Deprit, André
Overview
Works:  46 works in 74 publications in 3 languages and 188 library holdings 

Roles:  Author 
Classifications:  Q56, 531.01517 
Publication Timeline
.
Most widely held works by
André Deprit
Compression of ephemerides by discrete Chebyshev approximations by
André Deprit(
Book
)
4 editions published in 1979 in English and held by 53 WorldCat member libraries worldwide
Polynomial representations of astronomical ephemerides are usually derived from discrete leastsquares approximations. Ideally, to ensure a uniform distribution of the error, one should aim at a continuous Chebyshev approximation. This is feasible when the ephemeris is generated from a literal (analytical or semianalytical) development. But a discrete Chebyshev approximation is a realistic compromise. Application to the moon and geosynchronous satellites has given good results. On the whole, long ranges (several times the sidereal period) may be covered by polynomials of degree 30 to 50 with a moderate error. A lowdegree approximation over half the period usually delivers a high accuracy. Gibbs' phenomena, i.e. rapid oscillations of increasing amplitudes in the error curve at both ends of the approximation interval, are of course absent, contrary to what usually happens in a leastsquares approximation. (Author)
4 editions published in 1979 in English and held by 53 WorldCat member libraries worldwide
Polynomial representations of astronomical ephemerides are usually derived from discrete leastsquares approximations. Ideally, to ensure a uniform distribution of the error, one should aim at a continuous Chebyshev approximation. This is feasible when the ephemeris is generated from a literal (analytical or semianalytical) development. But a discrete Chebyshev approximation is a realistic compromise. Application to the moon and geosynchronous satellites has given good results. On the whole, long ranges (several times the sidereal period) may be covered by polynomials of degree 30 to 50 with a moderate error. A lowdegree approximation over half the period usually delivers a high accuracy. Gibbs' phenomena, i.e. rapid oscillations of increasing amplitudes in the error curve at both ends of the approximation interval, are of course absent, contrary to what usually happens in a leastsquares approximation. (Author)
Contribution à l'étude de l'algèbre des applications linéaires continues d'un espace localement convexe séparé: théorie
de Rieszthéorie spectrale by
André Deprit(
Book
)
10 editions published between 1958 and 1959 in French and held by 37 WorldCat member libraries worldwide
10 editions published between 1958 and 1959 in French and held by 37 WorldCat member libraries worldwide
Edición de documentos en LATEX by
André Deprit(
Book
)
4 editions published between 1990 and 1992 in Spanish and held by 18 WorldCat member libraries worldwide
4 editions published between 1990 and 1992 in Spanish and held by 18 WorldCat member libraries worldwide
Mécanique rationnelle by
André Deprit(
Book
)
2 editions published between 1967 and 1970 in French and held by 8 WorldCat member libraries worldwide
2 editions published between 1967 and 1970 in French and held by 8 WorldCat member libraries worldwide
A manifold of periodic orbits by
André Deprit(
Book
)
2 editions published in 1967 in English and held by 5 WorldCat member libraries worldwide
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 l of long period orbits, and this real analytical continuation is unique. The branch L sub 4 superscript l ends with an orbit traveled twice which is an element of L sub 4 superscript s. The branch L sub 4 superscript s meets its mirror image L sub 5 superscript s of short period orbits around L sub 5 on a symmetric orbit which is also an element of the branch L sub 3 of periodic solutions emanating from the collinear equilibrium L sub 3. Around L sub 4, the branches L sub 4 superscript l and L sub 4 superscript s are connected by bridges B(pL, qS) of periodic orbits which start from a long period orbit traveled p times and end with a short period one traveled q times. We have completely explored the bridges B(2L,3S), B(3L,4S) and B(4L,5S). (Author)
2 editions published in 1967 in English and held by 5 WorldCat member libraries worldwide
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 l of long period orbits, and this real analytical continuation is unique. The branch L sub 4 superscript l ends with an orbit traveled twice which is an element of L sub 4 superscript s. The branch L sub 4 superscript s meets its mirror image L sub 5 superscript s of short period orbits around L sub 5 on a symmetric orbit which is also an element of the branch L sub 3 of periodic solutions emanating from the collinear equilibrium L sub 3. Around L sub 4, the branches L sub 4 superscript l and L sub 4 superscript s are connected by bridges B(pL, qS) of periodic orbits which start from a long period orbit traveled p times and end with a short period one traveled q times. We have completely explored the bridges B(2L,3S), B(3L,4S) and B(4L,5S). (Author)
Natural orbits of the first kind in the restricted problem by
André Deprit(
Book
)
2 editions published in 1967 in English and held by 4 WorldCat member libraries worldwide
