On the elimination of short-period terms in second-order general planetary theory investigated by Hori's method View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

1973-12

AUTHORS

Jean Meffroy

ABSTRACT

The short-period terms of a second-order general planetary theory are removed through the Hori's method based on a development of the HamiltonianF in a Lie series which involves a determining functionS not depending upon mixed canonical variables as in the Von Zeipel's method but upon all the canonical variables resulting from the elimination of the short period terms ofF. Canonical variables adopted are the slow Delaunay variables. Eccentricitiesej and sines γj of the semi inclinations are respectively replaced by the Jacques Henrard variablesEj,Jj which lead to formulas remarkably simple.F is reduced to the sumF0+F1 of its terms of degrees 0,1 in small parameter ε of the order of the masses. Only one disturbing planet is considered.F1 is not calculated beyond its terms of degree 3 inEj,Ej,Jj, the determining functionS2 of degree 2 in ε not being therefore calculated beyond its terms of degree 2 inE′j,Ej,Jj and the expressions of slow Delaunay canonical variables of the disturbed planetP1 and the disturbing planetP2 in terms of the new slow Delaunay canonical variables ofP1 andP2 which result from the elimination of the short period terms ofF1 being therefore reduced to their terms of degree <1 in theE′j,E′j,J′j. Calculation of the principal partF1m ofF1 is carried out through Laplace coefficients and operatorD=α(d/dα) applied to Laplace coefficients, α ratio of the semi major axis ofP1 andP2. Eccentricitye2 of the disturbed planetP2 is assumed to be zero, such an assumption not restricting our aim which is to investigate the mechanism of the elimination of short period terms in a second order general planetary theory carried out through the Hori's method, not to perform the elimination of those terms for a complete second order general planetary theory. Expressions of the slow Delaunay canonical variables in terms of the new ones resulting from the elimination of the short period terms ofF1 are written down only for the disturbed planetP1.Small divisors in 1/E′1 and 1/E′12 appear in the longitude ϖ1 of perihelia ofP1. No small divisors appear in the other five slow Delaunay variables ofP1. The only Jacques Henrard variables which appear in the longitude Ω1 of the ascending node ofP1 are the Jj′j=1, 2 and no Jacques Henrard variables appear in the slow Delaunay canonical variablesX1,Y1,Z1, λ1. The solving of the ten canonical equations ofP1 andP2 in the slow Delaunay canonical variablesX′j,Y′1,Z′j,λ′j,ω′j,Ω′j resulting from the elimination of the short period terms ofF1 reduces to that of four canonical equations inZ′j,©′j and to six quadratures three of them expressing theX′j,Y′1 are constants and the three others expressingλ′j,ϖ′j as functions of timet. Solving of the four canonical equations inZ′j,Ω′j reduces to that of a first order non linear differential equation and to two quadratures. Sinceγ′1 is then constant, so is the Jacques Henrard variableE′1. If the eccentricitye2 ofP2 is no more assumed to be zero, additive small divisors inE′2/E′21 appear in longitude ϖ′1 of perihelia ofP1 and the solving of the twelve canonical equations ofP1 andP2 inX′j,Y′j,Z′j,λ′j ,ϖ′j,Ω′j is reduced to that of eight canonical equations inY′j,ϖ′j,Z′j,Ω′j and to four quadratures expressingX′j are constants andλ′j as functions oft. Those eight canonical equations split into two systems of four canonical equations, one of them inY′j,ϖ′j and the other one inZ′j,Ω′j. Each of those two systems is identical to the system inZ′j,Ω′j corresponding toe2=0 and its solving reduces to that of a first order non linear differential equation and to two quadratures identical to those of the casee2=0.Expressions ofX1,Y1,Z1,λ1,ϖ1,Ω1 as functions ofX′j,Y′1,Z′j,λ′j,ϖ′1,Ω′j;j=1, 2 are sums of sines and cosines of the multiples ofλ′j,ϖ′1,Ω′j for the terms arising from the indirect partF1j ofF1, Fourier series in those sines and cosines or products of two such Fourier series for the terms arising from the principal partF1m ofF1, coefficients of those sums and Fourier series having one of the eight forms:A,..., H being constants which depend upon ratio α. Numerical calculation of the constantsA,..., H arising from the terms ofF1j is easily carried out; that of theA,..., H arising from the terms ofF1m require more manipulations, Fourier series in sines and cosines of the multiples ofλ′j,ϖ′j,Ωij and products of two such Fourier series having then to be reduced to sums of a finite number of terms and treated through the methods of harmonic analysis. Divisors inp+qα3/2;p, q relative integers, or products of such divisors appear inA,..., H.the method extends to the case whenF1 is calculated beyond its terms of degree 3 in the Jacques Henrard variables.F1 being calculated up to its terms of degree 8 in the Jacques Henrard variables which is the precision required to eliminate the short period terms of a complete second order general planetary theory,S2 has to be calculated up to its terms of degree 7 and the expression of the slow Delaunay canonical variables ofP1 andP2 in terms of the slow Delaunay canonical variables ofP1 andP2 resulting from the elimination of the short period terms ofF1 