Three body problem

It is not known whether there exist oscillatory and capture orbits in the planar 3-body problem. Sitnikov proved such orbits exist for the restricted 3-body problem. Alekseev [I] extended this work and related it to the existence of homoclinic orbits. McGehee and Easton ]3] studied a model problem closely related to the planar 3-body problem and proved the existence of oscillatory and capture orbits near certain orbits biasymptotic to an invariant 3-sphere. The goal of this paper is to construct an invariant 3-sphere "at infinity" in the planar 3-body problem and to prove that the stable and unstable manifolds of this sphere are Lipschitz manifolds. To complete the study of oscillatory and capture orbits one must investigate the way in which the stable and unstable manifolds of the 3-sphere intersect each other. This is a difficult question which is not treated here. However, using methods of Melnikov and using one of the masses as a perturbation parameter such an investigation might be possible. The stable manifold theorem which is proven here is not standard since the invariant 3-sphere is not hyperbolic. Geometric methods of McGehee [ 4 ] are extended and applied to obtain the theorem. A special case of the system of equations that we study has the form J, =x: dY)lX, + E,(x, Y)L f, = 4 dY>l-x* + E,(x, Y)L (0.1) 4' = BY + F(x, Y), wherexER2,yES3,g:S3--' [O, 03). We assume that i, = By generates the Hopf flow on S3 and that the functions g, E,, E,, F are smooth with * Research partially supported by NSF MCS-8001526.

Symmetry is a fundamental principle in mathematics, physics, and biology, where it governs structure and invariance. Classical symmetry analysis focuses on exact group-theoretic descriptions, but rarely addresses how robust a symmetric configuration is to perturbations. In this work, we introduce a probabilistic framework for quantifying the stability of finite point-set symmetries under random deletions. Specifically, given a finite set of points with a prescribed nontrivial symmetry group, we define the probability P N that removing N points reduces the symmetry to the trivial group C 1. The complementary quantity S N = 1-P N serves as a measure of symmetry stability, providing a robustness profile of the configuration. We calculate S N explicitly for representative families of symmetric point sets, including linear arrays, polygons, polyhedra, directed necklace of points, and crystallographic unit cells. Our results demonstrate unexpected behaviors: the regular hexagon loses symmetry with a probability of 0.6 under the removal of three vertices, while cubes and tetrahedra exhibit the maximal robustness (S N = 1) for all admissible N. We further introduce a Shannon entropy of symmetry stability, which quantifies the overall uncertainty of symmetry breaking across all deletion sizes. This framework extends classical symmetry studies by incorporating randomness, linking group theory with probabilistic combinatorics, and suggesting applications ranging from crystallography to defect tolerance in physical systems.

We report on figure-eight choreographic solutions to a system of three identical particles interacting through a potential of Lennard-Jones-type, 1/r 12 -1/r 6 where r is a distance between the particles. By numerical search, we found there are a multitude of such solutions. A series of them are close to a figure-eight solutions to a homogeneous system with no 1/r 12 term in the potential. The rest are very different from them and have several points with large curvatures in their figure-eight orbits, at which particles are repelled. Here figure-eight choreographies are the periodic motion whose shape is symmetric in both horizontal and vertical axis, starting with an isosceles triangle configuration and going back to an isosceles triangle configuration with opposite direction through Euler configuration. Thus the lobe of this figure-eight may be complex shape and needs not to be convex.

Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian n-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for n ≥ 3.

Donald Saari conjectured that the N -body motion with constant configurational measure is a motion with fixed shape. Here, the configurational measure µ is a scale invariant product of the moment of inertia Namely, µ = I α/2 U . We will show that this conjecture is true for planar equal-mass three-body problem under the strong force potential i<j 1/|q i -q j | 2 .

