arXiv subjects

Guowei Yu

Publications and source records attributed to Guowei Yu.

18 recordsLinked to original sources

arXiv2025

Relative periodic solutions in spatial Kepler problem with symmetric perturbation

The spatial Kepler problem with a perturbation satisfying the rotational symmetry w.r.t. the $z$-axis and the reflection symmetry w.r.t. the $(x, y)$-plane, can be reduced to an Hamiltonian system with 2 degrees of freedom after fixing the angular momentum. For small enough perturbations, we show that for certain choices of energy and angular momentum, the corresponding energy surface is compact and diffeomorphic to $\mathbb{S}^3$, and on each compact energy surface there is a unique $z$-symmetric brake orbit, which forms a Hopf link with a planar relative periodic orbit. Moreover under some additional technical assumptions, by applying recent results from symplectic dynamics (\cite{CHHL23}) and Franks' Theorem, we prove there are infinitely many relative periodic orbits on each compact energy surface. These results can be applied to the motion of a satellite around a uniformly mass-distributed ellipsoid and the $n$-pyramidal problem, where one point mass moves along the $z$-axis and $n$ other equal point masses form a regular $n$-gon perpendicular to the $z$-axis.

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arXiv2025

Variational Construction of Homoclinic and Heteroclinic Orbits in the Planar Sitnikov Problem

The Sitnikov problem is a special case of the three-body problem. The system is known to be chaotic and has been studied by symbolic dynamics (J. Moser, Stable and random motions in dynamical systems, Princeton University Press, 1973). We study the limiting case of the Sitnikov problem as the eccentricity of the massive particles tends to 1. By variational method, we show the existence of infinitely many homoclinic and heteroclinic solutions in the planar Sitnikov problem. In a previous work, for certain periodic symbolic sequences, the second author showed the existence of periodic solutions realizing them. In this paper, we show the existence of homoclinic and heteroclinic solutions between some of these periodic orbits which realize certain non-periodic symbolic sequences.

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arXiv2025

Positive energy solutions in the anisotropic Kepler problem with homogeneous potential

We study positive energy solutions of the anisotropic Kepler problem with homogeneous potential. First some asymptotic property of positive energy solutions is obtained, as time goes to infinity. Afterwards, we prove the existence of hyperbolic solutions with given initial configuration and asymptotic behavior, when time goes to positive or negative infinity, and in the planar case, the existence of bi-hyperbolic solutions with given asymptotic behaviors, when time goes to both positive and negative infinities, under various conditions.

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arXiv2025

The problem of infinite Spin for parabolic and collision solutions in the planar $n$-body problem

In the planar $n$-body problem, the problem of infinite spin occurs for both parabolic and collision solutions. Recently Moeckel and Montgomery \cite{MM25} showed that there is no infinite spin for total collision solutions, when the reduced and normalized configuration converges to an isolated central configuration. Following their approach, we show it can not happen for both complete and partially parabolic solutions, under similar conditions. Our approach also allows us to generalize Moeckel and Montgomery's result to partial collision solutions under similar conditions.

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arXiv2024

Hyperbolic motions in the $N$-body problem with homogeneous potentials

In the $N$-body problem, a motion is called hyperbolic, when the mutual distances between the bodies go to infinity with non-zero limiting velocities as time goes to infinity. For Newtonian potential, in \cite{MV20} Maderna and Venturelli proved that starting from any initial position there is a hyperbolic motion with any prescribed limiting velocities at infinity. Recently based on a different approach, Liu, Yan and Zhou \cite{LYZ21} generalized this result to a larger class of $N$-body problem. As the proof in \cite{LYZ21} is quite long and technical, we give a simplified proof for homogeneous potentials following the approach given in the latter paper.

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arXiv2023

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

We study the spatial isosceles three-body problem from the perspective of Symplectic Dynamics. For certain choices of mass ratio, angular momentum, and energy, 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. In the case of large mass ratios, the Hopf link is non-resonant, forcing the existence of infinitely many periodic orbits. The rotation number of the Euler orbit plays a fundamental role in the existence of periodic orbits and their symmetries. We explore such symmetries in the Hill region and show that the Euler orbit is negative hyperbolic for an open set of parameters while it can never be positive hyperbolic. Finally, we address convexity and determine for each parameter whether the energy surface is strictly convex, convex, or non-convex. Dynamical consequences of this fact are then discussed.

