Equilateral Restricted Four-body Problem Research Articles

Computer assisted proof of homoclinic chaos in the spatial equilateral restricted four-body problem

We develop constructive, computer assisted arguments for proving the existence of transverse homoclinic connections for hyperbolic periodic orbits in Hamiltonian systems. These arguments are applied at a number of non-perturbative parameter and energy values in the spatial equilateral circular restricted four-body problem: a six dimensional vector field. The idea is to formulate the desired connecting orbits as solutions of certain two point boundary value problems for orbit segments which originate and terminate on the local stable/unstable manifolds attached to a periodic orbit. These boundary value problems are studied via a Newton-Kantorovich argument in an appropriate Cartesian product of Banach algebras of rapidly decaying sequences of Chebyshev coefficients. Perhaps the most delicate part of the problem is controlling the boundary conditions, which must lie on the local stable/unstable manifolds of the periodic orbit. For this portion of the problem we use a parameterization method to develop Fourier-Taylor approximations equipped with mathematically rigorous a-posteriori error bounds. This requires validated computation of a finite number of Fourier-Taylor coefficients via Newton-Kantorovich arguments in appropriate Cartesian product of rapidly decaying sequences of Fourier coefficients, followed by a fixed point argument to bound the tail terms of the Taylor expansion. After some geometric considerations in the energy manifold, transversality of the connection follows as a direct consequence of the Newton-Kantorovich argument.

Equilibrium points and their linear stability in the planar equilateral restricted four-body problem: a review and new results

In this work, we revisit the planar restricted four-body problem to study the dynamics of an infinitesimal mass under the gravitational force produced by three heavy bodies with unequal masses, forming an equilateral triangle configuration. We unify known results about the existence and linear stability of the equilibrium points of this problem which have been obtained earlier, either as relative equilibria or a central configuration of the planar restricted \((3 + 1)\)-body problem. It is the first attempt in this direction. A systematic numerical investigation is performed to obtain the resonance curves in the mass space. We use these curves to answer the question about the existing boundary between the domains of linear stability and instability. The characterization of the total number of stable points found inside the stability domain is discussed.

On the Equilibria of the Planar Equilateral Restricted Four-Body Problem with Radiation Pressure

On the Equilibria of the Planar Equilateral Restricted Four-Body Problem with Radiation Pressure

On the equilibria of the restricted four-body problem with triaxial rigid primaries - I. Oblate bodies

• We numerically explore the equilibria of the restricted four-body problem with triaxial rigid primaries. • We compute the linear stability of the libration points. • We determine how the triaxility parameters affect the dynamics of the equilibrium points. The equilateral restricted four-body problem, with triaxial rigid oblate bodies, is investigated. Using numerical methods we examine how the linear stability and the positions of the coplanar libration points are affected by the triaxility parameters of the primaries. In each case, we perform a rigorous and systematic analysis for determining the influence of the triaxility parameters σ 1 and σ 2 on the dynamics of the system. Our computations suggest the strong influence of these parameters by revealing additional cases, regarding the total number of equilibria, thus improving the findings of previous related works.

Exploring the Location and Linear Stability of the Equilibrium Points in the Equilateral Restricted Four-Body Problem

The planar version of the equilateral restricted four-body problem, with three unequal masses, is numerically investigated. By adopting the grid classification method we locate the coordinates, on the plane [Formula: see text], of the points of equilibrium, for all possible values of the masses of the primaries. The linear stability of the libration points is also determined, as a function of the masses. Our analysis indicates that linearly stable points of equilibrium exist only when one of the primaries has a considerably larger mass, with respect to the other two primary bodies, when the triangular configuration of the primaries is also dynamically stable.

Homoclinic dynamics in a spatial restricted four-body problem: blue skies into Smale horseshoes for vertical Lyapunov families

The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four-body problem admit certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection -- or homoclinic web. In the present work, the planar restricted four-body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way, we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale Tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddle-focus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycle-to-cycle homoclinics in the spatial problem are robust with respect to small changes in energy.

Homoclinic dynamics in a restricted four-body problem: transverse connections for the saddle-focus equilibrium solution set

We describe a method for computing an atlas for the stable or unstable manifold attached to an equilibrium point and implement the method for the saddle-focus libration points of the planar equilateral restricted four-body problem. We employ the method at the maximally symmetric case of equal masses, where we compute atlases for both the stable and unstable manifolds. The resulting atlases are comprised of thousands of individual chart maps, with each chart represented by a two-variable Taylor polynomial. Post-processing the atlas data yields approximate intersections of the invariant manifolds, which we refine via a shooting method for an appropriate two-point boundary value problem. Finally, we apply numerical continuation to some of the BVP problems. This breaks the symmetries and leads to connecting orbits for some nonequal values of the primary masses.

Spatial periodic orbits in the equilateral circular restricted four-body problem: computer-assisted proofs of existence

We use validated numerical methods to prove the existence of spatial periodic orbits in the equilateral restricted four-body problem. We study each of the vertical Lyapunov families (up to symmetry) in the triple Copenhagen problem, as well as some halo and axial families bifurcating from planar Lyapunov families. We consider the system with both equal and non-equal masses. Our method is constructive and non-perturbative, being based on a posteriori analysis of a certain nonlinear operator equation in the neighborhood of a suitable approximate solution. The approximation is via piecewise Chebyshev series with coefficients in a Banach space of rapidly decaying sequences. As by-product of the proof, we obtain useful quantitative information about the location and regularity of the solution.

