Escapes and Recurrence in a Simple Hamiltonian System

Many physical systems can be modeled as scattering problems. For example, the motions of stars escaping from a galaxy can be described using a potential with two or more escape routes. Each escape route is crossed by an unstable Lyapunov orbit. The region between the two Lyapunov orbits is where the particle interacts with the system. We study a simple dynamical system with escapes using a suitably selected surface of section. The surface of section is partitioned in different escape regions which are defined by the intersections of the asymptotic manifolds of the Lyapunov orbits with the surface of section. The asymptotic curves of the other unstable periodic orbits form spirals around various escape regions. These manifolds, together with the manifolds of the Lyapunov orbits, govern the transport between different parts of the phase space. We study in detail the form of the asymptotic manifolds of a central unstable periodic orbit, the form of the escape regions and the infinite spirals of the asymptotic manifolds around the escape regions. We compute the escape rate for different values of the energy. In particular, we give the percentage of orbits that escape after a finite number of iterations. In a system with escapes one cannot define a Poincaré recurrence time, because the available phase space is infinite. However, for certain domains inside the lobes of the asymptotic manifolds there is a finite minimum recurrence time. We find the minimum recurrence time as a function of the energy.This revised version was published online in October 2005 with corrections to the Cover Date.     

Twist and Non-Twist Bifurcations in a System of Coupled Oscillators

We consider a system of a harmonic and an unharmonic oscillator with a weak cubic coupling. We study the non-degenerate bifurcations of periodic orbits for the resonant tori of the unperturbed system for which the twist condition holds. We demonstrate that this system also exhibits for certain values of the small parameter non-twist bifurcations, where the rotation number of the Poincaré map attains a minimum value.     

A global solution in the resonance problem of Poincaré

Poincaré's procedure for the construction of a global solution for a particular class of resonance problem is investigated, with particular emphasis placed on those motions corresponding to circulation in the phase space. It is demonstrated that an error on Poincaré's part leads to an impractical, yet formally acceptable, procedure.The merits of alternative methods are discussed, with particular reference to the studies of Garfinkel and the author.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Differential Rotation Driven by Precession

The tidal force in the Earth–Moon system exerted on the Earth's equatorial bulge results in the Earth's precession. It was proposed a long time ago that the strong shear flow driven by the precession of the Earth may power the Earth's dynamo in its liquid core. We present a nonlinear analytical study investigating how the Poincaré force in a rotating, precessing spherical system drives a large-amplitude differential rotation which plays a major role in the modern theory of the geodynamo. The analysis is based on a perturbation approach in terms of the small Poincaré force parameter. It is found that the amplitude of the precession-driven differential rotation is consistent with that estimated from the geomagnetic secular variation.     

Third Integral and the Dynamics of the Galaxy in the Neighborhood of the Sun

Observational data on the dynamics of stars in the neighborhood of the sun indicate the existence of a third integral besides the integrals of the angular momentum and energy. The Poincaré integral is proposed as a third integral. The consequences of this assumption are derived and compared with available astrophysical data.     

Sur les solutions périodiques du mouvement plan de libration des satellites et des planètes

Résumé Ce papier présente une étude analytique du mouvement plan de rotation des satellites (et des planètes) dans leurs mouvements orbitaux. Les trois familles des solution périodiques sont obtenues par la méthode du prolongement analytique de Poincaré. Ensuite, la stabilité de ces solutions périodiques est discutée, et les équations approchées des courbes limites de stabilité sont données jusqu'au quatrième ordre.

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This paper presents an analytical study of the rotational motion of the satellites (and the planets) in their orbital planes. The three families of periodic solutions are obtained by the method of analytical continuation as formulated by Poincaré. The stability of these solutions are analyzed, and the approximate equations of the transition curves are obtained to the fourth order.

     

A theory of integration for the extended Delaunay method

In this paper a method for the integration of the equations of the extended Delaunay method is proposed. It is based on the equations of the characteristic curves associated with the partial differential equation of Delaunay-Poincaré. The use of the method of characteristics changes the partial differential equation for higher order approximations into a system of ordinary differential equations. The independent variable of the equations of the characteristics is used instead of the angular variables of the Jacobian methods and the averaging principle of Hori is applied to solve the equations for higher orders. It is well known that Jacobian methods applied to resonant problems generally lead to the singularity of Poincaré. In the ideal resonance problem, this singularity appears when higher order approximations of the librational motion are considered. The singularity of Poincaré is non-essential and is caused by the choice of the critical arguments as integration variables. The use of the independent variable of the equation of the characteristics in the place of the critical angles eliminates the singularity of Poincaré.     