have, therefore, to be calculated up to their terms of degree 5 in the Jacques Henrard variables. The short-period terms of a second-order general planetary theory are removed through the Hori's method based on a development of the HamiltonianF in a Lie series which involves a determining functionS not depending upon mixed canonical variables as in the Von Zeipel's method but upon all the canonical variables resulting from the elimination of the short period terms ofF. Canonical variables adopted are the slow Delaunay variables. Eccentricitiesej and sines γj of the semi inclinations are respectively replaced by the Jacques Henrard variablesEj,Jj which lead to formulas remarkably simple.F is reduced to the sumF0+F1 of its terms of degrees 0,1 in small parameter ε of the order of the masses. Only one disturbing planet is considered.F1 is not calculated beyond its terms of degree 3 inEj,Ej,Jj, the determining functionS2 of degree 2 in ε not being therefore calculated beyond its terms of degree 2 inE′j,Ej,Jj and the expressions of slow Delaunay canonical variables of the disturbed planetP1 and the disturbing planetP2 in terms of the new slow Delaunay canonical variables ofP1 andP2 which result from the elimination of the short period terms ofF1 being therefore reduced to their terms of degree <1 in theE′j,E′j,J′j. Calculation of the principal partF1m ofF1 is carried out through Laplace coefficients and operatorD=α(d/dα) applied to Laplace coefficients, α ratio of the semi major axis ofP1 andP2. Eccentricitye2 of the disturbed planetP2 is assumed to be zero, such an assumption not restricting our aim which is to investigate the mechanism of the elimination of short period terms in a second order general planetary theory carried out through the Hori's method, not to perform the elimination of those terms for a complete second order general planetary theory. Expressions of the slow Delaunay canonical variables in terms of the new ones resulting from the elimination of the short period terms ofF1 are written down only for the disturbed planetP1. Small divisors in 1/E′1 and 1/E′12 appear in the longitude ϖ1 of perihelia ofP1. No small divisors appear in the other five slow Delaunay variables ofP1. The only Jacques Henrard variables which appear in the longitude Ω1 of the ascending node ofP1 are the Jj′j=1, 2 and no Jacques Henrard variables appear in the slow Delaunay canonical variablesX1,Y1,Z1, λ1. The solving of the ten canonical equations ofP1 andP2 in the slow Delaunay canonical variablesX′j,Y′1,Z′j,λ′j,ω′j,Ω′j resulting from the elimination of the short period terms ofF1 reduces to that of four canonical equations inZ′j,©′j and to six quadratures three of them expressing theX′j,Y′1 are constants and the three others expressingλ′j,ϖ′j as functions of timet. Solving of the four canonical equations inZ′j,Ω′j reduces to that of a first order non linear differential equation and to two quadratures. Sinceγ′1 is then constant, so is the Jacques Henrard variableE′1. If the eccentricitye2 ofP2 is no more assumed to be zero, additive small divisors inE′2/E′21 appear in longitude ϖ′1 of perihelia ofP1 and the solving of the twelve canonical equations ofP1 andP2 inX′j,Y′j,Z′j,λ′j ,ϖ′j,Ω′j is reduced to that of eight canonical equations inY′j,ϖ′j,Z′j,Ω′j and to four quadratures expressingX′j are constants andλ′j as functions oft. Those eight canonical equations split into two systems of four canonical equations, one of them inY′j,ϖ′j and the other one inZ′j,Ω′j. Each of those two systems is identical to the system inZ′j,Ω′j corresponding toe2=0 and its solving reduces to that of a first order non linear differential equation and to two quadratures identical to those of the casee2=0. Expressions ofX1,Y1,Z1,λ1,ϖ1,Ω1 as functions ofX′j,Y′1,Z′j,λ′j,ϖ′1,Ω′j;j=1, 2 are sums of sines and cosines of the multiples ofλ′j,ϖ′1,Ω′j for the terms arising from the indirect partF1j ofF1, Fourier series in those sines and cosines or products of two such Fourier series for the terms arising from the principal partF1m ofF1, coefficients of those sums and Fourier series having one of the eight forms:A,..., H being constants which depend upon ratio α. Numerical calculation of the constantsA,..., H arising from the terms ofF1j is easily carried out; that of theA,..., H arising from the terms ofF1m require more manipulations, Fourier series in sines and cosines of the multiples ofλ′j,ϖ′j,Ωij and products of two such Fourier series having then to be reduced to sums of a finite number of terms and treated through the methods of harmonic analysis. Divisors inp+qα3/2;p, q relative integers, or products of such divisors appear inA,..., H. the method extends to the case whenF1 is calculated beyond its terms of degree 3 in the Jacques Henrard variables.F1 being calculated up to its terms of degree 8 in the Jacques Henrard variables which is the precision required to eliminate the short period terms of a complete second order general planetary theory,S2 has to be calculated up to its terms of degree 7 and the expression of the slow Delaunay canonical variables ofP1 andP2 in terms of the slow Delaunay canonical variables ofP1 andP2 resulting from the elimination of the short period terms ofF1 have, therefore, to be calculated up to their terms of degree 5 in the Jacques Henrard variables. More... »

