Analysis of Orbital Stability in the Three-Body Problem

Chapter 9 - The Restricted Three-Body Problem

This article presents the chaos synchronization problem of the restricted three body problem (RTBP) when the massive primary is supposed to be oblate spheroid and the smaller one is a uniform circular ring. The feedback controller for the stability of the closed-loop system are designed using the active control and adaptive control strategy. It is shown that the two methods have excellent performance, with the active control marginally outperforming the adaptive control in terms of transient analysis. Simulation results satisfy the theoretical findings. For validation of results by numerical simulations, the Mathematica 10 is used when the primaries are Saturn and Jupiter.

Dynamics in the center manifold of the collinear points of the restricted three body problem

9R7. Classical Many-Body Problems Amenable to Exact Treatments. (Solvable and/or Integrable and/or Linearizable…) in One-, Two-, and Three-Dimensional Space. Lecture Notes in Physics, Vol m66. - F Calogero (Dept of Phys, Univ of Rome “La Sapienza,” p Aldo Moro, Rome, 00185, Italy). Springer-Verlag, Berlin. 2001. 749 pp. ISBN 3-540-41764-8. $79.95.Reviewed by M Pascal (Lab de Modelisation en Mec, Univ Pierre et Marie Curie, Tour 66, 4 Place Jussieu, Paris, 75252 Cedex 05, France).This book is concerned with integrable problems in classical mechanics (excluding quantam or relativistic mechanics). These problems are related to the motion of a system of particles acted upon by several kinds of forces in one-, two-, or three-dimensions. More general Hamiltonian systems (not associated with a Lagrangian function) are also considered. In most cases, integrable Hamiltonian systems mean Liouville integrable systems. In almost all examples dealing with the motion of a set of particles, the masses are the same for all particles. Several methods to find integrable problems are described, but the main part of the book is devoted to the Lax pair technique. This book provides a rather exhaustive survey of results obtained in the past by scientists such as Moser, Toda, Flashka, and also by the author himself.The book involves five chapters followed by several appendices and a rather wide list of references. Each chapter includes a great amount of examples and exercises. Due to the rather abstract topics studied in the book, no figures are included in it. The first chapter is a short introduction giving some basic knowledge about Newtonian equations of motion, Hamiltonian formalism, and Liouville integrable systems. The main part of the book is contained in Chapter 2, which deals with one-dimensional motions of particles along a line or a circle. Liouville integrable problems are found by using the Lax pair technique. Several examples are obtained by solving three kinds of functional equations. From these integrable cases, other integrable systems are deduced by means of several transformations like change of variables, duplication and so on. Another method introduces a time dependant polynomial of degree N with respect to the variable X; the N zeros of this polynomial are interpreted as the positions of N particles (with the same masses) on a line. Assuming that the polynomial is solution of a linear partial derivative equation, several solvable many body problems are found. An extension of this approach is also presented, using the exact Lagrangian interpolation method. In Chapter 3, a generalized formulation of this last method is formulated for spaces of arbitrary dimensions. Chapter 4 deals with integrable systems in the plane. The method is based on the idea that integrable problems in two-dimensional space can be deduced from integrable problems in one-dimension assuming that the equations of motion hold also for complex values of the variables: it is the so called technique of complexification. A survey of integrable systems in the plane obtained by this method is given at the end of the chapter. The last chapter presents integrable many body systems in three-dimensional space. The idea is to identify some solvable evolution equations governing a time dependant matrix with the motion equations of a set of particles in three-dimensions. Again, several examples are presented. As a conclusion, Classical Many-Body Problems Amenable to Exact Treatments…, is not intended for a very large audience, but it provides a rather exhaustive survey about integrable problems in some special cases of particle dynamics. It can be useful for researchers involved in celestial mechanics or can be used as a background for teaching an undergraduate course.

Effects of Radiating Oblate Spheroid and Triaxial Rigid Body in the Restricted Three Body Problem

The irreversible process of ‘wave packet collapse’ in quantum measurement contradicts the reversible characteristics of the Schrödinger equation and the entire quantum mechanics. This contradiction is actually the origin of measurement problems in quantum mechanics. How to resolve this contradiction is related to the understanding of some philosophical concepts. The Copenhagen School embraces the reversible/irreversible contradictions in quantum measurement through the principle of complementarity. Both von Neumann's measurement hypothesis and Everett's many world explanation adhere to the ‘universal position of the Schrödinger equation’, but due to their detachment from the thermodynamic mechanism analysis of quantum measurement processes, they fall into the dilemma of subjectivism or possible state materialization in ontology. Decoherence theory is a natural extension of dissipative structure theory in quantum mechanics, which helps to solve quantum measurement problems by combining the study of classical conversion mechanisms between reversibility and irreversibility in statistical mechanics. Due to the common statistical roots of the time arrow problem in classical mechanics and quantum mechanics, decoherence theory cannot be seen as a measurement theory that fully implements the ‘universal Schrödinger position’. Many contradictions in quantum mechanics may need to be cleverly resolved through various contradictions in classical physics.