Periodic orbits of the Sitnikov problem via a Poincaré map

In this paper by means of a Poincaré map, we prove the existence of symmetric periodic orbits of the elliptic Sitnikov problem. Furthermore, using the presence of the Bernoulli shift as a subsystem of that Poincaré map, we prove that not all the periodic orbits of the Sitnikov problem are symmetric periodic orbits.This revised version was published online in October 2005 with corrections to the Cover Date.     

The development of the Poincaré-similar elements with eccentric anomaly as the independent variable

A new set of element differential equations for the perturbed two-body motion is derived. The elements are canonical and are similar to the classical canonical Poincaré elements, which have time as the independent variable. The phase space is extended by introducing the total energy and time as canonically conjugated variables. The new independent variable is, to within an additive constant, the eccentric anomaly. These elements are compared to the Kustaanheimo-Stiefel (KS) element differential equations, which also have the eccentric anomaly as the independent variable. For several numerical examples, the accuracy and stability of the new set are equal to those of the KS solution. This comparable accuracy result can probably be attributed to the fact that both sets have the same time element and very similar energy elements. The new set has only 8 elements, compared to 10 elements for the KS set. Both sets are free from singularities due to vanishing eccentricity and inclination.     

Classical periodic orbits and quantum mechanical eigenvalues and eigenfunctions

We have calculated several families of classical periodic orbits in simple Hamiltonian systems of two degrees of freedom and the corresponding quantum mechanical eigenvalues and eigenfuctions. We have found that in most cases the eigenfunctions have their maxima and minima on some simple periodic orbits. These periodic orbits are of several resonant types and can be either stable or unstable. In the latter case the quantum Poincaré surfaces of section are very different from the classical Poincaré surfaces of section.     

Periodic orbits of galactic motion

Poincaré's surface of section method is used to find and classify the main periodic orbits in a two-dimensional galactic potential first introduced by Hénon and Heiles. The stability of these periodic orbits is studied. Numerical integration with Bulirsch-Stoer method is used.     

The construction of a third order secular analytical J-S-U-N theory by Hori-Lie technique

We construct the outline of a third order secular theory for the four major planets. We apply the Hori-Lie technique to solve the problem. We take into consideration both parts of the perturbing function. Our canonical variables are those of H. Poincaré. Our periodic terms are the only 2:5 and 1:2 critical terms of J-S and U-N respectively. Terms of degree higher than the second in the Poincaré canonical variables H, K, P, Q are neglected.     

A second order Jupiter-Saturn planetary theory

We present a second order secular Jupiter-Saturn planetary theory through Poincaré canonical variables, von Zeipel's method and Jacobi-Radau referential. We neglect in our expansions terms of power higher than the fourth with respect to eccentricities and sines of inclinations. We assume that the disturbing function is composed of secular and critical terms only. We shall deriveF _2si and writeF _2s in terms of Poincaré canonical variables in Part II of this problem.     

A determination of secular terms in first-order planetary theory

The secular terms of the first-order planetary Hamiltonian is determined, by two methods, in terms of the variables of H. Poincaré, neglecting powers higher than the second in the eccentricity-inclination.     

Periodic solutions in Hill's relativistic problem

In this article the existence of periodic solutions in Hill's relativistic problem is demonstrated using Poincaré's small parameter method. This method guarantees the convergence of the series representing the periodic solutions.     

Periodic solutions in the Kovalevskaya case of a rigid body in rotation about a fixed point

The equation of motion of a rigid body in the Kovalevskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincaré the existence of a periodic solution is established.     

A new class of periodic solutions in the Kovalevskaya case of a rigid body in rotation about a fixed point

The equation of motion of a rigid body in Kovaleveskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincaré, the existence of a periodic solution is established.     

Sur une reduction des equations du mouvement du probleme des 3 corps

Résumé Dans cet article nous étudions, dans un premier temps, la réduction des équations du mouvement du problème plan des 3 corps en introduisant le groupe des similitudes planes dans la 1-forme de Poincaré. Ceci permet de dégager le cas des trajectoires de moment cinétique nul et d'énergie nulle. Nous envisageons ensuite la réduction du problème dans l'espace en établissant un lien remarquable avec le problème plan.

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In this article we first of all study the reduction of the equations of movement of the planar three body problem through the introduction of the group of similitude in Poincare's 1-form. This brings out the case of trajectories with zero angular momentum and zero energy. We then consider the reduction of the problem in space by establishing a remarkable link with the planar problem.

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Proceedings of the Sixth Conference on Mathematical methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.