PAGES

271-354

Journal

TITLE

Astrophysics and Space Science

ISSUE

2

VOLUME

25

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf00649176

DOI

http://dx.doi.org/10.1007/bf00649176

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1047530033


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    "description": "The short-period terms of a second-order general planetary theory are removed through the Hori's method based on a development of the HamiltonianF in a Lie series which involves a determining functionS not depending upon mixed canonical variables as in the Von Zeipel's method but upon all the canonical variables resulting from the elimination of the short period terms ofF. Canonical variables adopted are the slow Delaunay variables. Eccentricitiesej and sines \u03b3j of the semi inclinations are respectively replaced by the Jacques Henrard variablesEj,Jj which lead to formulas remarkably simple.F is reduced to the sumF0+F1 of its terms of degrees 0,1 in small parameter \u03b5 of the order of the masses. 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Eccentricitye2 of the disturbed planetP2 is assumed to be zero, such an assumption not restricting our aim which is to investigate the mechanism of the elimination of short period terms in a second order general planetary theory carried out through the Hori's method, not to perform the elimination of those terms for a complete second order general planetary theory. Expressions of the slow Delaunay canonical variables in terms of the new ones resulting from the elimination of the short period terms ofF1 are written down only for the disturbed planetP1.Small divisors in 1/E\u20321 and 1/E\u203212 appear in the longitude \u03d61 of perihelia ofP1. No small divisors appear in the other five slow Delaunay variables ofP1. The only Jacques Henrard variables which appear in the longitude \u03a91 of the ascending node ofP1 are the Jj\u2032j=1, 2 and no Jacques Henrard variables appear in the slow Delaunay canonical variablesX1,Y1,Z1, \u03bb1. The solving of the ten canonical equations ofP1 andP2 in the slow Delaunay canonical variablesX\u2032j,Y\u20321,Z\u2032j,\u03bb\u2032j,\u03c9\u2032j,\u03a9\u2032j resulting from the elimination of the short period terms ofF1 reduces to that of four canonical equations inZ\u2032j,\u00a9\u2032j and to six quadratures three of them expressing theX\u2032j,Y\u20321 are constants and the three others expressing\u03bb\u2032j,\u03d6\u2032j as functions of timet. Solving of the four canonical equations inZ\u2032j,\u03a9\u2032j reduces to that of a first order non linear differential equation and to two quadratures. Since\u03b3\u20321 is then constant, so is the Jacques Henrard variableE\u20321. If the eccentricitye2 ofP2 is no more assumed to be zero, additive small divisors inE\u20322/E\u203221 appear in longitude \u03d6\u20321 of perihelia ofP1 and the solving of the twelve canonical equations ofP1 andP2 inX\u2032j,Y\u2032j,Z\u2032j,\u03bb\u2032j ,\u03d6\u2032j,\u03a9\u2032j is reduced to that of eight canonical equations inY\u2032j,\u03d6\u2032j,Z\u2032j,\u03a9\u2032j and to four quadratures expressingX\u2032j are constants and\u03bb\u2032j as functions oft. Those eight canonical equations split into two systems of four canonical equations, one of them inY\u2032j,\u03d6\u2032j and the other one inZ\u2032j,\u03a9\u2032j. Each of those two systems is identical to the system inZ\u2032j,\u03a9\u2032j corresponding toe2=0 and its solving reduces to that of a first order non linear differential equation and to two quadratures identical to those of the casee2=0.Expressions ofX1,Y1,Z1,\u03bb1,\u03d61,\u03a91 as functions ofX\u2032j,Y\u20321,Z\u2032j,\u03bb\u2032j,\u03d6\u20321,\u03a9\u2032j;j=1, 2 are sums of sines and cosines of the multiples of\u03bb\u2032j,\u03d6\u20321,\u03a9\u2032j for the terms arising from the indirect partF1j ofF1, Fourier series in those sines and cosines or products of two such Fourier series for the terms arising from the principal partF1m ofF1, coefficients of those sums and Fourier series having one of the eight forms:A,..., H being constants which depend upon ratio \u03b1. Numerical