Fractal basins of attraction in the restricted four-body problem when the primaries are triaxial rigid bodies

The planar equilateral restricted four-body problem, formulated on the basis of Lagrange’s triangular solutions is used to determine the existence and locations of libration points and the Newton-Raphson basins of convergence associated with these libration points. We have supposed that all the three primaries situated on the vertices of an equilateral triangle are triaxial rigid bodies. This paper also deals with the effect of these triaxiality parameters on the regions of motion where the test particle is free to move. Further, the regions on the configuration plane filled by the basins of attraction are determined by using the multivariate version of the Newton-Raphson iterative system. The numerical study reveals that the triaxiality of the primaries is one of the most influential parameters in the four-body problem.

Revealing the basins of convergence in the planar equilateral restricted four-body problem

The planar equilateral restricted four-body problem where two of the primaries have equal masses is used in order to determine the Newton-Raphson basins of convergence associated with the equilibrium points. The parametric variation of the position of the libration points is monitored when the value of the mass parameter $m_3$ varies in predefined intervals. The regions on the configuration $(x,y)$ plane occupied by the basins of attraction are revealed using the multivariate version of the Newton-Raphson iterative scheme. The correlations between the attracting domains of the equilibrium points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We perform a thorough and systematic numerical investigation by demonstrating how the dynamical parameter $m_3$ influences the shape, the geometry and the degree of fractality of the converging regions. Our numerical outcomes strongly indicate that the mass parameter is indeed one of the most influential factors in this dynamical system.

Escape and collision dynamics in the planar equilateral restricted four-body problem

We consider the planar circular equilateral restricted four body-problem where a test particle of infinitesimal mass is moving under the gravitational attraction of three primary bodies which move on circular orbits around their common center of gravity, such that their configuration is always an equilateral triangle. The case where all three primaries have equal masses is numerically investigated. A thorough numerical analysis takes place in the configuration (x,y) as well as in the (x,y) space in which we classify initial conditions of orbits into four main categories: (i) bounded regular orbits, (ii) trapped chaotic orbits, (iii) escaping orbits and (iv) collision orbits. Interpreting the collision motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape and the collision basins and we managed to correlate them with the corresponding escape and collision times of orbits. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting dynamical system.

Nonlinear stability analysis in a equilateral restricted four-body problem

We consider the problem of the stability of equilibrium points in the restricted equilateral four-body problem where a particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses (called primaries) which move on circular orbits around their center of masses, such that their configuration is always an equilateral triangle (Lagrangian configuration). We consider the case of two bodies of equal mass, which in adimensional units is the parameter of the problem. We investigate the Birkhoff normal form around the linearly stable equilibrium points L 3 for μ∈(0,0.0027096302) and, L 5 and L 6 for μ∈(0,0.018858526). The quadratic part of the normalized Hamiltonian has no definite sign, so we examine a function which depends on the coefficients of the fourth order normal form and the stability follows from KAM theory if this function is not zero, otherwise the normal form has to be taken to sixth order or higher. This is the case for L 5 and L 6 for one value of μ. We calculate the sixth order normal form and establish the stability of the points in this case of the failure of the fourth order analysis. We also discuss the stability at the 2:1 resonances for the equilibrium point L 3 and the symmetrical L 5 and L 6 points.

Transport orbits in an equilateral restricted four-body problem

In this paper we consider a restricted equilateral four-body problem where a particle of negligible mass is moving under the Newtonian gravitational attraction of three masses (called primaries) which move on circular orbits around their center of masses such that their configuration is always an equilateral triangle (Lagrangian configuration). We consider the case of two bodies of equal masses, which in adimensional units is the parameter of the problem. We study numerically the existence of families of unstable periodic orbits, whose invariant stable and unstable manifolds are responsible for the existence of homoclinic and heteroclinic connections, as well as of transit orbits traveling from and to different regions. We explore, for three different values of the mass parameter, what kind of transits and energy levels exist for which there are orbits with prescribed itineraries visiting the neighborhood of different primaries.

Global Regularization of a Restricted Four-Body Problem

In the n-body problem, a collision singularity occurs when the position of two or more bodies coincide. By understanding the dynamics of collision motion in the regularized setting, a better understanding of the dynamics of near-collision motion is achieved. In this paper, we show that any double collision of the planar equilateral restricted four-body problem can be regularized by using a Birkhoff-type transformation. This transformation has the important property to provide a simultaneous regularization of three singularities due to binary collision.We present some ejection–collision orbits after the regularization of the restricted four-body problem (RFBP) with equal masses, which were obtained by numerical integration.

Dynamical Aspects of an Equilateral Restricted Four‐Body Problem

The spatial equilateral restricted four‐body problem (ERFBP) is a four body problem where a mass point of negligible mass is moving under the Newtonian gravitational attraction of three positive masses (called the primaries) which move on circular periodic orbits around their center of mass fixed at the origin of the coordinate system such that their configuration is always an equilateral triangle. Since fourth mass is small, it does not affect the motion of the three primaries. In our model we assume that the two masses of the primaries m2 and m3 are equal to μ and the mass m1 is 1 − 2μ. The Hamiltonian function that governs the motion of the fourth mass is derived and it has three degrees of freedom depending periodically on time. Using a synodical system, we fixed the primaries in order to eliminate the time dependence. Similarly to the circular restricted three‐body problem, we obtain a first integral of motion. With the help of the Hamiltonian structure, we characterize the region of the possible motions and the surface of fixed level in the spatial as well as in the planar case. Among other things, we verify that the number of equilibrium solutions depends upon the masses, also we show the existence of periodic solutions by different methods in the planar case.