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arXiv2022

Hyperbolic and Bi-hyperbolic solutions in the planar restricted $(N+1)$-body problem

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 $\theta_{\pm} \in [0, 2\pi)$, we prove that starting from any initial position $x$ at any initial time $t_x$, there are hyperbolic solutions $\gamma^{\pm}|_{[t_x, \pm \infty)}$ satisfying $\gamma^{\pm}(t_x) =x$ and $$ \lim_{t \to \pm \infty} \gamma^{\pm}(t) / |\gamma^{\pm}(t)| = e^{i \theta_{\pm} (\text{mod } 2\pi)}, \;\;\lim_{ t \to \pm \infty} \dot{\gamma}^{\pm}(t) = \pm \sqrt{2h} e^{i \theta_{\pm} (\text{mod } 2\pi)}.$$ Moreover we also prove the existence of a bi-hyperbolic solution $\gamma|_{\mathbb{R}}$ satisfying $$ \lim_{t \to \pm \infty} \gamma(t) / |\gamma(t)| = e^{i \theta_{\pm} (\text{mod } 2\pi)}, \;\;\lim_{ t \to \pm \infty} \dot{\gamma}(t) = \pm \sqrt{2h} e^{i \theta_{\pm} (\text{mod } 2\pi)}.$$

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arXiv2022

Regularizable collinear periodic solutions in the $n$-body problem with arbitrary masses

For $n$-body problem with arbitrary positive masses, we prove there are regularizable collinear periodic solutions for any ordering of the masses, going from a simultaneous binary collision to another in half of a period with half of the masses moving monotonically to the right and the other half monotonically to the left. When the masses satisfy certain equality condition, the solutions have extra symmetry. This also gives a new proof of the existence of Schubart orbit, when $n=3$.

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arXiv2022

Global Surfaces of Section and Periodic Orbits in The Spatial Isosceles Three Body Problem

We study the spatial isosceles three body problem, which is a system with two degrees of freedom after modulo the rotation symmetry. For certain choices of energy and angular momentum, we find some disk-like global surfaces of section with the Euler orbit as their common boundary, and a brake orbit passing through them. By considering the Poincar\'e maps of these global surfaces of section, we prove the existence of all kinds of different periodic orbits under certain assumption. Moreover, we are able to prove, for generic choices of masses, the system always has infinitely many periodic orbits. One of the key is to estimate the rotation numbers of the Euler orbit and the brake orbit with respect to the Poincar\'e map. For this, we establish formulas connected these numbers with the mean indices of the corresponding orbits using the Maslov-type index.

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arXiv2019

Chazy-Type Asymptotics and Hyperbolic Scattering for the $n$-Body Problem

We study solutions of the Newtonian $n$-body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as $t \rightarrow +\infty$ or as $t \rightarrow -\infty$. In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium points in a boundary manifold "at infinity". We show that the flow near these manifolds can be analytically linearized and use this to give a new proof of Chazy's classical asymptotic formulas. We also address the scattering problem, namely, for solutions which are hyperbolic in both forward and backward time, how are the limiting equilibrium points related? After proving some basic theorems about this scattering relation, we use perturbations of our manifold at infinity to study scattering "near infinity", that is, when the bodies stay far apart and interact only weakly.

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arXiv2019

An Index Theory for Collision, Parabolic and Hyperbolic Solutions of the Newtonian $n$-body Problem

In the Newtonian $n$-body problem for solutions with arbitrary energy, which start and end either at a total collision or a parabolic/hyperbolic infinity, we prove some basic results about their Morse and Maslov indices. Moreover for homothetic solutions with arbitrary energy, we give a simple and precise formula that relates the Morse indices of these homothetic solutions to the spectra of the normalized potential at the corresponding central configurations. Potentially these results could be useful in the application of non-action minimization methods in the Newtonian $n$-body problem.