calculation of the constantsA,..., H arising from the terms ofF1j is easily carried out; that of theA,..., H arising from the terms ofF1m require more manipulations, Fourier series in sines and cosines of the multiples of\u03bb\u2032j,\u03d6\u2032j,\u03a9ij and products of two such Fourier series having then to be reduced to sums of a finite number of terms and treated through the methods of harmonic analysis. Divisors inp+q\u03b13/2;p, q relative integers, or products of such divisors appear inA,..., H.the method extends to the case whenF1 is calculated beyond its terms of degree 3 in the Jacques Henrard variables.F1 being calculated up to its terms of degree 8 in the Jacques Henrard variables which is the precision required to eliminate the short period terms of a complete second order general planetary theory,S2 has to be calculated up to its terms of degree 7 and the expression of the slow Delaunay canonical variables ofP1 andP2 in terms of the slow Delaunay canonical variables ofP1 andP2 resulting from the elimination of the short period terms ofF1 have, therefore, to be calculated up to their terms of degree 5 in the Jacques Henrard variables. The short-period terms of a second-order general planetary theory are removed through the Hori's method based on a development of the HamiltonianF in a Lie series which involves a determining functionS not depending upon mixed canonical variables as in the Von Zeipel's method but upon all the canonical variables resulting from the elimination of the short period terms ofF. Canonical variables adopted are the slow Delaunay variables. Eccentricitiesej and sines \u03b3j of the semi inclinations are respectively replaced by the Jacques Henrard variablesEj,Jj which lead to formulas remarkably simple.F is reduced to the sumF0+F1 of its terms of degrees 0,1 in small parameter \u03b5 of the order of the masses. Only one disturbing planet is considered.F1 is not calculated beyond its terms of degree 3 inEj,Ej,Jj, the determining functionS2 of degree 2 in \u03b5 not being therefore calculated beyond its terms of degree 2 inE\u2032j,Ej,Jj and the expressions of slow Delaunay canonical variables of the disturbed planetP1 and the disturbing planetP2 in terms of the new slow Delaunay canonical variables ofP1 andP2 which result from the elimination of the short period terms ofF1 being therefore reduced to their terms of degree <1 in theE\u2032j,E\u2032j,J\u2032j. Calculation of the principal partF1m ofF1 is carried out through Laplace coefficients and operatorD=\u03b1(d/d\u03b1) applied to Laplace coefficients, \u03b1 ratio of the semi major axis ofP1 andP2. Eccentricitye2 of the disturbed planetP2 is assumed to be zero, such an assumption not restricting our aim which is to investigate the mechanism of the elimination of short period terms in a second order general planetary theory carried out through the Hori's method, not to perform the elimination of those terms for a complete second order general planetary theory. Expressions of the slow Delaunay canonical variables in terms of the new ones resulting from the elimination of the short period terms ofF1 are written down only for the disturbed planetP1. Small divisors in 1/E\u20321 and 1/E\u203212 appear in the longitude \u03d61 of perihelia ofP1. No small divisors appear in the other five slow Delaunay variables ofP1. The only Jacques Henrard variables which appear in the longitude \u03a91 of the ascending node ofP1 are the Jj\u2032j=1, 2 and no Jacques Henrard variables appear in the slow Delaunay canonical variablesX1,Y1,Z1, \u03bb1. The solving of the ten canonical equations ofP1 andP2 in the slow Delaunay canonical variablesX\u2032j,Y\u20321,Z\u2032j,\u03bb\u2032j,\u03c9\u2032j,\u03a9\u2032j resulting from the elimination of the short period terms ofF1 reduces to that of four canonical equations inZ\u2032j,\u00a9\u2032j and to six quadratures three of them expressing theX\u2032j,Y\u20321 are constants and the three others expressing\u03bb\u2032j,\u03d6\u2032j as functions of timet. Solving of the four canonical equations inZ\u2032j,\u03a9\u2032j reduces to that of a first order non linear differential equation and to two