We present a complete resolution to the Navier–Stokes existence and smoothness problem through a novel physical framework. The key insight is that singularities are artifacts of point-source abstraction, not nature. In both gravitational and hydrodynamic systems, continuity, causality, and finite propagation speed prevent divergence. We show that the refractive n-body solution—which achieves unprecedented stability without artificial softening or tuning—provides a direct mathematical and physical pathway to proving global regularity for incompressible flows. The core mechanism is a nonlinear feedback term which dynamically regulates velocity gradients.  This yields bounded energy growth, sub-exponential divergence, and long-term coherence—resolving turbulence not as chaos, but as structured modulation of a continuous field. All results emerge from first principles: phase continuity, Fermat’s principle, and Maxwellian electrodynamics—without postulates.

The three-body problem is classically unsolvable in closed form due to nonlinearity and sensitivity to initial conditions. CAT'S Theory reframes the system by applying the invariant law Reality = Pattern × Intent × Presence (P × I × P r). Pattern encodes lawful structure (gravitational dynamics), Intent enforces conservationdriven directional constraints, and Presence encodes the collapse to a single realized state at each timestep. This triadic approach converts chaotic divergence into a stepwise lawful sequence, enabling prediction by iterative collapse rather than indefinite projection. Simulation results show the method stabilizes trajectories and extends predictability compared to pattern-only schemes.

Suspending interpretive assumptions such as point-mass singularities and discrete forces, we present results from an ultra-long-term gravitational n-body simulation extending to 10,000,000 time units, achieving unprecedented stability without chaotic divergence or numerical artifacts. Guided by first principles—continuity of fields, local causality, and energy conservation—this classical refractive model treats gravity as emergent from vacuum variations, eliminating unphysical infinities and restoring deterministic evolution. Detailed derivations from Fermat's principle yield the Newtonian limit in weak fields, with velocity-dependent feedback in stronger regimes that damps instabilities. The simulation outcomes—linear trajectory growth in the refractive model versus anomalous flattening in Newtonian—imply a profound resolution to the n-body "problem": chaos as illusion, stability as nature's norm. Implications extend to cosmic structures, predicting mature galaxies at high redshift without ad-hoc entities. Logic demands Occam's razor: one responsive medium simplifies over multiple unobservables.

This paper presents a classical, singularity-free derivation of Lorentz symmetry from the physical requirement that the phase of a continuous electromagnetic wave must remain continuous across reference frames. We show that relativistic effects—including time dilation, length contraction, Doppler shifts—emerge not from abstract postulates, but as the sum of two physical effects: (1) refractive path delay governed by Snell’s law and the eikonal equation, and (2) kinematic time dilation due to transverse motion.

This paper presents a classical, singularity-free framework unifying gravity and electromagnetism through symmetric variations in the vacuum permittivity and permeability induced by mass. Gravitational effects-including perihelion precession, light bending, time dilation, and Shapiro delay-emerge from a flat-space refractive medium governed by electromagnetic principles. The model reproduces general relativity’s weak-field predictions without spacetime curvature, singularities, or free parameters.

The planar isosceles three-body problem has been reduced to a two-dimensional area preserving Poincaré map f. Using certain symmetry properties of the underlying differential equations and numerical integration, we offer a global description of f in the case of three equal masses. This description, which is based on the mapping of areas, immediately leads to the existence of various types of motion such as capture-escape, permanent capture, ejection-collision, etc., and their corresponding measures in the map domain. Moreover, this technique readily allows one to distinguish between so-called “fast” and “chaotic” scattering. Although capture-escape is the subset with the highest measure, there exist two important distinct invariant subsets under f where the solutions neither are captured nor lead to escape. The first set is a Cantor set which has zero measure and it is the outcome of the fact that f acts similar to the Smale horseshoe map in part of the domain. On this subset the ac...

Abstract. A formulation of the N-body problem is presented in which mi and ri ∈ Rd are the mass and the position vector of the i-th body, x = m1r1,..., mNrN) ∈ Rn and n = dN (d = 1, 2, 3). The configuration measure Z = |x|F, where F is the Poincare’s force function, which plays an important role in this formulation. The orbit plane is a two dimensional linear subspace of Rn spanned by the position vector x and the velocity vector ẋ. The N-body motion in Rn has been decomposed into an orbit in the orbit plane and the instantaneous orientation of the orbit plane. For a solution to stay on a level manifold of Z, it is necessary that the orbit in the orbit plane be elliptic (h &lt; 0), parabolic (h = 0)or hyperbolic (h&gt; 0) where h is the total energy. The instantaneous orientation of the orbit plane can be obtained by integration of certain differential equations. These possible solutions include the central configuration solutions in which the orbit plane is fixed in Rn. 1. introdu...

