On Co-Orbital Quasi-Periodic Motion in the Three-Body Problem (original) (raw)
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Symplectic structures and quasi–periodic motions in the planetary N–body problem
The 6n–dimensional phase space of the planetary (1 + n)–body problem (after the classical reduction of the total linear momentum) is shown to be foliated by symplectic leaves of dimension 6n − 2 invariant for the planetary Hamiltonian H; a global standard symplectic structure can be introduced so that H has the same form on each symplectic leaf ("partial reduction" corresponding to 3n − 1 degrees of freedom). The secular Hamiltonian in the partial reduced setting is studied and it is shown to preserve the classical sym-metries (parity, invariance by rotations, D'Alembert symmetries); Birkhoff normal form around the secular equilibrium (corresponding to co–planar and co–circular motions) up to order four is explicitly constructed in the limit of well separated semi–major axes and, in particular, the matrix of the second order Birkhoff invariants of the secular system is shown to be nonsingular ("full torsion" of the secular Hamiltonian in the partially reduced...
Periodic Solutions in Hamiltonian Systems, Averaging, and the Lunar Problem
SIAM Journal on Applied Dynamical Systems, 2008
We investigate the existence, characteristic multipliers, and stability of periodic solutions to a Hamiltonian vector field which is a small perturbation of a vector field tangent to the fibers of a circle bundle. Our primary examples are the planar lunar and spatial lunar problems of celestial mechanics, i.e., the restricted three-body problem where the infinitesimal is close to one of the primaries. By averaging the perturbation over the fibers of the circle bundle one obtains a Hamiltonian system on the reduced (orbit) space of the circle bundle. Our goal in the first part of the paper is to state and prove results which have hypotheses on the reduced system and have conclusions about the full system. Starting with the classical work of Reeb, we give a summary of lemmas, corollaries, and theorems about the existence, characteristic multipliers, and stability of periodic solutions to Hamiltonian systems which are perturbations of circle bundle flows. By reformulating the classical results in modern language and giving alternative proofs in place of the original proofs, we are able to infer new consequences of these classical results. The second part of the paper is devoted to applications of the general results. We apply these general results to the planar and spatial lunar problem. After scaling, the lunar problem is a perturbation of the Kepler problem, which after regularization is a circle bundle flow. We find the classical near-circular periodic solutions and the near-rectilinear periodic solutions. Then we compute their approximate multipliers and show that there is a "twist." However, the twist is too degenerate to apply the classical KAM theorem on invariant tori. We also find symmetric periodic solutions which are continuations of elliptic solutions of the Kepler problem.
Periodic orbits of the restricted three-body problem
Transactions of the American Mathematical Society, 1998
We prove, using a variational formulation, the existence of an infinity of periodic solutions of the restricted three-body problem. When the problem has some additional symmetry (in particular, in the autonomous case), we prove the existence of at least two periodic solutions of minimal period T , for every T > 0. We also study the bifurcation problem in a neighborhood of each closed orbit of the autonomous restricted three-body problem.
On the period of the periodic orbits of the restricted three body problem
Celestial Mechanics and Dynamical Astronomy, 2017
We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, 2T = kπ + Ω g where k is an integer, Ω is the region enclosed by the periodic orbit and g : R 2 → R is a function that only depends on the constant C known as the Jacobian integral; it does not depend on Ω. This theorem has a Keplerian flavor in the sense that it relates the period with the space "swept" by the orbit. As an application we prove that there is a neighborhood around L4 such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for L5.
The planetary N-body problem: symplectic foliation, reductions and invariant tori
Inventiones mathematicae, 2011
The 6n-dimensional phase space of the planetary (1+n)-body problem (after the classical reduction of the total linear momentum) is shown to be foliated by symplectic leaves of dimension (6n − 2) invariant for the planetary Hamiltonian H. Such foliation is described by means of a new global set of Darboux coordinates related to a symplectic (partial) reduction of rotations. On each symplectic leaf H has the same form and it is shown to preserve classical symmetries. Further sets of Darboux coordinates may be introduced on the symplectic leaves so as to achieve a complete (total) reduction of rotations. Next, by explicit computations, it is shown that, in the reduced settings, certain degeneracies are removed. In particular, full torsion is checked both in the partially and totally reduced settings. As a consequence, a new direct proof of Arnold's theorem [2] on the stability of planetary system (both in the partially and in the totally reduced setting) is easily deduced, producing Diophantine Lagrangian invariant tori of dimension (3n − 1) and (3n − 2). Finally, elliptic lower dimensional tori bifurcating from the secular equilibrium are easily obtained.
