Homographic Solutions Research Articles

Periodic and Quasi-Periodic Orbits in the Collinear Four-Body Problem: A Variational Analysis

This paper investigated the periodic and quasi-periodic orbits in the symmetric collinear four-body problem through a variational approach. We analyze the conditions under which homographic solutions minimize the action functional. We compute the minimal value of the action functional for these solutions and show that, for four equal masses organized in a linear configuration, these solutions are the minimizers of the action functional. Additionally, we employ numerical experiments using Poincaré sections to explore the existence and stability of periodic and quasi-periodic solutions within this dynamical system. Our results provide a deeper understanding of the variational principles in celestial mechanics and reveal complex dynamical behaviors, crucial for further studies in multi-body problems.

On the stability of ring relative equilibria in the N-body problem on with Hodge potential

AbstractIn this paper, we study the stability of the ring solution of the N-body problem in the entire sphere $\mathbb {S}^2$ by using the logarithmic potential proposed in Boatto et al. (2016, Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 472, 20160020) and Dritschel (2019, Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 377, 20180349), derived through a definition of central force and Hodge decomposition theorem for 1-forms in manifolds. First, we characterize the ring solution and study its spectral stability, obtaining regions (spherical caps) where the ring solution is spectrally stable for $2\leq N\leq 6$ , while, for $N\geq 7$ , the ring is spectrally unstable. The nonlinear stability is studied by reducing the system to the homographic regular polygonal solutions, obtaining a 2-d.o.f. Hamiltonian system, and therefore some classic results on stability for 2-d.o.f. Hamiltonian systems are applied to prove that the ring solution is unstable at any parallel where it is placed. Additionally, this system can be reduced to 1-d.o.f. by using the angular momentum integral, which enables us to describe the phase portraits and use them to find periodic ring solutions to the full system. Some of those solutions are numerically approximated.

Action minimizing orbits in the trapezoidal four body problem

<abstract><p>In this paper, we study the minimizing property for the isosceles trapezoid solutions of the four-body problem. We prove that the minimizers of the action functional restricted to homographic solutions are the Keplerian elliptical solutions, and this functional has a minimum equal to $ \frac{3}{2}(2\pi)^{2/3}T^{1/3}\left(\frac{\xi (a, b)}{\eta (a, b)}\right) ^{2/3} $. Further, we investigate the dynamical behavior in the trapezoidal four-body problem using the Poincaré surface of section method.</p></abstract>

Linear instability of elliptic rhombus solutions to the planar four-body problem

In this paper, we study the linear stability of the elliptic rhombus solutions, which are the Keplerian homographic solution with the rhombus central configurations in the classical planar four-body problems. Using ω-Maslov index theory and trace formula, we prove the linear instability of elliptic rhombus solutions if the shape parameter u and the eccentricity of the elliptic orbit e satisfy where u 2 ≈ 0.6633 and . Motivated on numerical results of the linear stability to the elliptic Lagrangian solutions in Martínez et al (2006 J. Differ. Equ. 226 619–651), we further analytically prove the linear instability of elliptic rhombus solutions for .

Morse Theory for S-balanced Configurations in the Newtonian n-body Problem

For the Newtonian (gravitational) n-body problem in the Euclidean d-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the n-body is a (constant up to rotations and scalings) central configuration. For dle 3, the only possible homographic motions are those given by central configurations. For d ge 4 instead, new possibilities arise due to the higher complexity of the orthogonal group mathrm {O}(d), as observed by Albouy and Chenciner (Invent Math 131(1):151–184, 1998). For instance, in mathbb {R}^4 it is possible to rotate in two mutually orthogonal planes with different angular velocities. This produces a new balance between gravitational forces and centrifugal forces providing new periodic and quasi-periodic motions. So, for dge 4 there is a wider class of S-balanced configurations (containing the central ones) providing simple solutions of the n-body problem, which can be characterized as well through critical point theory. In this paper, we first provide a lower bound on the number of balanced (non-central) configurations in mathbb {R}^d, for arbitrary dge 4, and establish a version of the 45^circ -theorem for balanced configurations, thus answering some of the questions raised in Moeckel (Central configurations, 2014). Also, a careful study of the asymptotics of the coefficients of the Poincaré polynomial of the collision free configuration sphere will enable us to derive some rather unexpected qualitative consequences on the count of S-balanced configurations. In the last part of the paper, we focus on the case d=4 and provide a lower bound on the number of periodic and quasi-periodic motions of the gravitational n-body problem which improves a previous celebrated result of McCord (Ergodic Theory Dyn Syst 16:1059–1070, 1996).