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arXiv2017

Connecting planar linear chains in the spatial $N$-body problem

The family of planar linear chains are found as collision-free action minimizers of the spatial $N$-body problem with equal masses under $D_N$ or $D_N \times \zz_2$-symmetry constraint and different types of topological constraints. This generalizes a previous result by the author in \cite{Y15c} for the planar $N$-body problem. In particular, the monotone constraints required in \cite{Y15c} are proven to be unnecessary, as it will be implied by the action minimization property. For each type of topological constraints, by considering the corresponding action minimization problem in a coordinate frame rotating around the vertical axis at a constant angular velocity $\om$, we find an entire family of simple choreographies (seen in the rotating frame), as $\om$ changes from $0$ to $N$. Such a family starts from one planar linear chain and ends at another (seen in the original non-rotating frame). The action minimizer is collision-free, when $\om=0$ or $N$, but may contain collision for $0 < \om < N$. However all possible collisions must be binary and each collision solution is $C^0$ block-regularizable. Moreover for certain types of topological constraints, based on results from \cite{BT04} and \cite{CF09}, we show that when $\om$ belongs to some sub-intervals of $[0, N]$, the corresponding minimizer must be a rotating regular $N$-gon contained in the horizontal plane. As a result, this generalizes Marchal's $P_{12}$ family of the three body problem to arbitrary $N \ge 3$.

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arXiv2017

Application of Morse index in weak force $N$-body problem

Due to collision singularities, the Lagrange action functional of the N-body problem in general is not differentiable. Because of this, the usual critical point theory can not be applied to this problem directly. Following ideas from \cite{BR91}, \cite{Tn93a} and \cite{ABT06}, we introduce a notion called weak critical point for such an action functional, as a generalization of the usual critical point. A corresponding definition of Morse index for such a weak critical point will also be given. Moreover it will be shown that the Morse index gives an upper bound of the number of possible binary collisions in a weak critical point of the $N$-body problem with weak force potentials including the Newtonian potential.

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arXiv2017

Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem

In the variational study of singular Lagrange systems, the zero energy solutions play an important role. In this paper we find a simple way of computing the Morse indices of these solutions for the planar anisotropic Kepler problem. In particular an interesting connection between the Morse indices and the oscillating behaviors of these solutions discovered by the physicist M. Gutzwiller is established.

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arXiv2016

Spatial double choreographies of the Newtonian $2n$-body problem

In this paper, for the spatial Newtonian $2n$-body problem with equal masses, by proving the minimizers of the action functional under certain symmetric, topological and monotone constraints are collision-free, we found a family of spatial double choreographies, which have the common feature that half of the masses are circling around the $z$-axis clockwise along a spatial loop, while the motions of the other half masses are given by a rotation of the first half around the $x$-axis by $\pi$. Both loops are simple, without any self-intersection, and symmetric with respect to the $xz$-plane and $yz$-plane. The set of intersection points between the two loops is non-empty and contained in the $xy$-plane. The number of such double choreographies grows exponentially as $n$ goes to infinity.

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arXiv2015

Simple choreographies of the planar Newtonian $N$-body Problem

In the $N$-body problem, a simple choreography is a periodic solution, where all masses chase each other on a single loop. In this paper we prove that for the planar Newtonian $N$-body problem with equal masses, $N \ge 3$, there are at least $2^{N-3} + 2^{[(N-3)/2]}$ different main simple choreographies. This confirms a conjecture given by Chenciner and etc. in \cite{CGMS02}.

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arXiv2015

Periodic Solutions of the Planar N-Center Problem with topological constraints

In the planar $N$-center problem, for a non-trivial free homotopy class of the configuration space satisfying certain mild condition, we show that there is at least one collision free $T$-periodic solution for any positive $T.$ We use the direct method of calculus of variations and the main difficulty is to show that minimizers under certain topological constraints are free of collision.

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arXiv2015

Shape Space Figure-$8$ Solution of Three Body Problem with Two Equal Masses

In a preprint by Montgomery \cite{Mo99}, the author attempted to prove the existence of a shape space Figure-$8$ solution of the Newtonian three body problem with two equal masses (it looks like a figure $8$ in the shape space, which is different from the famous Figure-$8$ solution with three equal masses \cite{CM00}). Unfortunately there is an error in the proof and the problem is still open. Consider the $\alpha$-homogeneous Newton-type potential, $1/r^{\alpha},$ using action minimization method, we prove the existence of this solution, for $\alpha \in (1,2)$; for $\alpha=1$ (the Newtonian potential), an extra condition is required, which unfortunately seems hard to verify at this moment.

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