arXiv subjects

Wentian Kuang

Publications and source records attributed to Wentian Kuang.

6 recordsLinked to original sources

arXiv2025

Birkhoff sections in 3-manifold with invariant toric foliation

In this paper, we study the Birkhoff sections in a 3-manifold foliated by invariant tori. We establish the necessary and sufficient conditions for various types of periodic orbits to serve as boundary orbits of a Birkhoff section. The construction relies on the dynamical behaviour of the flow combined with fundamental topological argument. As an application, we study the boundaries of toric domains and the energy hypersurfaces of separable Hamiltonian systems, providing conditions for the existence or non-existence of different types of Birkhoff sections. Additionally, we offer an alternative proof of part of the results presented in [23] and [14].

math.DS↗

arXiv2024

Periodic orbits of the Stark problem

The Stark problem is Kepler problem with an external constant acceleration. In this paper, we study the periodic orbits for Stark problem for both planar case and spatial case. We have conducted a detailed analysis of the invariant tori and periodic orbits appearing in the Stark problem, providing a more refined characterization of the properties of the orbits. Interestingly, there exists a family of circular orbits in the spatial case, some of which are quite stable with $L$ being fixed.

math.DS↗

arXiv2021

On shape sphere and rotation of three-body motion

For three-body problem, R.Montgomery [3] proved a reconstruction formula which calculates the overall rotation relating two similar triangle configurations if the initial triangular configuration is similar to the configuration formed at some later time. In this paper, we extend the formula so that it gives the angle of rotation for a single particle without requiring the similarity of initial and final configurations. The proof is different from that in [3] and uses fundamental calculus. Moreover, we answered a question proposed in [3].

math.DS↗

arXiv2020

Geometric properties of minimizers in the planar three-body problem with two equal masses

It it shown that each lobe of the figure-eight orbit is star-shaped, which implies the polar angle is monotone in each lobe. In general, it is not clear when a minimizer is star-shaped. In this paper, we study minimizers connecting two fixed-ends (i.e. the Bolza problem) in the planar three-body problem with two equal masses. We show that if the Jacobi coordinates of the two fixed-ends are in adjacent closed quadrants, then the corresponding minimizer must stay in two adjacent closed quadrants. If we further assume the two Jacobi coordinates are orthogonal on one of the fixed-ends, then the polar angles of the Jacobi coordinates in the minimizer have at most one critical point. If the two Jacobi coordinates are orthogonal on both ends, then the two polar angles must be monotone. These geometric properties can be applied to show the existence of two sets of periodic orbits.

math.DS↗

arXiv2017

Existence of prograde double-double orbits in the equal-mass four-body problem

By introducing simple topological constraints and applying a binary decomposition method, we show the existence of a set of prograde double-double orbits for any rotation angle $θ\in (0, π/7]$ in the equal-mass four-body problem. A new geometric argument is introduced to show that for any $θ\in (0, π/2)$, the action of the minimizer corresponding to the prograde double-double orbit is strictly greater than the action of the minimizer corresponding to the retrograde double-double orbit. This geometric argument can also be applied to study orbits in the planar three-body problem, such as the retrograde orbits, the prograde orbits, the Schubart orbit and the Hénon orbit.

math.DS↗

arXiv2017

The Broucke-Hénon orbit and the Schubart Orbit in the planar three-body problem with equal masses

In this paper, we study the variational properties of two special orbits: the Schubart orbit and the Broucke-Hénon orbit. We show that under an appropriate topological constraint, the action minimizer must be either the Schubart orbit or the Broucke-Hénon orbit. One of the main challenges is to prove that the Schubart orbit coincides with the action minimizer connecting a collinear configuration with a binary collision and an isosceles configuration. A new geometric argument is introduced to overcome this challenge.

math.DS↗