Special Session 42: Hamiltonian Dynamics and Celestial Mechanics

Reversing Symmetries and Periodic Motions in the Three-Body Problem on the Sphere

Abimael Bengochea Instituto Tecnologico Autonomo de Mexico Mexico Co-Author(s):    Ernesto Perez Chavela, Carlos Barrera Anzaldo

Abstract: We identify reversing symmetries for the three body problem on the sphere. This framework is applied to compute choreographies on the sphere, particularly those related to the eight figure choreography of the classical planar case. This approach is a useful tool for the search of periodic solutions and for studying the motion of bodies in spaces of constant curvature, highlighting the role of symmetry in such settings.

Mass constraints of some planar central configurations

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

Abstract: Determining relations between central configurations and masses is a challenging task in celestial mechanics. In this talk we will demonstrate some nontrivial techniques and criteria for the existence and uniqueness of some central configurations with 5 or 6 bodies. In particular, we will show criteria for rhomboidal and kite configurations with non-parallel masses.

A FAMILY OF LINEAR STABLE EQUILIBRIA IN THE SUN-EARTH-SAIL PROBLEM

Marcelo D Marchesin FEDERAL UNIVERSITY OF MINAS GERAIS Brazil Co-Author(s):

Abstract: The collinear libration point of the Sun-Earth Circular Restricted Three-Body Problem (CR3BP), L3 is located opposite to the Earth with respect to the Sun. Whereas several space missions have been launched to the other two collinear equilibrium points, i.e., L1 and L2, the region around L3 is so far unexploited essentially because of the severe communication limitations caused by Sun`s blocking location. By using an adequate size, location and attitude of a solar sail, the equilibrium point can be displaced from its original location to allow direct communication between the satellite and Earth. This paper presents several families of artificial equilibria located on the semi-space which is permanently opposite to Earth in relation to the Sun, but which allows direct communication with Earth. We present a family of such equilibria which are linearly stable and therefore very useful for space missions.

Dynamics and Symmetry: Minimizing brake orbits

Daniel Offin Queen`s University Canada Co-Author(s):    Henry Kavle

Abstract: The computation of the Conley-Zehnder index for periodic brake orbits of fixed energy E in classical mechanical systems can be resolved by decomposing the symplectic space of Jacobi fields along geodesic chords parameterized by Jacobi arclength, joining separated points on the Hill`s boundary. Our result gives this decomposition in terms of orthogonal Jacobi fields along the extended geodesic. In this talk, we will discuss properties of minimizing brake orbits.

Relative equilibria on spaces of constant curvature

Ernesto Perez Chavela ITAM Mexico Mexico Co-Author(s):    Toshiaki Fujiwara

Abstract: For $N=3$, it is well known that on the Euclidean space there are exactly five relative equilibria: three collinear (Euler relative equilibria) and two planar relative equilibria forming an equilateral triangle (Lagrange relative equilibria). What happen when we curved the Euclidian space in positive or negative way? The big difficulty to study relative equilibria on these spaces, that we call $RE$ by short, is the absence of the center of mass as a first integral, since many of the standard methods used in the classical case don`t apply any more. In this talk we will show a geometrical method to study $RE$ on these spaces, when the particles are moving under the influence of a general potential which only depends on the mutual distances among the masses. In particular we will show how to determine the rotation axis, which is a big obstacle to obtain $RE$. We restrict our analysis to the case $N=3$, where we give some new families of Euler and Lagrange $RE$ on the sphere for the cotangent potential (the natural extension of the Newtonian potential to the sphere).

C1 perturbations of a continuum of critical points

Antonio J Urena Universidad de Granada Spain Co-Author(s):    R. Ortega

Abstract: Given a real-valued function having a nondegenerate compact manifold of critical points, some of these points survive under small C2-perturbations. This is a well-known result in critical point theory. In 1986, Weinstein obtained the analogous conclusions when the perturbation is only C1 and the ambient space is a finite-dimensional manifold. In this work we present a complete proof for C1 perturbations in infinite-dimensional Hilbert spaces. This is a joint work with R. Ortega

On the finiteness of central and balanced configurations

Yuchen Wang Tianjin Normal University Peoples Rep of China Co-Author(s):    Lei Zhao

Abstract: Balanced configurations were defined by Albouy and Chenciner in 1996 as a natural generalization of central configurations in higher-dimensional spaces. Since the definition is purely algebraic, balanced configurations are also well-defined in the Euclidean plane. In this talk, we will present the generic finiteness of balanced configurations for the planar four-body problem. Our consideration is based on the singular sequence approach established by Albouy and Kaloshin.

Comparison principle of general Hamilton-Jacobi equations and applications

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

Abstract: In this talk, we prove a new type comparison principle for Hamilton-Jacobi equations. We also propose some applications in understanding the global structure of solutions.

Integrable Kepler billiards and the Poncelet Porism

Lei Zhao Dalian University of Technology Peoples Rep of China Co-Author(s):    Daniel Jaud

Abstract: In this talk I will review some recent works in the latest years concerning integrable Kepler billiards in the plane and some of their very nice geometrical properties. Some of these geometrical properties allow us to draw a direct link between their dynamics and the Poncelet porism, which in turn allows us to have a a relatively easy analysis of their dynamics in terms of elliptic curves. We shall work out an example of an integrable Kepler billiard system inside an ellipse in the plane with negative energy.