Three-body Problem Of Celestial Mechanics Research Articles

Chaos: From celestial mechanics to climate

Abstract Chaos is unpredictable behaviour arising in systems governed by deterministic laws. Since its ‘discovery’ in the 1960s by Edward Lorenz in his studies of meteorological systems, chaos has become an important way of understanding seemingly random behaviour in many other areas. These include the double pendulum, the climate, El Niño and electrical oscillators. Understanding chaos allows the limits of the predictability of a system to be determined. However, despite this recent activity this paper will show that the theory of chaos has a rich history and chaotic behaviour was originally identified in the 1890s by Henri Poincaré in his investigations of the three-body problem in celestial mechanics. But remarkably this paper will also show that chaotic behaviour can be found in such a simple problem as multiplication by 10 and it is closely linked to the question (studied by the ancient Greeks) of whether a number is rational or irrational.

Trigonometric Approximation of the Three-Body Problem: A Double Centroid Approach

_This manuscript introduces the Double Centroid Approximation, a trigonometric method to simplify the Three-body problem in celestial mechanics. By calculating the centroid of three celestial bodies without and then with motion, it offers a novel approach to determine their dynamic center of mass. Grounded in the Superconducting Field Theory, this method simplifies complex gravitational interactions and dark matter considerations. While applicable within our solar system, its principles may extend to other galaxies, highlighting the potential for broad astronomical applications. This concise methodology underscores the importance of triangulation in understanding celestial mechanics, providing a foundation for future research in the field._

Solution of the Dynamic Lame Problem

The well-known Lame problem, posed in 1852, involves solving the static equilibrium of a parallelepiped with free side surfaces subjected to action of opposite end forces. In this article, the same problem for a more complicated case of impacts of end forces is considered. An exact analytical solution of this problem is found. Emphasizing the particular difficulty of solving this problem, Lamé, in his book “Leçons sur la thorie mathematique de Ielasticite des corps solides” (Paris, 1852), wrote: “C’est une sorte d’engine aussi digne d’exercer la sagasite des analystes que le fameux problem des trios corps de la Mécanique celeste”,—“This is a kind of drive, as worthy of training the clairvoyance of analysts as the famous three-body problem of celestial mechanics.” At that time, this problem was the subject of a prize from the Paris Academy of Sciences, that was intended for the one who solved the Lamé problem. Despite this, to date, no solution has been found even for a static case of this problem, not to mention the complicated version of the problem.

Orbital Dynamics in the Restricted Three Body Problem: Overview of Recent Analytical Advances Obtained by Separating Rapid and Slow Subsystems in Non-Planar Configurations

Analytical solutions to a variety of simplified versions of the restricted three-body problem in celestial mechanics possess long running history that encompasses several centuries. Most of the successes were limited either to the planar configuration of the three bodies, to the motion around the Lagrange points, or to the Kozai–Lidov effect. We review some analytical advances obtained by separating rapid and slow subsystems as presented in recently published papers concerning the non-planar motion of the three bodies unrelated to the Lagrange points and to the Kozai–Lidov effect. Most (but not all) of the discussed advances correspond to the bound motion in the considered celestial systems.

Quantum effects on Lagrangian points and displaced periodic orbits in the Earth-Moon system

Recent work in the literature has shown that the one-loop long distance quantum corrections to the Newtonian potential imply tiny but observable effects in the restricted three-body problem of celestial mechanics, i.e., at the Lagrangian libration points of stable equilibrium the planetoid is not exactly at equal distance from the two bodies of large mass, but the Newtonian values of its coordinates are changed by a few millimeters in the Earth-Moon system. First, we assess such a theoretical calculation by exploiting the full theory of the quintic equation, i.e., its reduction to Bring-Jerrard form and the resulting expression of roots in terms of generalized hypergeometric functions. By performing the numerical analysis of the exact formulas for the roots, we confirm and slightly improve the theoretical evaluation of quantum corrected coordinates of Lagrangian libration points of stable equilibrium. Second, we prove in detail that also for collinear Lagrangian points the quantum corrections are of the same order of magnitude in the Earth-Moon system. Third, we discuss the prospects to measure, with the help of laser ranging, the above departure from the equilateral triangle picture, which is a challenging task. On the other hand, a modern version of the planetoid is the solar sail, and much progress has been made, in recent years, on the displaced periodic orbits of solar sails at all libration points, both stable and unstable. The present paper investigates therefore, eventually, a restricted three-body problem involving Earth, Moon and a solar sail. By taking into account the one-loop quantum corrections to the Newtonian potential, displaced periodic orbits of the solar sail at libration points are again found to exist.

Topological Decoupling Near Planar Parabolic Orbits

In two different three body problems, oscillatory orbits have been shown to exist for the three-body problem in Celestial Mechanics: Sitnikov, Alekseev, Moser, and McGehee considered a spatial problem which had one degree of freedom; Easton, McGehee, and Xia considered a planar problem which had at least three degrees of freedom. Both situations involve analyzing the motion as one particle with mass \(m_3\) goes to infinity while the other two masses stay bounded in elliptic motion. Motion with \(m_3\) at infinity corresponds to a periodic orbit in the first problem and the Hopf flow on \(S^3\) in the second problem, both of which are normally degenerately hyperbolic. The proof of the existence of oscillatory orbits uses stable and unstable manifolds for these degenerate cases. In order to get the symbolic dynamics which shows the existence of oscillation, the orbits which go near infinity need to be controlled for an unbounded length of time. In this paper, we prove that the flow near infinity for the Easton–McGehee example with three degrees of freedom is topologically equivalent to a product flow, i.e., a Grobman–Hartman type theorem in the degenerate situation.