quadratures. Since\u03b3\u20321 is then constant, so is the Jacques Henrard variableE\u20321. If the eccentricitye2 ofP2 is no more assumed to be zero, additive small divisors inE\u20322/E\u203221 appear in longitude \u03d6\u20321 of perihelia ofP1 and the solving of the twelve canonical equations ofP1 andP2 inX\u2032j,Y\u2032j,Z\u2032j,\u03bb\u2032j ,\u03d6\u2032j,\u03a9\u2032j is reduced to that of eight canonical equations inY\u2032j,\u03d6\u2032j,Z\u2032j,\u03a9\u2032j and to four quadratures expressingX\u2032j are constants and\u03bb\u2032j as functions oft. Those eight canonical equations split into two systems of four canonical equations, one of them inY\u2032j,\u03d6\u2032j and the other one inZ\u2032j,\u03a9\u2032j. Each of those two systems is identical to the system inZ\u2032j,\u03a9\u2032j corresponding toe2=0 and its solving reduces to that of a first order non linear differential equation and to two quadratures identical to those of the casee2=0. Expressions ofX1,Y1,Z1,\u03bb1,\u03d61,\u03a91 as functions ofX\u2032j,Y\u20321,Z\u2032j,\u03bb\u2032j,\u03d6\u20321,\u03a9\u2032j;j=1, 2 are sums of sines and cosines of the multiples of\u03bb\u2032j,\u03d6\u20321,\u03a9\u2032j for the terms arising from the indirect partF1j ofF1, Fourier series in those sines and cosines or products of two such Fourier series for the terms arising from the principal partF1m ofF1, coefficients of those sums and Fourier series having one of the eight forms:A,..., H being constants which depend upon ratio \u03b1. Numerical calculation of the constantsA,..., H arising from the terms ofF1j is easily carried out; that of theA,..., H arising from the terms ofF1m require more manipulations, Fourier series in sines and cosines of the multiples of\u03bb\u2032j,\u03d6\u2032j,\u03a9ij and products of two such Fourier series having then to be reduced to sums of a finite number of terms and treated through the methods of harmonic analysis. 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8 schema:description The short-period terms of a second-order general planetary theory are removed through the Hori's method based on a development of the HamiltonianF in a Lie series which involves a determining functionS not depending upon mixed canonical variables as in the Von Zeipel's method but upon all the canonical variables resulting from the elimination of the short period terms ofF. Canonical variables adopted are the slow Delaunay variables. Eccentricitiesej and sines γj of the semi inclinations are respectively replaced by the Jacques Henrard variablesEj,Jj which lead to formulas remarkably simple.F is reduced to the sumF0+F1 of its terms of degrees 0,1 in small parameter ε of the order of the masses. Only one disturbing planet is considered.F1 is not calculated beyond its terms of degree 3 inEj,Ej,Jj, the determining functionS2 of degree 2 in ε not being therefore calculated beyond its terms of degree 2 inE′j,Ej,Jj and the expressions of slow Delaunay canonical variables of the disturbed planetP1 and the disturbing planetP2 in terms of the new slow Delaunay canonical variables ofP1 andP2 which result from the elimination of the short period terms ofF1 being therefore reduced to their terms of degree <1 in theE′j,E′j,J′j. Calculation of the principal partF1m ofF1 is carried out through Laplace coefficients and operatorD=α(d/dα) applied to Laplace coefficients, α ratio of the semi major axis ofP1 andP2. Eccentricitye2 of the disturbed planetP2 is assumed to be zero, such an assumption not restricting our aim which is to investigate the mechanism of the elimination of short period terms in a second order general planetary theory carried out through the Hori's method, not to perform the elimination of those terms for a complete second order general planetary theory. Expressions of the slow Delaunay canonical variables in terms of the new ones resulting from the elimination of the short period terms ofF1 are written down only for the disturbed planetP1.Small divisors in 1/E′1 and 1/E′12 appear in the longitude ϖ1 of perihelia ofP1. No small divisors appear in the other five slow Delaunay variables ofP1. The only Jacques Henrard variables