Lindstedt's method is applied to expand analytically the natural families of periodic orbits of the first kind in the restricted problem of three bodies. The mass ratio is kept as a literal variable; the series proceed in the powers of Hill's ratio of periods. They cover all four cases at once, namely the direct or retrograde orbits for either inferior planets or for satellites. When the mass ratio is given a numerical value, the expansions contain less terms and thus can be carried up to degree 17 within a 32K core of an IBM 7094. For the system SunJupiter, the initial conditions provided by the series have been corrected to yield the beginnings of all four natural families, and the characteristic exponents have been computed. Comparison between the values computed out of the series and their improvement by numerical integrations and successive variational corrections show that, within an accuracy of one part in ten thousand, the series represent moderately well even the orbit of J VIII; the retrograde planetary orbits are fairly well covered up to 90% of the distance from SunJupiter, whereas the direct planetary orbits are covered only up to 60% of that distance. (Author)
2 editions published in 1967 in English and held by 4 WorldCat member libraries worldwide
Lindstedt's method is applied to expand analytically the natural families of periodic orbits of the first kind in the restricted problem of three bodies. The mass ratio is kept as a literal variable; the series proceed in the powers of Hill's ratio of periods. They cover all four cases at once, namely the direct or retrograde orbits for either inferior planets or for satellites. When the mass ratio is given a numerical value, the expansions contain less terms and thus can be carried up to degree 17 within a 32K core of an IBM 7094. For the system SunJupiter, the initial conditions provided by the series have been corrected to yield the beginnings of all four natural families, and the characteristic exponents have been computed. Comparison between the values computed out of the series and their improvement by numerical integrations and successive variational corrections show that, within an accuracy of one part in ten thousand, the series represent moderately well even the orbit of J VIII; the retrograde planetary orbits are fairly well covered up to 90% of the distance from SunJupiter, whereas the direct planetary orbits are covered only up to 60% of that distance. (Author)
Natural families of periodic orbits by
André Deprit(
Book
)
2 editions published in 1966 in English and held by 4 WorldCat member libraries worldwide
In reference to any solution of a conservative dynamical system with two degrees of freedom, Hill's equation is generalized to encompass nonnecessarily 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. Although the algorithm makes no assumption of symmetry on the periodic orbits to be continued, special attention is paid to the symmetric orbits, but only to show how in these cases the method can be simplified substantially. (Author)
2 editions published in 1966 in English and held by 4 WorldCat member libraries worldwide
In reference to any solution of a conservative dynamical system with two degrees of freedom, Hill's equation is generalized to encompass nonnecessarily 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. Although the algorithm makes no assumption of symmetry on the periodic orbits to be continued, special attention is paid to the symmetric orbits, but only to show how in these cases the method can be simplified substantially. (Author)
Stability of the triangular Lagrangian points by
André Deprit(
Book
)
2 editions published in 1966 in English and held by 4 WorldCat member libraries worldwide
Leontovich has proved that the triangular equilibrium positions in the planar Restricted Problem of Three Bodies are stable for almost all admissible mass ratios. It is shown here that the set of exceptional mass ratios for which stability remains to be proved or invalidated contains only one point besides the critical mass ratios of order two and three. (Author)
2 editions published in 1966 in English and held by 4 WorldCat member libraries worldwide
Leontovich has proved that the triangular equilibrium positions in the planar Restricted Problem of Three Bodies are stable for almost all admissible mass ratios. It is shown here that the set of exceptional mass ratios for which stability remains to be proved or invalidated contains only one point besides the critical mass ratios of order two and three. (Author)
The trojan manifold in the system EarthMoon by
André Deprit(
Book
)
2 editions published in 1967 in English and held by 4 WorldCat member libraries worldwide
The Trojan manifold is defined as the analytical manifold of periodic orbits which contains the triangular equilibrium L4 as a singularity. Identification of the EarthMoon 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 of equilibrium, usually denoted by L4, is stable, and it generates two natural families of periodic orbits. (Author)
2 editions published in 1967 in English and held by 4 WorldCat member libraries worldwide