Nonplanar Second Species Periodic and Chaotic Trajectories for the Circular Restricted Three-Body Problem

For the circular restricted three-body problem of celestial mechanics with small secondary mass, we prove the existence of uniformly hyperbolic invariant sets of non-planar periodic and chaotic almost collision orbits. Poincare conjectured existence of periodic ones and gave them the name "second species solutions". We obtain large subshifts of finite type containing solutions of this type.

Stellar Motions in Galactic Satellites

The study of the motions of the stars that belong to a galactic satellite (i.e. a globular cluster or a dwarf galaxy orbiting a larger one) has some similarities, as well as significant differences, with that of the restricted three-body problem of celestial mechanics. The high percentage of chaotic orbits present in some models is of particular interest because it rises, on the one hand, the question of the origin of those chaotic motions and, on the other hand, the question of whether an equilibrium stellar system can be built when the bulk of the stars that make it up behave chaotically.

Invariant rotational curves in Sitnikov's Problem

The Sitnikov's Problem is a Restricted Three-Body Problem of Celestial Mechanics depending on a parameter, the eccentricity,e. The Hamiltonian,H(z, v, t, e), does not depend ont ife=0 and we have an integrable system; ife is small the KAM Theory proves the existence of invariant rotational curves, IRC. For larger eccentricities, we show that there exist two complementary sequences of intervals of values ofe that accumulate to the maximum admissible value of the eccentricity, 1, and such that, for one of the sequences IRC around a fixed point persist. Moreover, they shrink to the planez=0 ase tends to 1.

An analysis of truncated fractal growths in the stability boundaries of three-node swing equations

A second-order nonautonomous ordinary differential equation (the transiently forced pendulum system), is studied and the existence of fractally intertwined stability domains is reported for certain ranges of parameters in the time-dependent forcing function. The analysis is extended to a modified version of Sitnikov's three-body problem in celestial mechanics. Parallels are drawn with the three-body problem and the swing dynamics of a three-generator electric power system. >

Analytical solution of the Lagrange quintic equation in the three-body problem in celestial mechanics

The Lagrange equation is a quintic (fifth-degree) equation appearing in a stationary solution of the three-body problem in celestial mechanics. This equation has one positive root, which is the only required in the above problem. Here we apply an elementary real integral formula for the closed-form solution of nonlinear equations to the solution of the Lagrange equation. Two different approaches to this solution are described in detail and numerical results (obtained by both approaches) are displayed. Beyond the Lagrange equation, the present results are also applicable to the solution of other nontrivial quintic and higher-degree polynomial equations.

Contributions to the elliptic restricted three-body problem

Two contributions are given for the elliptic case of the restricted three-body problem of Celestial Mechanics. The first contribution consists in ascertaining analytically the eigenvalues about a Lagrangian triangular point as a power series of the eccentricity. This has entailed the determination of a coordinate transformation, whose coefficients are functions of the true anomaly, for the purpose of reducing the linear portion of the differential system to diagonal form. For each power of the eccentricity, a linear system of differential equations was found that yielded the coefficients of the coordinate transformation. The eigenvalues were calculated by imposing the absence of pure secular terms. The second contribution is the establishing of series solutions for the equations of motion about a triangular point by appropriately modifying a procedure devised by this author for the circular problem. Solutions are given in the form of double power series, whose coefficients (functions of the true anomaly) can be found recursively by solving linear differential systems. The two variables of the power series can be obtained in a similar way as solutions of an auxiliary nonlinear differential system of appropriate form. It was further required that the distance between the Lagrangian point and the initial point on the orbit be considered a small parameter.

A class of periodic solutions for the restricted three-body problem

Periodic orbits have been established in the planar case of the restricted three-body problem of Celestial Mechanics, which can be generated from circular Keplerian orbits and are valid for any mass ratio of the finite bodies. Extending a procedure by Siegel, already applied by this author to Hill's lunar theory, a periodic series solution has been obtained for the equation of motion, whose numerical coefficients can be calculated from a consistent set of recurrent relations. Fourier series representations have been given for the coordinates of the third body with respect to the synodic reference, where the coefficients of the trigonometric functions appear as power series of a parameter related to the period of the orbit. Approximate expansions for the radius vector, angular velocity, magnitude of velocity vector, and the Jacobi constant of motion have been furnished which exhibit how the motion occuring along a periodic orbit of this class deviates from a uniform, circular one. Using the majorant method of series and adapting a theorem on implicit functions, the series solution has been proved to be convergent for small values of the period but for an arbitrary mass ratio of the two primary bodies. The results include as a limiting case Hill's periodic solutions. The procedure is fit for extensive numerical work.