There has been recently a growing interest in cislunar missions, in particular for supporting human deep space exploration. Understanding the dynamical environment between various cislunar orbits is therefore useful. The current study is focused on finding efficient transfer trajectory options between a NearRectilinear Halo Orbit (NRHO) and a Distant Retrograde Orbit (DRO) in the Earth-Moon system. A general methodology is introduced to design these transfers in a systematic way, including the use of solar perturbations and lunar flybys. Representative solutions are presented and compared in terms of delta-v and flight time, including a transfer requiring 56 m/s only.

Many academic studies in spaceflight dynamics rely on simplified dynamical models, such as restricted three-body models or averaged forms of the equations of motion of an orbiter. In practice, the end result of these preliminary orbit studies needs to be transformed into more realistic models, in particular to generate good initial guesses for high-fidelity trajectory optimization tools like Mystic. This paper reviews and extends some of the approaches used in the literature to perform such a task, and explores the inherent trade-offs of such a transformation with a view toward automating it for the case of ballistic arcs. Sample test cases in the libration point regimes and small body orbiter transfers are presented.

Various researchers have proposed the use of electrodynamic tethers for power generation and capture from interplanetary transfers. The effect of tether forces on periodic orbits in Jupiter-satellite systems are investigated. A perturbation force is added to the restricted three-body problem model and a series of simplifications allows development of a conservative system that retains the Jacobi integral. Expressions are developed to find modified locations of equilibrium positions. Modified families of Lyapunov orbits are generated as functions of tether size and Jacobi integral. Zero velocity curves and stability analyses are used to evaluate the dynamical properties of tether-modified orbits.

Symmetry is a fundamental principle in mathematics, physics, and biology, where it governs structure and invariance. Classical symmetry analysis focuses on exact group-theoretic descriptions, but rarely addresses how robust a symmetric configuration is to perturbations. In this work, we introduce a probabilistic framework for quantifying the stability of finite point-set symmetries under random deletions. Specifically, given a finite set of points with a prescribed nontrivial symmetry group, we define the probability 𝑃 𝑁 that removing N points reduces the symmetry to the trivial group 𝐶 1. The complementary quantity 𝑆 𝑁 = 1-𝑃 𝑁 , serves as a measure of symmetry stability, providing a robustness profile of the configuration. We calculate 𝑆 𝑁 explicitly for representative families of symmetric point sets, including linear arrays, polygons, polyhedra, and crystallographic unit cells. Our results demonstrate unexpected behaviors: the regular hexagon loses symmetry with probability 0.6 under removal of three vertices, while cubes and tetrahedra exhibit maximal robustness (𝑆 𝑁 = 1) for all admissible N. We further introduce a Shannon entropy of symmetry stability, which quantifies the overall uncertainty of symmetry breaking across all deletion sizes. This framework extends classical symmetry studies by incorporating randomness, linking group theory with probabilistic combinatorics, and suggesting applications ranging from crystallography to defect tolerance in physical systems.

The graph theory-based approach to the three-body problem is introduced. Vectors of linear and angular momenta of the particles form the vertices of the graph. Scalar products of the vectors of the linear and angular momenta define the colors of the links connecting the vertices. The bi-colored, complete graph emerges. This graph is called the "momenta graph". According to the Ramsey theorem, this graph contains at least one mono-chromatic triangle. This is true even for chaotic motion of three bodies; thus, illustrating the idea supplied by the Ramsey theory, total chaos is impossible. Coloring of the graph is independent on the rotation of frames; however, it is sensitive to Galilean transformations. The coloring of the momenta graph remains the same for general linear transformations of vectors with a positive-definite matrix. For a given motion, changing the order of the vertices does not change the number and distribution of monochromatic triangles. Symmetry of the momenta graph is addressed. The symmetry group remains the same for general linear transformation of vectors of the linear and angular momenta with a positive-definite matrix. Conditions defining conservation of the coloring of the momenta graph are addressed. The notion of the stereographic momenta graph is introduced. Shannon entropy of the momenta graph is calculated. The particular configurations of bodies are addressed, including the Lagrange configuration and the figure eight-shaped motion. The suggested approach is generalized for the quantum field theory with the Pauli-Lubanski pseudo-vector. The suggested coloring procedure is the Lorenz invariant.

