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11 pages, 296 KiB

Open AccessArticle

On the Periodic Orbits of the Perturbed Two- and Three-Body Problems

by Elbaz I. Abouelmagd, Juan Luis García Guirao and Jaume Llibre

Galaxies 2023, 11(2), 58; https://doi.org/10.3390/galaxies11020058 - 18 Apr 2023

Cited by 8 | Viewed by 2062

In this work, a perturbed system of the restricted three-body problem is derived when the perturbation forces are conservative alongside the corresponding mean motion of two primaries bodies. Thus, we have proved that the first and second types of periodic orbits of the rotating Kepler problem can persist for all perturbed two-body and circular restricted three-body problems when the perturbation forces are conservative or the perturbed motion has its own extended Jacobian integral. Full article

11 pages, 301 KiB

Open AccessArticle

Periodic Orbits of Quantised Restricted Three-Body Problem

by Elbaz I. Abouelmagd, Juan Luis García Guirao and Jaume Llibre

Universe 2023, 9(3), 149; https://doi.org/10.3390/universe9030149 - 15 Mar 2023

Cited by 7 | Viewed by 1528

Abstract

In this paper, perturbed third-body motion is considered under quantum corrections to analyse the existence of periodic orbits. These orbits are studied through two approaches to identify the first (second) periodic-orbit types. The essential conditions are given in order to prove that the circular (elliptical) periodic orbits of the rotating Kepler problem (RKP) can continue to the perturbed motion of the third body under quantum corrections where a massive primary body has excessive gravitational force over the smaller primary body. The primaries moved around each other in circular (elliptical) orbits, and the mass ratio was assumed to be sufficiently small. We prove the existence of the two types of orbits by using the terminologies of Poincaré for quantised perturbed motion. Full article

(This article belongs to the Section Foundations of Quantum Mechanics and Quantum Gravity)

19 pages, 333 KiB

Open AccessArticle

Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits

by Vladislav N. Kovalnogov, Ruslan V. Fedorov, Andrey V. Chukalin, Theodore E. Simos and Charalampos Tsitouras

Mathematics 2021, 9(23), 3071; https://doi.org/10.3390/math9233071 - 29 Nov 2021

Cited by 7 | Viewed by 1233

Abstract

The family of Numerov-type methods that effectively uses seven stages per step is considered. All the coefficients of the methods belonging to this family can be expressed analytically with respect to four free parameters. These coefficients are trained through a differential evolution technique in order to perform best in a wide range of Keplerian-type orbits. Then it is observed with extended numerical tests that a certain method behaves extremely well in a variety of orbits (e.g., Kepler, perturbed Kepler, Arenstorf, Pleiades) for various steplengths used by the methods and for various intervals of integration. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

9 pages, 324 KiB

Open AccessArticle

Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits

by Yu-Cheng Shen, Chia-Liang Lin, Theodore E. Simos and Charalampos Tsitouras

Mathematics 2021, 9(12), 1342; https://doi.org/10.3390/math9121342 - 10 Jun 2021

Cited by 7 | Viewed by 2858

Abstract

We consider a family of explicit Runge–Kutta pairs of orders six and five without any additional property (reduced truncation errors, Hamiltonian preservation, symplecticness, etc.). This family offers five parameters that someone chooses freely. Then, we train them in order for the presented method to furnish the best results on a couple of Kepler orbits, a certain interval and tolerance. Consequently, we observe an efficient performance on a wide range of orbital problems (i.e., Kepler for a variety of eccentricities, perturbed Kepler with various disturbances, Arenstorf and Pleiades). About

1.8

digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Physics, Chemistry, Biology, and Economics)

12 pages, 279 KiB

Open AccessFeature PaperArticle

Periodic Orbits of Third Kind in the Zonal J₂ + J₃ Problem

by M. Teresa de Bustos, Antonio Fernández, Miguel A. López, Raquel Martínez and Juan A. Vera

Symmetry 2019, 11(1), 111; https://doi.org/10.3390/sym11010111 - 18 Jan 2019

Cited by 1 | Viewed by 3006

Abstract

In this work, the periodic orbits’ phase portrait of the zonal

J_{2} + J_{3}

problem is studied. In particular, we center our attention on the periodic orbits of the third kind in the Poincaré sense using the averaging theory of dynamical [...] Read more.

In this work, the periodic orbits’ phase portrait of the zonal

J_{2} + J_{3}

problem is studied. In particular, we center our attention on the periodic orbits of the third kind in the Poincaré sense using the averaging theory of dynamical systems. We find three families of polar periodic orbits and four families of inclined periodic orbits for which we are able to state their explicit expressions. Full article

(This article belongs to the Special Issue New Trends in Dynamics)

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