| Analytical Lunar Ephemeris: The Mean Motions, |
AUG 1970 |
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| Authors:
Andre Deprit; Jacques Henrard; Arnold Rom; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICAL AND INFORMATION SCIENCES LAB
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 | The Main Problem of Lunear Theory has been solved in a completely literal form; at places the development reaches terms of order 21 in the secularized Hamiltonian. In order to assess the accuracy reached in this manner, the constants are given the numerical values they have in the Improved Lunar Ephemeris and the series for the mean motions of node and perigee are evaluated. There is agreement on the principal ... |
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| ANALYTICAL LUNAR EPHEMERIS. 1. DEFINITION OF THE MAIN PROBLEM, |
MAR 1970 |
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| Authors:
Andre Deprit; Jacques Henrard; Arnold Rom; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| A new set of phase variables is proposed and justified to develop automatically by computer the solar terms in Lunar Theory. The dependence of the perturbation function on the mass ratio Moon/(Earth + Moon) is completely elucidated. A recursive procedure is proposed to develop that function so as to keep explicit all its d'Alembert characteristics. The perturbation series obtained by computer is compared with Delaunay's development. (Author) |
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| ON A PERTURBATION THEORY USING LIE TRANSFORMS, |
NOV 1969 |
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| Authors:
Jacques Henrard; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| Kamel has recently extended to non-Hamiltonian equations a perturbation theory using Lie transforms. It is shown in this report extension can be approached from an intrinsic viewpoint, which reformulation leads to a simpler algorithm. Kamel's contribution is then completed by establishing rules for inverting the transformation generated by the perturbation theory, and for composing two such transformations. (Author) |
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| THE HALOING EFFECT OF THE THIRD INTEGRAL, |
NOV 1969 |
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| Authors:
Andre Deprit; Jacques Henrard; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| Whether Contopoulos' galactic system is separable (unlikely) or not (likely), the fact is that there exists a vicinity of the equilibrium in which numerical integration of high accuracy cannot separate the system from its image through a Birkhoff's normalization of high order. To all practical purposes, Stellar Dynamics is then justified in pretending that the model is, in that region, structured by a so-called third integral. (Author) |
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| CONCERNING THE GENEALOGY OF LONG PERIOD FAMILIES AT L SUBSCRIPT 4, |
SEP 1969 |
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| Authors:
Jacques Henrard; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| The family of long period orbits at L sub 4 does not evolve in a continuous manner with the mass ratio. Discontinuities appear not only at the mass ratios for which the long period at the equilibrium is a multiple of the short period, but also at mass ratios where the global analysis of the family detects singular bifurcation orbits. Extended numerical continuations carried with very great accuracy enable us ... |
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| THE TROJAN MANIFOLD -SURVEY AND CONJECTURES-. |
SEP 1969 |
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| Authors:
Andre Deprit; Jacques Henrard; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| Recent results concerning the families of periodic orbits emanating from the triangular equilibrium L sub 4 are interpreted in an attempt to establish the evolution of these manifolds as the mass ratio varies from Routh's critical value down to its value in the system Sun-Jupiter. (Author) |
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| PERIODIC ORBITS EMANATING FROM A RESONANT EQUILIBRIUM, |
JUN 1969 |
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| Authors:
Jacques Henrard; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| For a conservative Hamiltonian system with two degrees of freedom, in the case where the two frequencies at an equilibrium of the elliptic type are commensurable or close to being so, completely canonical transformations can be formally constructed in explicit terms under the form of Lie transforms to the effect that it renders one angle coordinate ignorable and gives to the transformed Hamiltonian the form of what Garfinkel calls an ... |
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| CONSTRUCTION OF ORBITS ASYMPTOTIC TO A PERIODIC ORBIT, |
APR 1968 |
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| Authors:
Andre Deprit; Jacques Henrard; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| In the immediate neighborhood of an unstable periodic orbit, the families of orbits asymptotic to it may be expanded in power series of an orbital parameter, the coefficients being successive variations of increasing order from the generating orbit. When the dynamical system is Hamiltonian, conservative and with two degrees of freedom, the intrinsic components of these variations are shown to be solutions of a recurrent sequence consisting at each step ... |
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| A MANIFOLD OF PERIODIC ORBITS, |
JUN 1967 |
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| Authors:
Andre Deprit; Jacques Henrard; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| In the restricted problem of three bodies, when the mass ratio is such that the characteristic exponents at L sub 4 are equal in pair, the triangular equilibrium is a point of ramification in the analytical manifold of periodic orbits emanating from L sub 4: the branch L sub 4 superscript s of short period orbits can be continued through L sub 4 by the branch L sub 4 superscript ... |
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| THE TROJAN MANIFOLD IN THE SYSTEM EARTH-MOON, |
FEB 1967 |
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| Authors:
A. Deprit; Jacques Henrard; Julian Palmore; J. F. Price; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| The Trojan manifold is defined as the analytical manifold of periodic orbits which contains the triangular equilibrium L4 as a singularity. Identification of the Earth-Moon system is made to a planar Restricted Problem of Three Bodies. The barycentric synodical coordinate system and the units of length, time and mass are chosen as defined by Wintner; the mass ratio is taken equal to about 0.012150. For this value, the triangular configuration ... |
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| NATURAL FAMILIES OF PERIODIC ORBITS, |
OCT 1966 |
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| Authors:
Andre Deprit; Jacques Henrard; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| In reference to any solution of a conservative dynamical system with two degrees of freedom, Hill's equation is generalized to encompass non-necessarily isoenergetic displacements as well as the isonergetic displacements caused by a variation of a parameter. This new variational equation is made the foundation of a methodical procedure for continuing numerically natural families of periodic orbits. The method consists of two steps-- an isoenergetic corrector and a tangential predictor. ... |
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| TROJAN ORBITS. II. BIRKHOFF'S NORMALIZATION, |
AUG 1966 |
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| Authors:
Andre Deprit; Jacques Henrard; A. R. M. Rom; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| In the Restricted Problem of Three Bodies for the system Sun-Jupiter, all terms of short and long period up to degree 13 are eliminated from the Hamiltonian expanded in power series in the neighborhood of an equilateral center of libration. The normalizing canonical transformation expresses the Cartesian phase variables as double Fourier series in two angle coordinates whose coefficients are power series in the square roots of two action momenta. ... |
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| NORMALIZATION AT L4, |
JUL 1966 |
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| Authors:
Andre Deprit; Jacques Henrard; A. R. M. Rom; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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| Birkhoff's normalization about a stable equilibrium for a conservative Hamiltonian system with two degrees of freedom can be implemented in a direct way: the necessary canonical transformation is built up explicitly, without inversions, by the method of undetermined coefficients. This procedure has been programmed symbolically for an electronic computer. It has been applied up to order 13 to the equilateral center of libration in the Restricted Problem of Three Bodies ... |
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Reports by Author