Consider the Restricted Planar Circular 3 Body Problem with both realistic mass ratio and Jacobi constant for the Sun-Jupiter pair. We prove the existence of all possible combinations of past and future final motions. In particular, we obtain the existence of oscillatory motions. All the constructed trajectories cross the orbit of Jupiter but avoid close encounters with it. The proof relies on the method of correctly aligned windows and is computer assisted.

We present a diffusion mechanism for time-dependent perturbations of autonomous Hamiltonian systems introduced in . This mechanism is based on shadowing of pseudo-orbits generated by two dynamics: an 'outer dynamics', given by homoclinic trajectories to a normally hyperbolic invariant manifold, and an 'inner dynamics', given by the restriction to that manifold. On the inner dynamics the only assumption is that it preserves area. Unlike other approaches, does not rely on the KAM theory and/or Aubry-Mather theory to establish the existence of diffusion. Moreover, it does not require to check twist conditions or non-degeneracy conditions near resonances. The conditions are explicit and can be checked by finite precision calculations in concrete systems (roughly, they amount to checking that Melnikov-type integrals do not vanish and that some manifolds are transversal). As an application, we study the planar elliptic restricted three-body problem. We present a rigorous theorem that shows that if some concrete calculations yield a non zero value, then for any sufficiently small, positive value of the eccentricity of the orbits of the main bodies, there are orbits of the infinitesimal body that exhibit a change of energy that is bigger than some fixed number, which is independent of the eccentricity. We verify numerically these calculations for values of the masses close to that of the Jupiter/Sun system. The numerical calculations are not completely rigorous, because we ignore issues of round-off error and do not estimate the truncations, but they are not delicate at all by the standard of numerical analysis. (Standard tests indicate that we get 7 or 8 figures of accuracy where 1 would be enough). The code of these

We consider the elliptic three body problem as a perturbation of the circular problem. We show that for sufficiently small eccentricities of the elliptic problem, and for energies sufficiently close to the energy of the libration point L 2 , a Cantor set of Lyapounov orbits survives the perturbation. The orbits are perturbed to quasi-periodic invariant tori. We show that for a certain family of masses of the primaries, for such tori we have transversal intersections of stable and unstable manifolds, which lead to chaotic dynamics involving diffusion over a short range of energy levels. Some parts of our argument are nonrigorous, but are strongly backed by numerical computations.

We prove, under suitable non-resonance and non-degeneracy "twist" conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricityinclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the "planets". The proofs are based on averaging theory, KAM theory and variational methods 1 .

Photodetachment of the positronium negative ion, a bound state of one positron and two electrons, has been observed. Development of a method to produce the ions efficiently using a Na coated tungsten surface has enabled the first observation of the photodetachment. The obtained lower limit of the photodetachment cross section for the wavelength of 1064 nm is consistent with the theoretical calculations reported so far. The experimental field developed in the present work gives new opportunities to explore the quantum mechanical three-body problem and to develop energy-tunable positronium beams.

The aim of this work is to explore and find new closed-form nematicon solutions for different nonlinearities which occur in nematic liquid crystals (NLC) along with proposing optical system application that utilizes NLC nonlinearities. In particular, Lie point symmetry method is employed to scrupulously inspect and acquire solutions for some interesting cases of nonlinearities which have not been fully examined in literature such as quadratic, generalized dual-power law, and eighth-order nonlinearities. A variety of different nematicon dynamics are observed, including bright solitons, dark solitons and periodic behaviors. The explicit solution form for each dynamical behavior is obtained and the solution dependence on model parameters is investigated. The proposed optical system enables the flexible realization of different types of NLC nonlinearities. To the best of our knowledge, this is the first time to attain explicit exact nontrivial solutions for the particular cases of generalized dual-power law and eighth-order nonlinearities.

