Problems Of Celestial Mechanics Research Articles

Novel techniques to analyze dynamical properties of quantum chaos with peculiar evidence of hybrid systems confinement

Recent results demonstrate the dynamical peculiarities of the quantum chaos within the hybrid systems by chaotic parameters and probe the pattern formation under the influence of condensation. The complex dynamic behavior of the considered systems was determined with numerical simulation and presented an efficient technique that studied fractional systems comprising chaos-coherence fractions. The findings divulge the peculiar association between the coherence structure and the correlations at finite relative momenta. Thus the present study helps to explore the partially chaos hybrid systems in order to stimulate the experimental applications of nonlinear phenomena. The coherent-chaotic parameters can be measured by examining the chaos peculiarities that possess explicit relations with the condensations to demonstrate the environs of the physical systems. We investigate the influence of the multiplicities, chaos, momentum and temperature of the nonlinear system on the coherent-chaotic normalized correlations. The chaotic parameters are suppressed considerably with the coherence fraction and it appears numerically zero at maximum condensation and one at ideal chaos emissions. We procure that the meaningful parameters decrease significantly with the multiplicity of the nonlinear systems and increase with the momentum in the specified regimes. The identical multiplicity leads to contemplating the coherence and thus the normalized chaotic parameters within its spectacular influences exhibit significance worth contemplating in earnest. The findings underscore the significance of cogitating correlations in deciphering the nonlinear system characteristics and bestowing extraordinary perceptiveness into the convoluted essence of complex systems. The contemplated methodology can be applied to evaluating and analyzing the nonlinear systems and such an innovative approach computes the problems of celestial mechanics, heartbeats and chemical reactions in engineering and medical fields.

Chaotic Phenomena for Generalised N-centre Problems

We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological conjugation with the archetypal chaotic dynamical system, the Bernoulli shift. After providing infinitely many geometrically distinct and collision-less periodic solutions, we encode them in bi-infinite sequences of symbols. Solutions are obtained as minimisers of the Maupertuis functional in suitable free homotopy classes of the punctured surface, without any collision regularisation. For any sufficiently large value of the energy, we prove that the generalised N-centre problem admits a symbolic dynamics. Moreover, when the Jacobi-Maupertuis metric curvature is negative, we construct chaotic invariant subsets.

Модернизация интегратора R0 для вычисления эфемерид астероидов: учет релятивистских и негравитационных эффектов

Currently, there are many tools for calculating the trajectories of asteroids, comets, and meteor particles in the Solar System. However, only a portion of them are available to the astronomical community. Many of them do not always meet the requirements for accuracy and calculation speed. Since there is no universal solution here, the development of new methods that combine speed and accuracy following specific problems of celestial mechanics does not lose its importance. There is an attempt to combine the advantages of two trajectory calculation methods in this work. The first of them was developed at Saint Petersburg State University to quickly calculate the trajectories of a large number of particles and was implemented as a program R0. A significant disadvantage of this R0 realization is relatively low accuracy. The second one is designed to calculate the trajectory with maximum accuracy and is implemented in the form of the ASSIST software package. However, it provides lower performance. One of the reasons is the usage of complex motion equations. Some simplified models are compared with data obtained using NASA services. An assessment of the applicability of the upgraded R0 program is given. However, only part of them are available to the astronomical community. Many of them do not always meet the requirements for accuracy and calculation speed. Since there is no universal solution here, the development of new methods that combine speed and accuracy following specific problems of celestial mechanics does not lose its importance. There is an attempt to combine the advantages of two trajectory calculation methods in this work. The first of them was developed at Saint Petersburg State University to quickly calculate the trajectories of a large number of particles and was implemented as a program R0. A significant disadvantage of this R0 realization is relatively low accuracy. The second one is designed to calculate the trajectory with maximum accuracy and is implemented in the form of the ASSIST software package. However, it provides lower performance. One of the reasons is the usage of complex motion equations. Some simplified models are compared with data obtained using NASA services. An assessment of the applicability of the upgraded R0 program is given.

Trigonometric Approximation of the Three-Body Problem: A Double Centroid Approach

_This manuscript introduces the Double Centroid Approximation, a trigonometric method to simplify the Three-body problem in celestial mechanics. By calculating the centroid of three celestial bodies without and then with motion, it offers a novel approach to determine their dynamic center of mass. Grounded in the Superconducting Field Theory, this method simplifies complex gravitational interactions and dark matter considerations. While applicable within our solar system, its principles may extend to other galaxies, highlighting the potential for broad astronomical applications. This concise methodology underscores the importance of triangulation in understanding celestial mechanics, providing a foundation for future research in the field._

Theory and applications of fast Lyapunov indicators to model problems of celestial mechanics

Theory and applications of fast Lyapunov indicators to model problems of celestial mechanics

Numerical studies to detect chaotic motion in the full planar averaged three-body problem

In this paper, the author deals with a well-known problem of Celestial Mechanics, namely the three-body problem. A numerical analysis has been done in order to prove existence of chaotic motions of the full-averaged problem in particular configurations. Full because all the three bodies have non-negligible masses and averaged because the Hamiltonian describing the system has been averaged with respect to a fast angle. A reduction of degrees of freedom and of the phase-space is performed in order to apply the notion of covering relations and symbolic dynamics.