which appear in the longitude Ω1 of the ascending node ofP1 are the Jj′j=1, 2 and no Jacques Henrard variables appear in the slow Delaunay canonical variablesX1,Y1,Z1, λ1. The solving of the ten canonical equations ofP1 andP2 in the slow Delaunay canonical variablesX′j,Y′1,Z′j,λ′j,ω′j,Ω′j resulting from the elimination of the short period terms ofF1 reduces to that of four canonical equations inZ′j,©′j and to six quadratures three of them expressing theX′j,Y′1 are constants and the three others expressingλ′j,ϖ′j as functions of timet. Solving of the four canonical equations inZ′j,Ω′j reduces to that of a first order non linear differential equation and to two quadratures. Sinceγ′1 is then constant, so is the Jacques Henrard variableE′1. If the eccentricitye2 ofP2 is no more assumed to be zero, additive small divisors inE′2/E′21 appear in longitude ϖ′1 of perihelia ofP1 and the solving of the twelve canonical equations ofP1 andP2 inX′j,Y′j,Z′j,λ′j ,ϖ′j,Ω′j is reduced to that of eight canonical equations inY′j,ϖ′j,Z′j,Ω′j and to four quadratures expressingX′j are constants andλ′j as functions oft. Those eight canonical equations split into two systems of four canonical equations, one of them inY′j,ϖ′j and the other one inZ′j,Ω′j. Each of those two systems is identical to the system inZ′j,Ω′j corresponding toe2=0 and its solving reduces to that of a first order non linear differential equation and to two quadratures identical to those of the casee2=0.Expressions ofX1,Y1,Z1,λ1,ϖ1,Ω1 as functions ofX′j,Y′1,Z′j,λ′j,ϖ′1,Ω′j;j=1, 2 are sums of sines and cosines of the multiples ofλ′j,ϖ′1,Ω′j for the terms arising from the indirect partF1j ofF1, Fourier series in those sines and cosines or products of two such Fourier series for the terms arising from the principal partF1m ofF1, coefficients of those sums and Fourier series having one of the eight forms:A,..., H being constants which depend upon ratio α. Numerical calculation of the constantsA,..., H arising from the terms ofF1j is easily carried out; that of theA,..., H arising from the terms ofF1m require more manipulations, Fourier series in sines and cosines of the multiples ofλ′j,ϖ′j,Ωij and products of two such Fourier series having then to be reduced to sums of a finite number of terms and treated through the methods of harmonic analysis. Divisors inp+qα3/2;p, q relative integers, or products of such divisors appear inA,..., H.the method extends to the case whenF1 is calculated beyond its terms of degree 3 in the Jacques Henrard variables.F1 being calculated up to its terms of degree 8 in the Jacques Henrard variables which is the precision required to eliminate the short period terms of a complete second order general planetary theory,S2 has to be calculated up to its terms of degree 7 and the expression of the slow Delaunay canonical variables ofP1 andP2 in terms of the slow Delaunay canonical variables ofP1 andP2 resulting from the elimination of the short period terms ofF1 have, therefore, to be calculated up to their terms of degree 5 in the Jacques Henrard variables. The short-period terms of a second-order general planetary theory are removed through the Hori's method based on a development of the HamiltonianF in a Lie series which involves a determining functionS not depending upon mixed canonical variables as in the Von Zeipel's method but upon all the canonical variables resulting from the elimination of the short period terms ofF. Canonical variables adopted are the slow Delaunay variables. Eccentricitiesej and sines γj of the semi inclinations are respectively replaced by the Jacques Henrard variablesEj,Jj which lead to formulas remarkably simple.F is reduced to the sumF0+F1 of its terms of degrees 0,1 in small parameter ε of the order of the masses. Only one disturbing planet is considered.F1 is not calculated beyond its terms of degree 3 inEj,Ej,Jj, the determining functionS2 of degree 2 in ε not being therefore calculated beyond its terms of degree 2 inE′j,Ej,Jj and the expressions of slow Delaunay canonical variables of the disturbed planetP1 and the disturbing planetP2 in terms of the new slow Delaunay canonical variables ofP1 andP2 which result