We apply a quark model developed in earlier work to the spectrum of baryons with strangeness -2 and -3. The model describes a number of well-established baryons successfully, and application to cascade baryons allows the quantum numbers of some known states to be deduced.

Geophysical data processing for the AFS-IP-LB method requires algorithms that handle noisy, rectangular datasets (e.g., 4 × 169) with high precision and computational efficiency. The HybridGeoFilter framework integrates the RALF-1 filter, Faddeev equations, and T-Matrix method to achieve an SNR of 18-28 dB, runtime of 20-30 seconds for n = 10, 000, and memory usage of 60-90 MB. By combining refined Faddeev preprocessing for multibody scattering, T-Matrix computation with conjugate gradient solvers, and randomized coordinate descent in RALF-1, the framework offers robust noise suppression and signal enhancement. Optimized for outer multiprocessing, it resolves issues like indexing errors, dimension mismatches, and numerical instability. Applications include seismic imaging, electromagnetic surveying, gravimetric analysis, magnetotellurics, and interdisciplinary fields like astrophysics and medical imaging. Comparative analysis shows a 3-8x speedup over traditional methods like Lippmann-Schwinger, with superior precision. Future directions include GPU acceleration and real-time 3D processing.

The restricted rhomboidal five-body problem (RRFBP) is a problem in which four positive masses, called the primaries, move two by two in circular motions such that their configuration is always a rhombus, the fifth mass being small and not influencing the motion of the four primaries. In our model, we assume that the fifth mass is in the same plane

We consider a system of five mass points r1, r2, r3, and r4 with masses m1 = m2 = m and \documentclass[12pt]{minimal}\begin{document}$m_3= m_4= \tilde{m}$\end{document}m3=m4=m̃ moving about a single massive body r0 with mass m0 at its center which is assumed to be the origin of the coordinates system. We assume that the central body r0 makes a generalized force on the four mass points and that such a force is generated by a Manev&#39;s type potential, i.e., characterized by a potential of the form \documentclass[12pt]{minimal}\begin{document}$\frac{1}{r}+\frac{\epsilon }{r^2}$\end{document}1r+εr2, on the other hand is assumed that the attraction between the bodies r1, r2, r3, and r4 is of the Newtonian type. This model represents several cases, for instance, when the central body is a spheroid or a radiating source. First, we prove the existence of three different relative rhomboidal solutions, and its (central) configuration is as follows: (1) the rhombus is a square and all primar...

We show that any bounded zero-angular momentum solution for the Newtonian three-body problem must suffer infinitely many eclipses, or collinearities, provided that it does not suffer a triple collision. Motivation for the result comes from the dream of building a symbolic dynamics for the three-body problem, one whose symbols 1, 2, 3 representing the three types of eclipses. The proof involves the conformal geometry of the shape sphere.

A reduced periodic orbit is one which is periodic modulo a rigid motion. If such an orbit for the planar N-body problem is collision-free then it represents a conjugacy class in the projective colored braid group. Under a 'strong force' assumption which excludes the original 1/r Newtonian potential we prove that in most conjugacy classes there is a collision-free reduced periodic solution to Newton's 1 N-body equations. These are the classes that are "tied" in the sense of Gordon [?]. We give explicit homological conditions which insure that a class tied. The method of proof is the direct method of the calculus of variations. For the three-body problem we obtain qualitative information regarding the shape of our solutions which leads to a partial symbolic dynamics.