Old perturbative methods for a new problem in Celestial Mechanics: the space debris dynamics

Perturbative methods have been developed and widely used in the XVIII and XIX century to study the behavior of N-body problems in Celestial Mechanics. Such methods apply to nearly-integrable Hamiltonian systems and they have the remarkable property to be constructive. A well-known application of perturbative techniques is represented by the construction of the so-called proper elements, which are quasi-invariants of the dynamics, obtained by removing the perturbing function to higher orders. They have been used to identify families of asteroids; more recently, they have been used in the context of space debris, which is the main core of this work. We describe the dynamics of space debris, considering a model including the Earth’s gravitational attraction, the influence of Sun and Moon, and the Solar radiation pressure. We construct a Lie series normalization procedure and we compute the proper elements associated to the orbital elements. To provide a concrete example, we analyze three different break-up events with nearby initial orbital elements. We use the information coming from proper elements to successfully group the fragments; the clusterization is supported by statistical data analysis and by machine learning methods. These results show that perturbative methods still play an important role in the study of the dynamics of space objects.

Artificial equilibrium points and their linear stability analysis in the solar sail problem with triaxial second primary

Motion of a solar sail or a spacecraft equipped with a solar sail is one of the interesting problem of celestial mechanics. This paper deals with stationary solution of solar sail problem with radiating first primary and triaxial second primary under the frame of circular restricted three body problem (RTBP). First, equations of motion is derived with triaxiality and sail lightness number and then artificial equilibrium points (AEPs) are determined. It is noticed that impact of triaxiality parameters (σ1andσ2) of second primary is less than that of sail lightness number (β) due to which a considerable deviations in the positions of AEPs are occurred. Further, linear stability test is performed for all AEPs and it is found that similar to equilibrium points in classical RTBP, collinear AEPs are unstable for all mass parameter μ∈(0,1/2] and triangular AEPs are stable in the range 0<μ<μc=0.03758902. Moreover, stability range decreases with the values of β and σ1 but increases with σ2. After this we analyze the periodic orbits of the solar sail in the vicinity of AEPs and it is noticed that nature of motion is unaffected but time period is influenced from the sail number β and triaxiality parameters σ1 and σ2. The findings are useful to describe the more generalized solar sail problem with other perturbations such as oblateness, albedo, drag forces etc.

The Number of Relative Equilibria in the PCR4BP

AbstractThe aim of this paper is to present a new, analytical, method for computing the exact number of relative equilibria in the planar, circular, restricted 4-body problem of celestial mechanics. The new approach allows for a very efficient computer-aided proof, and opens a potential pathway to proving harder instances of the n-body problem.

Spectral stability, spectral flow and circular relative equilibria for the Newtonian n-body problem

For the Newtonian (gravitational) n-body problem in the Euclidean d-dimensional space, d≥2, the simplest possible periodic solutions are provided by circular relative equilibria, (RE) for short, namely solutions in which each body rigidly rotates about the center of mass and the configuration of the whole system is constant in time and central (or, more generally, balanced) configuration. For d≤3, the only possible (RE) are planar, but in dimension four it is possible to get truly four dimensional (RE).A classical problem in celestial mechanics aims at relating the (in-)stability properties of a (RE) to the index properties of the central (or, more generally, balanced) configuration generating it. In this paper, we provide sufficient conditions that imply the spectral instability of planar and non-planar (RE) in R4 generated by a central configuration, thus answering some of the questions raised in [14, Page 63]. As a corollary, we retrieve a classical result of Hu and Sun [11] on the linear instability of planar (RE) whose generating central configuration is non-degenerate and has odd Morse index, and fix a gap in the statement of [6, Theorem 1] about the spectral instability of planar (RE) whose (possibly degenerate) generating central configuration has odd Morse index.The key ingredient is a new formula of independent interest that allows to compute the spectral flow of a path of symmetric matrices having degenerate starting point, and a symplectic decomposition of the phase space of the linearized Hamiltonian system along a given (RE) which is inspired by Meyer and Schmidt's planar decomposition [13] and which allows us to rule out the uninteresting part of the dynamics corresponding to the translational and (partially) to the rotational symmetry of the problem.

