arXiv subjects

Yangshanshan Liu

Publications and source records attributed to Yangshanshan Liu.

5 recordsLinked to original sources

arXiv2026

Concave Kite Central Configurations in the Planar Four-Body Problem with Three Equal Masses

We present a complete classification of concave kite central configurations in the planar 4-body problem with three equal masses. There are two different types of central configurations when the fourth mass lies inside or outside the triangle formed by the other three. Using a rigorous computer-assisted analytical method and a fixed coordinate system, we show that the central configurations in each case form a one-parameter family and obtain a complete classification of these configurations. In addition, we rigorously show the existence and types of the bifurcation points in the reduced space. We also provide two numerical global bifurcation pictures in the entire planar 4-body configuration space as the mass ratio varies from $0$ to $+\infty$, including symmetric and asymmetric concave central configurations with three equal masses.

math.DS↗

arXiv2026

Symmetric Central Configurations in the Concave 4-Body Problem with Two Pairs of Equal Masses

We establish the existence of a single-parameter family of the concave kite central configurations in the 4-body problem with two pairs of equal masses. In such configurations, one pair of masses must lie on the base of an isosceles triangle, and the other pair on its symmetric axis with one mass positioned inside the triangle formed by the other three. Using a rigorous computer-assisted analytical approach, we prove that for any non-negative mass ratio, the number of such configurations is either zero, one, or two, thereby providing a complete classification of this family. Furthermore, we show that the unique configuration corresponding to a specific mass ratio is a fold-type bifurcation point within the reduced subspace. We also give a clear and complete bifurcation picture for both symmetric and asymmetric cases of this concave type across the entire planar 4-body configuration space.

math.DS↗

arXiv2024

On the uniqueness of the strictly convex quadrilateral central configuration with a fixed angle

The conjecture of the existence and the uniqueness of the strictly convex quadrilateral central configuration for the Newtonian 4-body problem is one of the most-talked open problems in the study of the classical n-body problems in celestial mechanics. MacMillan and Bartky first gave its general existence in the 1930s and a particular case for its uniqueness. Still, the general case has yet to be solved perfectly since it was considered by Sim'{o} and Yoccoz in the 1980s and was first mentioned by Albouy and Fu in 2008 in the formal publication. Using coordinates of mutual distances and Morse's critical point theory, we give the (at most) uniqueness of the planar strictly convex 4-body central configuration when the angle of one pair of the opposite sides is given.

math-ph↗

arXiv2024

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

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.

math.DS↗

arXiv2023

Stacked central configurations with a homogeneous potential in $\mathbb{R}^3$

In this paper we generalize some results in \cite{Yu2021} concerning stacked central configurations. We can deal with the general homogeneous potential $U_α$ (containing the vortex case) in $\mathbb{R}^3$. We give the admissible set of $α$ for a convex central configuration (with respect to the Newtonian potential i.e. $α=3$). We discuss some properties of the regular $n$-gon co-circular central configurations. We also find that the stacked property is particular for central configurations by studying the S-balanced configuration case.

math.DS↗