We investigate the planar three-body problem in the range where one mass, say the 'sun' is very far from the other two, call them 'earth' and 'moon'. We show that "stutters" : two consecutives eclipses in which the moon lies on the line between the earth and sun, occur for an open set of initial conditions. In these motions the moon reverses its sense of rotation about the earth. The mechanism is a kind of tidal torque (see the 'key equation'). The motivation is to better understand the limits of variational methods. The methods of proof are classical estimates and bounds in this asymptotic regime. Consider the planar three-body problem with Newtonian force law. A syzygy is a collinear configuration of the three masses. Double collisions count as syzygies, but triple collisions do not count. The rationale behind this counting is that we can analytically families of solutions through binary collisions but not through triple collision. Noncollision syzygies come in three flavors, 1, 2, and 3 , depending on which of the three masses, 1, 2, or 3, lies between the other two at the moment of syzygy. We associate to any non-collinear, collision-free solution its sequences of syzygies, listed in the time order of appearance. For example, a "tight binary" in which 1 and 2 move in a bound, nearly Keplerian orbit while 3 moves far away will execute the syzygy sequence . . . 121212 . . .. A recent theorem of one of us asserts that when the energy is negative and the angular momentum is zero then every solution has syzygies, with the single exception of the Lagrange homothety solution. So syzygies are ubiquitous in this case. A stutter is a syzygy sequence in which the same letter repeats itself, one or more times in a row: for example: 112. From a variational perspective, stutters are perverse. (See the next section.) We would like to know how prevalent stutters are. Certainly there are some.

In the planar three-body problem, we study solutions with zero initial velocity (brake orbits). Following such a solution until the three masses become collinear (syzygy), we obtain a continuous, flow-induced Poincaré map. We study the image of the map in the set of collinear configurations and define a continuous extension to the Lagrange triple collision orbit. In addition we provide a variational characterization of some of the resulting brake-to-syzygy orbits and find simple examples of periodic brake orbits.

Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a fixed eight-shaped curve. Setting aside collinear motions, the only other known motion along a fixed curve in the inertial plane is the "Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every "Euler configuration" in which one of the bodies sits at the midpoint of the segment defined by the other two (Figure

Breaking symmetry is crucial in many areas of physics, mathematics, biology and engineering. We address symmetry of regular convex polygons, non-convex regular polygons (stars) and symmetric Jordan curves/domains. We demonstrate that removing of a single point from the boundary of the regular convex and non-convex polygons and symmetrical Jordan curves reduces the symmetry group of the polygon to the trivial 𝐶1 group, when the point does not belong to the axis of symmetry of the polygon. The same is true for solid and open 2D regular convex polygons and symmetric Jordan curves. The only exception is a circle. Removing of a single point from the boundary of a circle gives rise to the curve characterized by 𝐶2 group. Symmetry of circles is reduced to the trivial 𝐶1 group by removing a triad of non-symmetrical points. The same is true for a solid circle. The "effort" necessary for breaking symmetry of a circle is maximal. 3D generalization of the theorem is trivial. Thus, classification of symmetrical curves following the minimal number of points necessary for breaking their symmetry becomes possible. The demonstrated theorem shows that the symmetry group action on curves and domains becomes trivial when an asymmetric perturbation is introduced, when the curve is not a circle. Information interpretation of the demonstrated theorem, related to the Landauer principle is introduced.

This paper introduces a new conceptual model for understanding the dynamic and unpredictable interactions of gravitational bodies in space. Rather than focusing solely on three isolated objects, this model-called the Planetorial Chaos System-embraces the full complexity of planetary environments where moons, binary stars, and other influences contribute to a chaotic yet structured behavior.

We propose a survey of Michel Hénon works devoted to studying periodic solutions of the well-known celestial mechanics problem -restricted three-body problem. The description of the main results obtained by Hénon is given in comparison with results of russian mathematician Alexander Bruno. Finaly, a survey of Hénon's works on Hill problem is given as well, and, more over, the authors propose some generalization of Hill problem that makes possible to provide the description of its families of periodic orbits in the form of common network.

We study different families of even periodic solutions in the classical Sitnikov problem that emanate from the circular case as the eccentricity is increased. The families can be classified by the number N of full revolutions of the primaries and labelled by the number of zeroes p of the vertical coordinate of the massless body in half a period. We give a linear stability criterion of these branches depending on even N , based on the sign for the initial slope of the discriminant function for the associated Hill's equation, in a computable interval of eccentricities. All families for N = 2 are linearly stable for small and computable e. The results show a fundamental symmetry-driven difference between the even and odd N cases.