Classification of regular and chaotic motions in Hamiltonian systems with deep learning

This paper demonstrates the capabilities of convolutional neural networks (CNNs) at classifying types of motion starting from time series, without any prior knowledge of the underlying dynamics. The paper applies different forms of deep learning to problems of increasing complexity with the goal of testing the ability of different deep learning architectures at predicting the character of the dynamics by simply observing a time-ordered set of data. We will demonstrate that a properly trained CNN can correctly classify the types of motion on a given data set. We also demonstrate effective generalisation capabilities by using a CNN trained on one dynamic model to predict the character of the motion governed by another dynamic model. The ability to predict types of motion from observations is then verified on a model problem known as the forced pendulum and on a relevant problem in Celestial Mechanics where observational data can be used to predict the long-term evolution of the system.

Efficient and Accurate KAM Tori Construction for the Dissipative Spin–Orbit Problem Using a Map Reduction

We consider the dissipative spin–orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and drift. Our goal is to construct quasi-periodic solutions with fixed frequency, satisfying appropriate conditions. With the goal of applying rigorous KAM theory, we compute such quasi-periodic solutions with very high precision. To this end, we have developed a very efficient algorithm. The first step is to compute very accurately the return map to a surface of section (using a high-order Taylor’s method with extended precision). Then, we find an invariant curve for the return map using recent algorithms that take advantage of the geometric features of the problem. This method is based on a rapidly convergent Newton’s method which is guaranteed to converge if the initial error is small enough. So, it is very suitable for a continuation algorithm. The resulting algorithm is quite efficient. We only need to deal with a one-dimensional function. If this function is discretized in N points, the algorithm requires \(O(N \log N) \) operations and O(N) storage. The most costly step (the numerical integration of the equation along a turn) is trivial to parallelize. The main goal of the paper is to present the algorithms, implementation details and several sample results of runs. We also present both a rigorous and a numerical comparison of the results of averaged and not averaged models.

Three New Models of Layered Inhomogeneous Elliptical Galaxies

To solve some problems of celestial mechanics and astrophysics, three new models of an elliptical galaxy (EG) have been created, which are in good agreement with modern understanding of the structure of such galaxies. Based on these models, the total gravitational (potential) energy and rotational kinetic energy of an EG, as well as the velocity dispersion at a distance of its effective radius, are determined. A new method is proposed for determining the average values of the scale radius of an EG and the density at its center, as well as the average value of its key parameter density $\beta$ and its value at a distance corresponding to the effective radius of the galaxy. The results obtained are applied to sixty EGs and presented in the form of tables for ten galaxies.

К ОБРАТНОЙ ЗАДАЧЕ НЕБЕСНОЙ МЕХАНИКИ

The link between exterior solutions to the Einstein gravitational field equations such as the exact Erez-Rosen metric and approximate Hartle-Thorne metric is established here for the static case in the limit of linear mass quadrupole moment (Q) and second order terms in total mass (M). To this end, the Geroch-Hansen multipole moments are calculated for the Erez-Rosen and Hartle-Thorne solutions in order to find the relationship among the parameters of both metrics. The coordinate transformations are sought in a general form with two unknown functions in the corresponding limit of ~Q and ~M^2. By employing the perturbation theory, the approximate Erez-Rosen metric is written in the same coordinates as the Hartle-Thorne metric. By equating the radial and azimuthal components of the metric tensor of both solutions the sought functions are found in a straightforward way. It is shown that the approximation ~Q and ~M^2, which is used throughout the article, is physical and suitable for solving most problems of celestial mechanics in post-Newtonian physics. This approximation does not require the use of the Zipoy-Voorhees transformation, which is a necessary strict mathematical requirement in the ~Q approximation, i.e. when no other approximations are made. This implies that the explicit form of the coordinate transformations depends entirely on the approximation that is adopted in each particular case. The results obtained here are in agreement with the previous results in the literature and can be applied to different astrophysical goals. The paper pursues not only pure scientific, but also academic purposes and can be used as an auxiliary and additional material to the special courses of general theory of relativity, celestial mechanics and relativistic astrophysics.

Orbital Dynamics in the Restricted Three Body Problem: Overview of Recent Analytical Advances Obtained by Separating Rapid and Slow Subsystems in Non-Planar Configurations

Analytical solutions to a variety of simplified versions of the restricted three-body problem in celestial mechanics possess long running history that encompasses several centuries. Most of the successes were limited either to the planar configuration of the three bodies, to the motion around the Lagrange points, or to the Kozai–Lidov effect. We review some analytical advances obtained by separating rapid and slow subsystems as presented in recently published papers concerning the non-planar motion of the three bodies unrelated to the Lagrange points and to the Kozai–Lidov effect. Most (but not all) of the discussed advances correspond to the bound motion in the considered celestial systems.