The Trojan manifold is defined as the analytical manifold of periodic orbits which contains the triangular equilibrium L4 as a singularity. Identification of the EarthMoon 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 of equilibrium, usually denoted by L4, is stable, and it generates two natural families of periodic orbits. (Author)
Asymptotic representation of the cycle of Van der Pol's equation for small damping coefficients by
André Deprit(
Book
)
2 editions published in 1967 in English and held by 3 WorldCat member libraries worldwide
The limit cycle of Van der Pol's equation is expanded as a power series of the damping coefficient epsilon, the coefficients being finite trigonometric sums with the time as argument. Because the development was implemented in a fully automatic way on a computer, it has been pushed to a high power of epsilon. Hence, for epsilon as large as 0.75, the estimates given by the series for the amplitude and the period agree to 15 decimal places with their correct values computed on integrating by recurrent power series Van der Pol's equation and its associated variational equation. When epsilon reaches 1.75, the agreement is still within 0.001. (Author)
2 editions published in 1967 in English and held by 3 WorldCat member libraries worldwide
The limit cycle of Van der Pol's equation is expanded as a power series of the damping coefficient epsilon, the coefficients being finite trigonometric sums with the time as argument. Because the development was implemented in a fully automatic way on a computer, it has been pushed to a high power of epsilon. Hence, for epsilon as large as 0.75, the estimates given by the series for the amplitude and the period agree to 15 decimal places with their correct values computed on integrating by recurrent power series Van der Pol's equation and its associated variational equation. When epsilon reaches 1.75, the agreement is still within 0.001. (Author)
Computerized expansions in elliptic motion by
André Deprit(
Book
)
2 editions published in 1967 in English and held by 3 WorldCat member libraries worldwide
The functions of the Keplerian elliptic motion are expanded with respect to mean anomaly and the eccentricity by applying Poincare's method of continuation in a direct manner to the equations of motion. Two algorithms are proposed; both lead to programs by which the classical expansions are constructed symbolically and automatically on a computer in double precision arithmetic. Even to as high a degree as 30 in the eccentricity, the procedures are remarkably swift. Special care is taken here in describing significant error controls at each step of the recurrences
2 editions published in 1967 in English and held by 3 WorldCat member libraries worldwide
The functions of the Keplerian elliptic motion are expanded with respect to mean anomaly and the eccentricity by applying Poincare's method of continuation in a direct manner to the equations of motion. Two algorithms are proposed; both lead to programs by which the classical expansions are constructed symbolically and automatically on a computer in double precision arithmetic. Even to as high a degree as 30 in the eccentricity, the procedures are remarkably swift. Special care is taken here in describing significant error controls at each step of the recurrences
La géométrie affiné et ses structures by
André Deprit(
Book
)
3 editions published between 1962 and 1966 in French and held by 3 WorldCat member libraries worldwide
3 editions published between 1962 and 1966 in French and held by 3 WorldCat member libraries worldwide
Birkhoff's normalization(
Book
)
2 editions published in 1969 in English and held by 3 WorldCat member libraries worldwide
Birkhoff's normalizing canonical transformation at an equilibrium of elliptic type with no internal resonance can be built explicitly and recursively, without partial inversions or substitutions, by means of Lie transforms. Invariant sections and ordinary families of periodic orbits for truncated normalized systems are analyzed in detail. (Author)
2 editions published in 1969 in English and held by 3 WorldCat member libraries worldwide
Birkhoff's normalizing canonical transformation at an equilibrium of elliptic type with no internal resonance can be built explicitly and recursively, without partial inversions or substitutions, by means of Lie transforms. Invariant sections and ordinary families of periodic orbits for truncated normalized systems are analyzed in detail. (Author)
Construction of orbits asymptotic to a periodic orbit by
André Deprit(
Book
)
2 editions published in 1968 in English and held by 3 WorldCat member libraries worldwide
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 of a nonhomogeneous Hill's equation for the normal displacements and of a quadrature for the tangential components. Accordingly, Floquet's theory reduces the construction of the asymptotic series to a chain of routine operations, like trigonometric interpolations and elementary primitives. In the case of the restricted problem of three bodies, even the development of the force function about the periodic orbit can be implemented automatically on a computer. (Author)
2 editions published in 1968 in English and held by 3 WorldCat member libraries worldwide