Central Configurations and Action Minimizing Orbits in Kite Four-Body Problem

In the current article, we study the kite four-body problems with the goal of identifying global regions in the mass parameter space which admits a corresponding central configuration of the four masses. We consider two different types of symmetrical configurations. In each of the two cases, the existence of a continuous family of central configurations for positive masses is shown. We address the dynamical aspect of periodic solutions in the settings considered and show that the minimizers of the classical action functional restricted to the homographic solutions are the Keplerian elliptical solutions. Finally, we provide numerical explorations via Poincaré cross-sections, to show the existence of periodic and quasiperiodic solutions within the broader dynamical context of the four-body problem.

Homographic Solutions of the N-body Generalized Lennard-Jones System

In this paper, we obtain the existence of non-planar circular homographic solutions and non-circular homographic solutions of the $(2+N)$- and $(3+N)$-body problems of the Lennard-Jones system. These results show the essential difference between the Lennard-Jones potential and the Newton's potential of universal gravitation.

Linear stability of the elliptic relative equilibrium with (1 + n)-gon central configurations in planar n-body problem

We study the linear stability of -gon elliptic relative equilibrium (ERE for short), that is the Kepler homographic solution with the -gon central configurations. We show that for and any eccentricity , the -gon ERE is stable when the central mass m is large enough. Some linear instability results are given when m is small.

Instability for a Family of Homographic Periodic Solutions in the Parallelogram Four Body Problem

We consider the question of stability for the homographic family of rhombus solutions within the four degree of freedom parallelogram four body problem. Our approach to demonstrate spectral instability for the entire parameter ranges of mass ratio and eccentricity \(e \ge 0\) , rests on a topological invariant which is the Maslov index. The analysis begins with a holonomically constrained system of rhombus configurations. This is a three degree of freedom problem and the homographic solutions we are studying belong to the constraint set as well as the unconstrained parallelogram system. Minimizing the constrained action functional over a homology class of rhombus loops brings us to the homographic family. After reducing by the angular momentum to remove degenerate eigenvalues of the Poincare map, we use the minimization property together with an analysis of Lagrangian subspaces in the unreduced space to demonstrate the hyperbolicity (assuming nondegeneracy of the variational problem) on the reduced energy surface of the constrained system. In the last section we argue that the unconstrained homographic solutions of the parallelogram four body problem inherit this nonlinear instability.

Parametric Stability in a Sitnikov-Like Restricted P-Body Problem

We consider the dynamics of an infinitesimal particle under the gravitational action of P primaries of equal masses. These move in an elliptic homographic solution of the P-body problem and the infinitesimal particle moves along the straight line perpendicular to their plane of motion and passing through the common focus of the ellipses. In this work we consider the parametric stability of the infinitesimal mass located at the focus of the ellipses. We construct the boundary curves of the stability/instability regions in the plane of the parameters \(\mu \) and \(\epsilon \), which are the mass of each primary and the eccentricity of the elliptic orbit, respectively.

Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems

In this paper, we prove that the linearized system of elliptic triangle homographic solution of planar charged three-body problem can be transformed to that of the elliptic equilateral triangle solution of the planar classical three-body problem. Consequently, the results of Martínez, Samà and Simó (2006) [15] and results of Hu, Long and Sun (2014) [6] can be applied to these solutions of the charged three-body problem to get their linear stability.

Linear Stability of Elliptic Lagrangian Solutions of the Planar Three-Body Problem via Index Theory

It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter \({\beta=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2 \in [0, 9]}\) and the eccentricity \({e \in [0, 1)}\) . We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle [0, 9] × [0, 1), aside from perturbation methods for e > 0 small enough, blow-up techniques for e sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full (β, e) range [0, 9] × [0, 1) via the ω-index theory of symplectic paths for ω belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the ω-index decreasing property of the solutions in β for fixed \({e\in [0, 1)}\) , we prove the existence of three curves located from left to right in the rectangle [0, 9] × [0, 1), among which two are −1 degeneracy curves and the third one is the right envelope curve of the ω-degeneracy curves, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter (β, e) passes through each of these three curves. Interesting symmetries of these curves are also observed. The linear stability of the singular case when the eccentricity e approaches 1 is also analyzed in detail.

Hyperbolicity of elliptic Lagrangian orbits in the planar three body problem

The linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three body problem depends on the mass parameter β = 27(m 1 m 2 +m 2 m 3 +m 3 m 1)/(m 1 +m 2 +m 3)2 ∈ [0, 9] and the eccentricity e ∈ [0, 1). In this paper we use Maslov-type index to study the stability of these solutions and prove that the elliptic Lagrangian solutions is hyperbolic for β > 8 with any eccentricity.