This diagram has been adapted from Henon (1965) [1] and Szebehely (1967) [2]. It shows the stable orbit zones for retrograde coplanar S-type planets of negligible mass in a binary star system in which the two stars M 1 and M 2 have equal masses and circular orbits. The diagram is in a corotating reference frame. It is remarkable that the two zones, within which planetary orbits are stable, are separated by a zone where the orbits are slightly unstable ("légèrement instables"). An upcoming article will examine these zones using orbit simulation software. [1] Henon, M. (1965), Ann. Astr. 28, 992, fig 12. [2] Szebehely, V. (1967), Theory of Orbits, fig 9.49.

If even an extremely simple arrangement like the three-body system is unpredictable chaos, how can we have any faith in discovering the laws of the complicated universe?"-Liu Cixin, The Three-Body Problem – This article examines the instability of three-party project setups-like customer, contractor, and subcontractor-using metaphors from physics and cinema, including the three-body problem and The Good, the Bad and the Ugly. It shows how such arrangements often lead to unpredictable outcomes due to conflicting goals, hidden agendas, and poor communication. Real-world examples, including the Gorch Fock refit and a corporate PMO case, highlight how misalignment and lack of trust can derail projects. The solution, the article argues, lies in empathy – not as a soft skill, but as a core professional capability to understand all parties and guide projects back to stability.

The Three-Body Problem is a science fiction (SF) novel written by Chinese author Liu Cixin and published in China in 2006. Since its publication, the novel has been a great success in its country of origin. In this article, we will compare the Chinese audiovisual adaptation by Tencent (2023) and the Western adaptation by Netflix (2024).

The stability limit of coplanar hierarchical triple systems is numerically studied. Systems we investigated consist of two equal mass bodies initially on a circular orbit and third body with various masses, which at the maximum are equal to the mass of the binary. In order to estimate the stability limit, we use an empirically-found fact that the system is quasiperiodic if the initial eccentricity of the outer binary is less than some critical value, otherwise the third body eventually escapes. We make an analytical expression for the stability limit in terms of the ratio of the orbital radii and find that the expression improves the previous criteria. The resultant expression also suggests that the ratio of the orbital radii rapidly approaches to a certain value (e.g. ∼2, in an initially circular outer binary) as the mass of the third-body tends to zero.

We consider the non-canonical Hamiltonian dynamics of a gyrostat in the n- body problem. Using the symmetries of the system we carry out a reduction process in two steps, giving explicitly at each step the Poisson structure of the reduced system. Next, we obtain general properties of the relative equilibria of the problem and if we restrict to different approximations of the gravitational potential function, some particular cases are studied and, by means of energy-Casimir and spectral methods, sufficient and necessary conditions of stability can be obtained. We extend some results by Fanny and Badaoui (1998) and by Mondéjar, Vigueras and Ferrer (2001). In (3) the case of a rigid body in the three-body problem, in terms of the global variables in the unreduced problem, was considered, and in (12) the case of a gyrostat in the three-body problem was studied, but working now in the reduced system; in this way natural simplifications in the conditions of the equilibria appear. As a par...

We consider the non-canonical Hamiltonian dynamics of a gyrostat in the three body problem. By means of geometric-mechanics methods we study the approxi- mate Poisson dynamics that arises when we develop the potential in series of Leg- endre and truncate this in an arbitrary order k. Working in the reduced problem, the existence and number of equilibria, that we denominate planar rotation type in analogy with classic results on the topic, is considered. Necessary and sufficient conditions for their existence in a approximate dynamics of order k is obtained and we give explicit expressions of this equilibria, useful for the later study of the sta- bility of the same ones. A complete study of the planar rotation type equilibria is made in approximate dynamics or order zero and one. AMS Subject Classification: 34J15, 34J20, 53D17, 70F07, 70K42, 70H14

Interior resonance periodic orbits around the Sun in the Sun-Jupiter photogravitational restricted three-body problem using the method of Poincaré surface of section are studied. The nature, stability and location of these periodic orbits with interior resonances of 2:1, 3:2 and 4:3 are generated for different values of Jacobi constant C. It is found that with the increase in the value of C, these orbits transform to tidal lock, which is a rare case of resonance with 1:1. The period of time for these orbits is found to decrease with the increase in the Sun's radiation pressure. The oblateness of Jupiter is found to increase the period of time for these orbits marginally.

