restricted three-body problem

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

Abstract: The problem of nonintegrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincar\`{e} in the nineteenth century: He showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the first one and is functionally independent of the Hamiltonian. When the mass of the second body becomes zero, the restricted three-body problem reduces to the two-body Kepler problem. We prove the nonintegrability of the restricted three-body problem both in the planar and spatial cases for any nonzero mass of the second body. Our basic tool of the proofs is a technique developed here for determining whether perturbations of integrable systems which may be non-Hamiltonian are not meromorphically integrable near resonant periodic orbits such that the first integrals and commutative vector fields also depend meromorphically on the perturbation parameter. The technique is based on generalized versions due to Ayoul and Zung of the Morales-Ramis and Morales-Ramis-Sim\`{o} theories.

A symplectic dynamics approach to the spatial isosceles three-body problem

Guowei Yu Chern Institute of Math, Nankai University Peoples Rep of China Co-Author(s):    Xijun Hu, Lei Liu, Yuwei Ou, Pedro Salomao

Abstract: In this talk, we consider the spatial isosceles three body problem. For certain choices of energy and angular momentum, the dynamics on the energy surface is equivalent to a Reeb flow on the tight three-sphere. We find a Hopf link formed by the Euler orbit and a symmetric brake orbit, which spans an open book decomposition whose pages are annulus-like global surfaces of section. The convexity and non-convexity of the energy surface will also be discussed. Then we will adress the dynamical consequences of these facts, in particular the existence of periodic solutions.

Selection principle of generalized Hamilton-Jacobi equations

Jianlu ZHANG Academy of Mathematics and Systems Science Peoples Rep of China Co-Author(s):

Abstract: In 1987, Lions firstly proposed the homogenization for Hamilton-Jacobi equations, which revealed the significance of effective Hamiltonian in controlling the large time behavior of solutions. He also pointed out a vanishing discount procedure which is equivalent in obtaining the effective Hamiltonian, yet the convergence of solutions in this procedure was unknown until recently. In a bunch of joint works, we verified this convergence by using dynamical techniques.