Contribution to the solution of the three‐body problem in power series form

Sconzo, Pasquale

doi:10.1002/asna.19672900405

Cited by 3 publications

(3 citation statements)

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“…Muchos resultados obtenidos acerca de la dinámica de homeomorfismos definidos en el anillo están motivados históricamente por sistemas mecánicos y problemas geométricos. Uno de los problemas famosos en la física es el problema de los tres cuerpos, el cual ha sido estudiado ampliamente con resultados interesantes [55]- [61]. Este problema consiste en determinar, en cualquier instante, las velocidades y las posiciones de tres cuerpos de cualquier masa, sometidos entre sí a una atracción mutua y partiendo de unas velocidades y posiciones dadas.…”

Section: El Número De Rotación Y Homeomorfismos En El Anillounclassified

Sobre el número de rotación en el círculo y el anillo

Sánchez Salazar

2023

Rev. Fac. Cienc. Básicas

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Uno de los conceptos y herramientas fundamentales para el estudio de la existencia de órbitas periódicas es el número de rotación, que fue introducido por Henri Poincaré hacia el año de 1913, y juega un papel fundamental en la descripción cualitativa de sistemas dinámicos. En este trabajo, se hará una revisión detallada de las propiedades del número de rotación en el círculo y el anillo, y presentaremos algunas aplicaciones a los difeomorfismos Morse-Smale, el flujo billiard y el mapa de retorno geodésico.

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Section: El Número De Rotación Y Homeomorfismos En El Anillounclassified

Sobre el número de rotación en el círculo y el anillo

Sánchez Salazar

2023

Rev. Fac. Cienc. Básicas

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“…In fact, it is known for a long time by the (thus traditionnal) Taylor-series-method users that the X k may be determined more efficiently by iteration provided f (x, t) be simple enough (see, e.g. [114,115,116], see also [41, pp. 47-49]).…”

Section: Getting the Raw Taylor Seriesmentioning

confidence: 99%

Status of the differential transformation method

Bervillier

2012

Applied Mathematics and Computation

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Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages -numerical routines-for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated. It is concluded that, for the sake of clarity, when the DTM is applied to ODEs, it should be mentioned that the DTM exactly coincides with the traditional Taylor method, contrary to what is currently written.

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“…Sconzo has further employed FORMAC to verify Kozai's second order artificial satellite theory (Kozai 1962)) and Klumpp and Kalan (1970) have used FORMAC for computations arising from the Apollo 12 mission. Sconzo (1967) gives two machine methods for the solution of the three body problem in series. Gustavson (1966)' using a system of his own design, has computed certain formal integrals of motion for a problem in galactic structure, while Hamid (1968) has worked on general planetary theory.…”

Section: Direct Applicationsmentioning

confidence: 99%

Applications of algebraic manipulation programs in physics

Barton

Fitch

1972

Rep. Prog. Phys.

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Barton andJ P Fitch algebraic manipulation is given. Further applications of such systems to mathematics, chemistry and engineering are briefly mentioned in the text and relevant work is referenced in the bibliography but the main emphasis is placed on applications in theoretical physics. However, the simple examples indicate, and the applications in the physical sciences confirm, that algebra systems are capable of exploitation over a much wider area than is covered in the present review.This review was completed in December 1971.

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Contribution to the solution of the three‐body problem in power series form

Cited by 3 publications

References 8 publications

Sobre el número de rotación en el círculo y el anillo

Sobre el número de rotación en el círculo y el anillo

Status of the differential transformation method

Applications of algebraic manipulation programs in physics

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