Cet article présente une méthode numérique simplifiée, fondée sur une triangulation vectorielle, pour estimer les positions spatio-temporelles de trois corps en interaction gravitationnelle. La méthode repose sur une approximation géométrique du centre de gravité des corps, y compris ceux de forme non sphérique, à l’aide d’un ensemble de vecteurs maximaux internes. Après avoir déterminé ces centres, les forces gravitationnelles entre les corps sont calculées à l’aide de la loi de Newton, et les déplacements sont modélisés par des vecteurs de force résultante appliqués selon un pas infinitésimal. Cette approche, bien que non conventionnelle, propose une alternative pédagogique et intuitive aux méthodes d’intégration numérique classiques du problème des trois corps.

The graph theory based approach to the three-body problem is introduced. Vectors of linear and angular momenta of the particles form the vertices of the graph. Scalar products of the vectors of the linear and angular momenta define the colors of the links connecting the vertices. The bi-colored, complete graph emerges. This graph is called the "momenta graph". According to the Ramsey theorem this graph contains at least one mono-chromatic triangle. The momenta graph contains at least one mono-chromatic triangle even for a chaotic motion of the particles. Coloring of the graph is independent on the rotation of frames; however, it is sensitive to Galilean transformations. The coloring of the momenta graph remains the same for general linear transformations of vectors with a positive-definite matrix. For a given motion, changing the order of the vertices does not change the number and distribution of monochromatic triangles. Symmetry of the momenta graph is addressed. The symmetry group remains the same for general linear transformation of vectors of the linear and angular momenta with a positive-definite matrix. Conditions defining conservation of the coloring of the momenta graph are addressed. The notion of the stereo-graphic momenta graph is introduced. The suggested approach is generalized for the quantum field theory with the Pauli-Lubanski pseudo-vector. The suggested coloring procedure is Lorentz invariant.

When the planar circular restricted 3-body problem is periodically perturbed, most unstable periodic orbits become invariant tori. However, 2D Poincaré sections no longer work to find their manifolds’ intersections; new methods are needed. In this study, we first review a method of restricting the intersection search to only certain manifold subsets. We then implement this search using Julia and OpenCL, representing the manifolds as triangular meshes and gaining a 30x speedup using GPUs. We finally show how to use manifold parameterizations to refine the approximate connections found in the mesh search. We demonstrate the tools on the planar elliptic RTBP.

The calculation of the many-electron (screened) Coulomb and exchange integrals is a very common task to perform in modern electronic structure theory. While an analytical treatment of the complete integrals is too complicated to be attempted directly, essential insight can be gained by focusing on the most significant contributions, as determined by a physical criterion. The monopole term of the screened Coulomb interaction is particularly important, since it defines the Hubbard interaction 𝑈 among a set of localized electrons, which is an essential parameter for effective models of strongly correlated systems. Here, we derive an analytical solution for the matrix elements of the screened Coulomb interaction on a plane-wave basis, which is routinely used in the most common methods for electronic structure calculations. Screening is treated using a Yukawa potential, which is suitable to describe the valence electrons in the Fermi level region. For the solution of the integrals, the plane waves and the Yukawa potential are first expanded in Bessel functions of first and second kind. Then, by means of the lower and upper incomplete gamma functions, we are able to obtain closed-form integrals of a series that remains convergent for realistic parameters. Our exact solution for the radial integrals of the monopole term can find usage in plane-wave codes for electronic structure calculations, both as an output tool as well as within the computational cycle, as e.g., for many-body extensions of density-functional theory. Considering the importance of Bessel functions in solid state physics and electronic structure theory, it is also easy to foresee that our solution to the various integrals across this work may become useful to several other problems in the field.