On the motion of a three-body system on hypersurface of proper energy

Based on the fact that for hamiltonian system there exists equivalence between phase trajectories and geodesic trajectories on the Riemannian manifold, the classical three-body problem is formulated in the framework of six ordinary differential equations (ODEs) of the second order on the energy hypersurface of body system. It is shown that in the case when the total interaction potential of the body system depends on the relative distances between particles, the three of six geodesic equations describing rotations of formed by three bodies triangle are solved exactly. Using this fact, it is shown that the three-body problem can be described in the limits of three nonlinear ODEs of canonical form, which in phase space is equivalent to the autonomous sixth-order system. The equations of geodesic deviations on the manifold Open image in new window (the space of relative distances between particles) are derived in an explicit form. A system of algebraic equations for finding the homographic solutions of restricted three-body problem is obtained. The initial and asymptotic conditions for solution of the classical scattering problem are found.

A minimizing property of homographic solutions

In this paper, we prove that the homographic solutions to the rhombus four body problem are the variational minimizers of the Lagrangian action restricted on a holonomically constrained rhombus loop space.

Central configurations and homographic solutions for the quasihomogeneous N-body problem

For the N-body problem given by quasihomogeneous force functions of the form C1/ra + C2/rb (C1, C2 real, and 0 < a ⩽ b), we prove that the simultaneous and extraneous central configurations are the only types of central configurations. Then we focus on attractive-attractive force functions (C1 and C2 positive) and analyze the connection between central configurations and homographic solutions. Namely, we extend the classical Lagrange's theorem (a solution is homographicif and only if the bodies form equivalent central configurations for all times) to attractive-attractive quasihomogeneous problems, and give sufficient conditions for the construction of homographic solutions.

ω-Index Theory and Linear Stability of Elliptic Lagrangian Solutions of the Classical Three-Body Problem

Abstract It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter β = 27(m1m2 + m2m3 + m3m1)/(m1 + m2 + m3)2 ∈ [0, 9] and the eccentricity ℯ ∈ [0, 1). We are not aware of any existing analytical method which relates the linear stability of these solutions to these two parameters directly in the full rectangle [0, 9] × [0, 1), besides perturbation methods for ℯ > 0 small enough, blow-up techniques for e sufficiently close to 1, and numerical studies. In the recent paper [5], X. Hu, S. Sun and the author introduced a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full (β, ℯ) range [0, 9] × [0, 1) via the ω-index theory of symplectic paths for ω belonging to the unit circle of the complex plane, and the theory of linear operators. This is an expository article on [5]. In Section 1 of this paper we give a brief review on the history of this stability problem, and describe recent results obtained in [5]. Then in Section 2, after reviewing the ω-index theory in [15], the ω-index decreasing property of the elliptic Lagragian solutions in β for fixed ℯ ∈ [0, 1) proved in [5] is introduced, then the existence of three curves located from left to right in the parameter rectangle [0, 9] × [0, 1) is explained. Among these three curves, two are precisely the −1 degeneracy curves and the third one is the left boundary curve of the hyperbolic subregion in [0, 9] × [0, 1). Then in Section 3, it is explained why the linear stability pattern of such an elliptic Lagrangian solution qβ,ℯ changes if and only if the parameter (β, ℯ) passes through one of these three curves. Some open problems on related topics are suggested at the end of this paper.

On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$

The problem of three bodies with equal masses in $\mathbb{S}^2$ is known to haveLagrangian homographic orbits. We study the linear stability and also a'practical'' (or effective) stability of these orbits on the unit sphere.

Polygonal homographic orbits in spaces of constant curvature

We prove that the geometry of the 2-dimensional n n -body problem for spaces of constant curvature κ ≠ 0 \kappa \neq 0 , n ≥ 3 n\geq 3 , does not allow for polygonal homographic solutions, provided that the corresponding orbits are irregular polygons of non-constant size.

Lagrange-Pizzetti theorem for the quasihomogeneous N-body problem

We consider the N-body problem given by quasihomogeneous force functions of the form C1/r a + C2/r b (C1, C2, a, b positive constants and a ⩽ b) and address the fundamentals of homographic solutions. Generalizing techniques of the classical N-body problem, we prove that if the homographic solution is not flat, then it is homothetic; if the homographic solution is flat, then it is planar. Using them, we prove an analogue of the Lagrange-Pizzetti theorem for quasihomogeneous potentials.