Ejection–Collision Orbits in Two Degrees of Freedom Problems in Celestial Mechanics

In a general setting of a Hamiltonian system with two degrees of freedom and assuming some properties for the undergoing potential, we study the dynamics close and tending to a singularity of the system which in models of N-body problems corresponds to total collision. We restrict to potentials that exhibit two more singularities that can be regarded as two kind of partial collisions when not all the bodies are involved. Regularizing the singularities, the total collision transforms into a 2-dimensional invariant manifold. The goal of this paper is to prove the existence of different types of ejection–collision orbits, that is, orbits that start and end at total collision. Such orbits are regarded as heteroclinic connections between two equilibrium points and are mainly characterized by the partial collisions that the trajectories find on their way. The proof of their existence is based on the transversality of 2-dimensional invariant manifolds and on the behavior of the dynamics on the total collision manifold; both of them are thoroughly described.

An improved C++ Poisson series processor with its applications

One of major problems in celestial mechanics is the management of the long developments in Fourier or Poisson series used to describe the perturbed motion in the planetary system. In this work we will develop a software package suitable for managing these objects. This package includes the ordinary arithmetic operations with Poisson series—such as sum or product—as well as the most commonly used functions sin , cos , or exp , among others. Derivation or integration procedures with respect to time of these objects have been implemented and inversion or serialization procedures are also attainable to obtain such series. All the programming has been accomplished overloading the arithmetic and functional operators natural to C++ with the intention of allowing the programmer to work in a more friendly way and treating the series as if they were mere numbers. In this work we extend the processor in order to obtain the solution to perturbed problems which solution is in the form of a Poisson series. These algorithms have been written in C++.

Parabolic orbits in Celestial Mechanics: a functional‐analytic approach

We prove the existence of half-entire parabolic solutions, asymptotic to a prescribed central configuration, for the equation \begin{equation*} \ddot{x} = \nabla U(x) + \nabla W(t,x), \qquad x \in \mathbb{R}^{d}, \end{equation*} where $d \geq 2$, $U$ is a positive and positively homogeneous potential with homogeneity degree $-\alpha$ with $\alpha\in\mathopen{]}0,2\mathclose{[}$, and $W$ is a (possibly time-dependent) lower order term, for $\vert x \vert \to +\infty$, with respect to $U$. The proof relies on a perturbative argument, after an appropriate formulation of the problem in a suitable functional space. Applications to several problems of Celestial Mechanics (including the $N$-centre problem, the $N$-body problem and the restricted $(N+H)$-body problem) are given.

ON APPROXIMATION OF ALMOST-PERIODIC SOLUTIONS FOR A NON-LINEAR COUNTABLE SYSTEM OF DIFFERENTIAL EQUATIONS BY QUASI-PERIODIC SOLUTIONS FOR SOME LINEAR SYSTEM

It is well-known that many applied problems in different areas of mathematics, physics, and technology require research into questions of existence of oscillating solutions for differential systems, which are their mathematical models. This is especially true for the problems of celestial mechanics. Novadays, by oscillatory motions in dynamical systems, according to V. V. Nemitsky, we call their recurrent motions. As it is known from Birkhoff theorem, trajectories of such motions contain minimal compact sets of dynamical systems. The class of recurrent motions contains, in particular, both quasi-periodic and almost-periodic motions. There are renowned fundamental theorems by Amerio and Favard related to existence of almost-periodic solutions for linear and non-linear systems. It is also of interest to research the behavior of a dynamical system’s motions in a neighborhood of a recurrent trajectory. It became understood later, that the question of existence of such trajectories is closely related to existence of invariant tori in such systems, and the method of Green-Samoilenko function is useful for constructing such tori. Here we consider a non-linear system of differential equations defined on Cartesian product of the infinite-dimensional torus T∞ and the space of bounded number sequences m. The problem is to find sufficient conditions for the given system of equations to possess a family of almost-periodic in the sense of Bohr solutions, dependent on the parameter ψ ∈ T∞, every one of which can be approximated by a quasi-periodic solution of some linear system of equations defined on a finite-dimensional torus.

First kind symmetric periodic solutions and their stability for the Kepler problem and anisotropic Kepler problem plus generalized anisotropic perturbation

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solutions and also we characterize its type of stability. As an application of this general situation, we discuss the existence of symmetric periodic solutions for the anisotropic Kepler problem plus a generalized anisotropic perturbation, (shortly, p-AKPQ problem) and for the Kepler problem plus a generalized anisotropic perturbation (shortly, p-KPQ problem), as continuation of circular orbits of the two-dimensional Kepler problem. To get this objective, we consider different types of perturbations and then we apply our main result.