The Three Body Problem The three body problem studies the motion of three masses whose gravitational attraction have an effect on each other. The dynamics of the three-body problem are essentially different from those of two bodies, because in the latter case, an analytical solution may be found that admits orbits in the form of conic sections. This problem has been studied at great length and is the basis of most of today’s orbit planning and trajectory design for satellites. However, the two-body problem is valid on close to a single massive body, compared to which the target body (the object whose motion is desired) is essentially a massless particle. In deep space, when there may be two or more massive bodies to affect the motion of our test particle, the two-body solution obviously fails. It then becomes essential to study the three-body problem.

In this paper we have considered the restricted three body problem (R3BP) in which the more massive primary is triaxial, the less massive primary and infinitesimal body are oblate spheroids, and are encompassed by a belt of homogenous material points. Analytically and numerically, we have studied the effects of triaxiality of the more massive primary, oblateness of both the less massive primary and infinitesimal body and the gravitational potential generated by the belt on the location of the triangular libration points L 4,5 and their linear stability. L 4,5 do not form equilateral triangles with the primaries in the presence of all or any of the aforementioned perturbations. Due to triaxiality of the more massive primary and oblateness of the infinitesimal body the triangular libration points are seen to move away from the line linking the primaries, whereas they shift closer to the line owing to the oblateness of the less massive primary and the potential from the belt. The range 0 < μ < μ c of stability of the triangular points is reduced in the presence of any of the perturbations, except when considering the potential from the belt the range increases, where μ c is the critical mass ratio. The oblateness of a test particle (of infinitesimal mass) shifts the location of its libration positions away from the primaries and reduces its range of stability.

Radar interference into LTE base stations in the 3.5 GHz band

The rectilinear elliptic restricted Three Body Problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity $e'=1$, but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter $\mu=0.5$ (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke's computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to $\mu$ and $e'<1$. Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449.

The orbital dynamics in the vicinity of a binary asteroid system has been studied extensively, motivated by the special dynamical environment and possible exploration missions. Equilibrium points, periodic orbits, and invariant manifolds have been investigated in many studies based on the model of the Restricted Full Three Body Problem (RF3BP). In this paper, a new semi-analytical orbital dynamical model around the primary of a binary system is developed as a perturbed two-body problem. The solution includes the effect of the primary's oblateness and the secondary's third-body gravity. The semi-analytical dynamical model, also denoted as the averaged model, is obtained by using the averaging process and Lagrange planetary equations in terms of the Milankovitch orbital elements. This semi-analytical model enables much faster orbital propagations than the non-averaged counterpart, and is particularly useful in orbital stability analysis and the design of long-term passively stable orbits and orbits with specific performance, e.g. frozen orbits. The applicability of the semi-analytical model is then discussed. Two parameters describing relative magnitudes of both perturbations w.r.t. the primary's point mass gravity and the third parameter related to the orbital period ratio w.r.t. the secondary are defined to provide indicators for the validity of the averaged model. The validity boundaries in terms of the three parameters are given based on numerical simulations, by comparing with the full orbital model. The application to a real binary system, 2003 YT1, has shown that the averaged solution has a high precision in the long-term orbital propagation.

Bogolubov's classical example of statistical relaxation in a many-dimensional linear oscillator is discussed. The relation of the discovered relaxation mechanism to quantum dynamics as well as to some new problems in classical mechanics is considered.

As is known [134][135][136][137][138][139][140][141][142][143][144], classical mechanics as a branch of physics arose from the needs of sciences and practice and has a long history of development.The important significance of classical mechanics is determined by the contribution of the prominent scientists of past time: J. Kepler, Galileo Galilei, I.

A quantitative comparison of the dissociation probabilities of H2 on the W(100) surface obtained from quantum and classical dynamical simulations is presented. Four dimensions are employed, and the vibrationally adiabatic approximation is used to circumvent problems in classical mechanics associated with nonconservation of zero-point energy. For normal incidence, we find good agreement between the classical and quantum dissociation probabilities. Both are dominated by strong steering into geometries favorable for dissociation. Exciting either initial rotations or parallel translations reduces the effectiveness of steering, thus reducing the dissociation probability. For the former, the good agreement between classical and quantum still holds, however, for the latter this is not so, and the disagreement becomes progressively greater as the initial parallel momentum is increased. To understand this, we employ two new visualization techniques. For the classical dynamics, we use swarms of color-coded trajectories to illustrate the detailed motion of an ensemble. While for the quantum system, we project time-dependent wave packets onto a local basis set, adiabatic in the internal (rotational and parallel translational) degrees of freedom, plotting the results on the correlation diagram. Examination of these shows that the behavior of adiabatic states populated for normal incidence is dominated by a combination of steering and orientational hindering, both classical phenomena. In contrast, there are very large avoided crossings in the states populated at off-normal incidence, leading to motion which cannot be reproduced by a classical system.

This work will discuss the effects of angular momentum for the multipactor electrons in a coaxial line. In particular, analytical modeling of a multipactor discharge in the two-dimensional annular cross-sectional area of a coaxial transmission line will be considered. The strongly coupled governing equations of motion for the electrons may be decoupled and combined using techniques from the central-force problem in classical mechanics. A closed form approximate analytical formula for the multipactor electron trajectory will then be obtained from the resulting simplified orbital equation.