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 of a nonhomogeneous Hill's equation for the normal displacements and of a quadrature for the tangential components. Accordingly, Floquet's theory reduces the construction of the asymptotic series to a chain of routine operations, like trigonometric interpolations and elementary primitives. In the case of the restricted problem of three bodies, even the development of the force function about the periodic orbit can be implemented automatically on a computer. (Author)
Periodic Trojan orbits for the resonace 1/12 by
André Deprit(
Book
)
1 edition published in 1968 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 1968 in English and held by 2 WorldCat member libraries worldwide
Canonical transformations depending on a small parameter by
André Deprit(
Book
)
2 editions published in 1968 in English and held by 2 WorldCat member libraries worldwide
The concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicitly on the small parameter. Lie transforms define naturally a class of canonical mappings in the form of power series in the small parameter. The formalism generates nonconservative as well as conservative transformations. Perturbation theories based on it offer three substantial advantages: they yield the transformation of state variables in an explicit form; its inverse in an explicit form results basically from elementary quadratures; in a function of the original variables, substitution of the new variables consists simply of an iterative procedure involving only explicit chains of Poisson brackets. (Author)
2 editions published in 1968 in English and held by 2 WorldCat member libraries worldwide
The concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicitly on the small parameter. Lie transforms define naturally a class of canonical mappings in the form of power series in the small parameter. The formalism generates nonconservative as well as conservative transformations. Perturbation theories based on it offer three substantial advantages: they yield the transformation of state variables in an explicit form; its inverse in an explicit form results basically from elementary quadratures; in a function of the original variables, substitution of the new variables consists simply of an iterative procedure involving only explicit chains of Poisson brackets. (Author)
Numerical integration of an orbit and its concomitant variations by recurrent power series(
Book
)
1 edition published in 1965 in English and held by 2 WorldCat member libraries worldwide
Power series expansions (with coefficients obtained by recurrence formulas) are more efficient than other integration procedures for computing concurrently an orbit and the resolvent matrix of its variational equations, in the Restricted Problem of Three Bodies. For the same requirements on accuracy, the series expansions use only about 30 per cent of the computing time of the multistep procedures, and only 12 to 15 per cent of the computing time of the RungeKuttaNystrom method. (Author)
1 edition published in 1965 in English and held by 2 WorldCat member libraries worldwide
Power series expansions (with coefficients obtained by recurrence formulas) are more efficient than other integration procedures for computing concurrently an orbit and the resolvent matrix of its variational equations, in the Restricted Problem of Three Bodies. For the same requirements on accuracy, the series expansions use only about 30 per cent of the computing time of the multistep procedures, and only 12 to 15 per cent of the computing time of the RungeKuttaNystrom method. (Author)
Symmetric doubly asymptotic orbits in the restricted threebody problem by
André Deprit(
Book
)
1 edition published in 1965 in English and held by 1 WorldCat member library worldwide
1 edition published in 1965 in English and held by 1 WorldCat member library worldwide
THE HALOING EFFECT OF THE THIRD INTEGRAL(
Book
)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
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 socalled third integral. (Author)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
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 socalled third integral. (Author)
Analytical lunar ephemeris. 1. definition of the main problem(
Book
)
1 edition published in 1970 in English and held by 1 WorldCat member library worldwide
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)
1 edition published in 1970 in English and held by 1 WorldCat member library worldwide
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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 Poplarchek, Walter
 Pickard, Henry
 Naval Research Laboratory (U.S.). Space Sciences Division Technical Staff
 BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
 Elipe, A. (Antonio)
 Ferrer, Sebastián
 Henrard, Jacques
 Boeing Scientific Research Laboratories
 Rom, Arnold
 Rouche, Nicolas
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Associated Subjects
AsteroidsOrbits Asymptotic expansions Celestial mechanics Chebyshev approximation Chebyshev polynomials Combinatorial dynamics Differentiable dynamical systems Differential equations, NonlinearAsymptotic theory Ephemerides Functional analysis Lagrangian points Lie groups Mechanics, Analytic Orbits Perturbation (Mathematics) Polynomials Stability Threebody problem Transformations (Mathematics) Twobody problem