Symplectic structures and quasi–periodic motions in the planetary N–body problem (original) (raw)
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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.
On Symplectic Reduction in Classical Mechanics
Philosophy of Physics, 2007
This Chapter expounds the modern theory of symplectic reduction in finitedimensional Hamiltonian mechanics. This theory generalizes the well-known connection between continuous symmetries and conserved quantities, i.e. Noether's theorem. It also illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. The exposition emphasises how the theory provides insights about the rotation group and the rigid body. The theory's device of quotienting a state space also casts light on philosophical issues about whether two apparently distinct but utterly indiscernible possibilities should be ruled to be one and the same. These issues are illustrated using "relationist" mechanics.
On Co-Orbital Quasi-Periodic Motion in the Three-Body Problem
SIAM Journal on Applied Dynamical Systems
Within the framework of the planar three-body problem we establish the existence of quasi-periodic motions and KAM 4-tori related to the co-orbital motion of two small moons about a large planet where the moons move in nearly circular orbits with almost equal radii. The approach is based on a combination of normal form and symplectic reduction theories and the application of a KAM theorem for high-order degenerate systems. To accomplish our results we need to expand the Hamiltonian of the three-body problem as a perturbation of two uncoupled Kepler problems. This approximation is valid in the region of phase space where co-orbital solutions occur.
Singularly weighted symplectic forms and applications to asteroid motion
Celestial Mechanics & Dynamical Astronomy, 1995
New techniques to study Hamiltonian systems with Hamiltonian forcing are proposed. They are based on singularly weighted symplectic forms and transformations which preserve these forms. Applications pertaining to asteroid motion are outlined. These involve the presence of both Jupiter and Saturn.
Mass-weighted symplectic forms for the n-body problem
1998
Abstract Mass-weighted symplectic forms provide a unified framework for the treatment of both finite and vanishingly small masses in the N-body problem. These forms are introduced, compared to previous approaches, and their properties are discussed. Applications to symplectic mappings, the definition of action-angle variables for the Kepler problem, and Hamiltonian perturbation theory are outlined
The Global Phase Space for the 2- and 3-Dimensional Kepler Problems
Qualitative Theory of Dynamical Systems, 2009
We determine the foliations of the phase space of four particular integrable Hamiltonian systems obtained from the Kepler problem, namely the sidereal and the synodical Kepler Problem in the plane (R 2) and in the space (R 3). These problems differ in their formulation by the choice of the referentials and by the dimension of the phase space. These four Kepler problems have played a main role in Celestial Mechanics. Their importance is justified: First, the study of an integrable problem allow us to obtain information about a non-integrable problem sufficiently close to the integrable one. In fact this is the principle of perturbation theory. Second, from the point of view of the applications, the sidereal is basic for the computation of the planetary ephemerides and the synodical is the limit case of the non-integrable restricted circular 3-body problem when one of the masses of the two primaries tends to zero. We determine the foliations of the phase space of these four Kepler problems by the orbits (i.e. we characterize their global flow), and by fixing one, two or three independent first integrals in involution; of course, at most three for the two spatial problems, and at most two for the two planar problems.
Planetary Birkhoff normal forms
The Birkhoff normal form for the (secular) planetary problem is investigated. Unique-ness is discussed and, in particular, it is shown the the classical Poincaré variables and the rps–variables (introduced in [4]), after a trivial lift, leed to the same Birkhoff normal form; as a corollary the Birkhoff normal form (in Poincaré variables) is de-generate at all orders. Explicit asymptotic formulae for the Birkhoff invariants of first and second order are provided. The symplectically reduced planar and three– body case are discussed separately; in particular it is shown that the normal form obtained in the Jacobi's reduction–of–the–nodes setting coincide with the normal form in the totally reduced setting of [4].
The Lie–Poisson structure of the reduced n -body problem
Nonlinearity, 2013
The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a result we obtain a reduced system with a Lie-Poisson structure which is isomorphic to sp(2n − 2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.
Qualitative features of Hamiltonian systems through averaging and reduction
In this work we analyze the existence and stability of periodic solutions to a Hamil-tonian vector field which is a small perturbation of a vector field tangent to the fibers of a circle bundle. 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. First we state results which have hypotheses on the reduced system and have con-clusions about the full system. The second part is devoted to the application of the general results to the spatial lunar problem of celestial mechanics, i.e. the restricted three-body problem where the infinitesimal is close to one of the primaries. After scaling, the lunar problem is a perturbation of the Kepler problem, which after regu-larization is a circle bundle flow. We prove the existence of four families of periodic solutions for any small regular perturbation of the spatial Kepler problem: we find the classical near circular periodic solutions and the ne...