from the elimination of the short period terms ofF1 being therefore reduced to their terms of degree <1 in theE′j,E′j,J′j. Calculation of the principal partF1m ofF1 is carried out through Laplace coefficients and operatorD=α(d/dα) applied to Laplace coefficients, α ratio of the semi major axis ofP1 andP2. Eccentricitye2 of the disturbed planetP2 is assumed to be zero, such an assumption not restricting our aim which is to investigate the mechanism of the elimination of short period terms in a second order general planetary theory carried out through the Hori's method, not to perform the elimination of those terms for a complete second order general planetary theory. Expressions of the slow Delaunay canonical variables in terms of the new ones resulting from the elimination of the short period terms ofF1 are written down only for the disturbed planetP1. Small divisors in 1/E′1 and 1/E′12 appear in the longitude ϖ1 of perihelia ofP1. No small divisors appear in the other five slow Delaunay variables ofP1. The only Jacques Henrard variables which appear in the longitude Ω1 of the ascending node ofP1 are the Jj′j=1, 2 and no Jacques Henrard variables appear in the slow Delaunay canonical variablesX1,Y1,Z1, λ1. The solving of the ten canonical equations ofP1 andP2 in the slow Delaunay canonical variablesX′j,Y′1,Z′j,λ′j,ω′j,Ω′j resulting from the elimination of the short period terms ofF1 reduces to that of four canonical equations inZ′j,©′j and to six quadratures three of them expressing theX′j,Y′1 are constants and the three others expressingλ′j,ϖ′j as functions of timet. Solving of the four canonical equations inZ′j,Ω′j reduces to that of a first order non linear differential equation and to two quadratures. Sinceγ′1 is then constant, so is the Jacques Henrard variableE′1. If the eccentricitye2 ofP2 is no more assumed to be zero, additive small divisors inE′2/E′21 appear in longitude ϖ′1 of perihelia ofP1 and the solving of the twelve canonical equations ofP1 andP2 inX′j,Y′j,Z′j,λ′j ,ϖ′j,Ω′j is reduced to that of eight canonical equations inY′j,ϖ′j,Z′j,Ω′j and to four quadratures expressingX′j are constants andλ′j as functions oft. Those eight canonical equations split into two systems of four canonical equations, one of them inY′j,ϖ′j and the other one inZ′j,Ω′j. Each of those two systems is identical to the system inZ′j,Ω′j corresponding toe2=0 and its solving reduces to that of a first order non linear differential equation and to two quadratures identical to those of the casee2=0. Expressions ofX1,Y1,Z1,λ1,ϖ1,Ω1 as functions ofX′j,Y′1,Z′j,λ′j,ϖ′1,Ω′j;j=1, 2 are sums of sines and cosines of the multiples ofλ′j,ϖ′1,Ω′j for the terms arising from the indirect partF1j ofF1, Fourier series in those sines and cosines or products of two such Fourier series for the terms arising from the principal partF1m ofF1, coefficients of those sums and Fourier series having one of the eight forms:A,..., H being constants which depend upon ratio α. Numerical calculation of the constantsA,..., H arising from the terms ofF1j is easily carried out; that of theA,..., H arising from the terms ofF1m require more manipulations, Fourier series in sines and cosines of the multiples ofλ′j,ϖ′j,Ωij and products of two such Fourier series having then to be reduced to sums of a finite number of terms and treated through the methods of harmonic analysis. Divisors inp+qα3/2;p, q relative integers, or products of such divisors appear inA,..., H. the method extends to the case whenF1 is calculated beyond its terms of degree 3 in the Jacques Henrard variables.F1 being calculated up to its terms of degree 8 in the Jacques Henrard variables which is the precision required to eliminate the short period terms of a complete second order general planetary theory,S2 has to be calculated up to its terms of degree 7 and the expression of the slow Delaunay canonical variables ofP1 andP2 in terms of the slow Delaunay canonical variables ofP1 andP2 resulting from the elimination of the short period terms ofF1 have, therefore, to be calculated up to their terms of degree 5 in the Jacques Henrard variables.
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