Related papers: Super-hyperbolic orbits and non-collision singular…
In this paper, we prove the existence of noncollision singularities in a planar four-body problem in a model different from [J. Xue,Acta Math.V224(2)253-388, 2020.]. In this model, the acceleration can be arbitrarily fast and the masses can…
In this paper, we use variational minimizing method to prove the existence of hyperbolic solution with a prescribed positive energy for N-body type problems with strong forces. Firstly, we get periodic solutions using suitable constraints,…
We consider the 4-body problem in spaces of constant curvature and study the existence of spherical and hyperbolic rectangular solutions, i.e. equiangular quadrilateral motions on spheres and hyperbolic spheres. We focus on relative…
The existence of hyperbolic orbits is proved for a class of restricted three-body problems with a fixed energy by taking limit for a sequence of periodic solutions which are obtained by variational methods.
We generalize the Newtonian n-body problem to spaces of curvature k=constant, and study the motion in the 2-dimensional case. For k>0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal.…
We prove for the $N$-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level $h>0$ of the motion can also be chosen arbitrarily. Our approach is…
Analytical methods are used to prove the existence of a periodic, symmetric solution with singularities in the planar 4-body problem. A numerical calculation and simulation are used to generate the orbit. The analytical method easily…
This paper shows the existence of a periodic orbit with singularity in the symmetric collinear four body problem. In each period of the orbit, there is a binary collision (BC) between the inner two bodies and a simultaneous binary collision…
The existence of hyperbolic orbits is proved for a class of singular Hamiltonian systems with repulsive potentials by taking limit for a sequence of periodic solutions which are the minimizers of variational functional
The existence of hyperbolic orbits is proved for a class of singular Hamiltonian systems $\ddot{u}(t)+\nabla V(u(t))=0$ by taking limit for a sequence of periodic solutions which are the variational minimizers of Lagrangian actions.
Bautista and Morales proved the existence of periodic orbits in singular-hyperbolic attracting sets(*). In this paper, we extend their result to singular-hyperbolic Lyapunov stable sets. ((*)"Existence of periodic orbits for…
Consider the planar restricted $(N+1)$-body problem with trajectories of the $N(\ge 2)$ primaries forming a collision-free periodic solution of the $N$-body problem, for any positive energy $h$ and directions $\theta_{\pm} \in [0, 2\pi)$,…
This is a short survey on finite-volume hyperbolic four-manifolds. We describe some general theorems and focus on the concrete examples that we found in the literature. The paper contains no new result.
The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $n$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $n \geq 2$,…
In this paper, we first describe how we can arrange any bodies on Figure-Eight without collision in a dense subset of $[0,T]$ after showing that the self-intersections of Figure-Eight will not happen in this subset. Then it is reasonable…
In the context of the Newtonian N-body problem, we prove the existence of a partially hyperbolic motion with prescribed positive energy and any initial collisionless configuration. Moreover, it is a free time minimizer of the respective…
We consider the three body problem on $S^1$ under the cotangent potential. We first construct homothetic orbits ending in singularities, including total collision singularity and collision-antipodal singularity. Then certain symmetrical…
We present an exact solution of the equations for orbit determination of a two body system in a hyperbolic or parabolic motion. In solving this problem, we extend the method employed by Asada, Akasaka and Kasai (AAK) for a binary system in…
Using the variational method, Chenciner and Montgomery (2000 Ann. Math. 152 881--901) proved the existence of an eight-shaped periodic solution of the planar three-body problem with equal masses. Just after the discovery, Gerver have…
We show the existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture of Claude LeBrun. This is achieved by showing, using results in geometric and arithmetic group theory, that certain hyperbolic…