Journal of Differential Equations, 2011
In order to analyse the dynamics of a given Hamiltonian system in the space defined as the Cartesian product of two spheres, we propose to combine Delaunay coordinates with Poincaré-like coordinates. The coordinates are of local character and have to be selected accordingly with the type of motions one has to take into consideration, so we distinguish the following types: (i) rectilinear motions; (ii) circular and equatorial motions; (iii) circular nonequatorial motions; (iv) non-circular equatorial motions; and (v) non-circular and non-equatorial motions. We apply the theory to study the dynamics of the reduced flow of a generalised Størmer problem that is modelled as a perturbation of the Kepler problem. After using averaging and reduction theories, the corresponding flow is analysed on the manifold S 2 × S 2 , calculating the occurring equilibria and their stability. Finally, the flow of the original problem is reconstructed, concluding the existence of some families of periodic solutions and KAM tori.
Families of Periodic Orbits for the Spatial Isosceles 3-Body Problem
SIAM Journal on Mathematical Analysis, 2004
We study the families of periodic orbits of the spatial isosceles 3-body problem (for small enough values of the mass lying on the symmetry axis) coming via the analytic continuation method from periodic orbits of the circular Sitnikov problem. Using the first integral of the angular momentum, we reduce the dimension of the phase space of the problem by two units. Since periodic orbits of the reduced isosceles problem generate invariant two-dimensional tori of the nonreduced problem, the analytic continuation of periodic orbits of the (reduced) circular Sitnikov problem at this level becomes the continuation of invariant two-dimensional tori from the circular Sitnikov problem to the nonreduced isosceles problem, each one filled with periodic or quasi-periodic orbits. These tori are not KAM tori but just isotropic, since we are dealing with a three-degrees-of-freedom system. The continuation of periodic orbits is done in two different ways, the first going directly from the reduced circular Sitnikov problem to the reduced isosceles problem, and the second one using two steps: first we continue the periodic orbits from the reduced circular Sitnikov problem to the reduced elliptic Sitnikov problem, and then we continue those periodic orbits of the reduced elliptic Sitnikov problem to the reduced isosceles problem. The continuation in one or two steps produces different results. This work is merely analytic and uses the variational equations in order to apply Poincaré's continuation method.
Quasi-Periodic Orbits of the Restricted Three-Body Problem Made Easy
AIP Conference Proceedings, 2007
A new fully numerical method is presented which employs multiple Poincaré sections to find quasi-periodic orbits. The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasi-periodic orbits and the minimal memory required to store these orbits. This method reduces the calculation of the search for the two-dimensional invariant torus to a search for the closed orbits, which are the intersection of the invariant torus with the Poincaré sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant torus is reduced to maps between the consecutive Poincaré maps. A Newton iteration scheme makes use of the invariancy of the circles of the maps on these Poincaré sections in order to find the Fourier coefficient that define the circles to any given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasi-periodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi-Halo and Lissajous families of the Sun-Earth Restricted Three-Body Problem (RTBP) around the L1 and L2 libration points are obtained via this method. Results are compared with the existing literature.
Aerospace, 2022
In this work, we investigate the behavior of low-energy trajectories in the dynamical framework of the spatial elliptic restricted 4-body problem, developed using the Hamiltonian formalism. Introducing canonical transformations, the Hamiltonian function in the neighborhood of the collinear libration point L1 (or L2), can be expressed as a sum of three second order local integrals of motion, which provide a compact topological description of low-energy transits, captures and quasiperiodic libration point orbits, plus higher order terms that represent perturbations. The problem of small denominators is then applied to the order three of the transformed Hamiltonian function, to identify the effects of orbital resonance of the primaries onto quasiperiodic orbits. Stationary solutions for these resonant terms are determined, corresponding to quasiperiodic orbits existing in the presence of orbital resonance. The proposed model is applied to the Jupiter-Europa-Io system, determining quasi...
Pseudo periodic orbits of the planar collision restricted 3-body problem in rotating coordinates
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By using the continuation method of Poincaré, we characterize the periodic circular orbits and the symmetric periodic elliptic orbits that can be prolonged from the planar Kepler problem in rotating coordinates to pseudo periodic orbits of the planar restricted 3-body problem in rotating coordinates with the two primaries moving in an elliptic collision orbit.