From: Manuel Date: Sat, 7 Jul 2012 03:56:55 +0000 (+0200) Subject: Gravitation : le problème des deux corps avec PSTricks : partie 3 X-Git-Url: https://melusine.eu.org/syracuse/G/git/?p=pst-eqdf.git;a=commitdiff_plain;h=b5129bd75f4eb3edd3dde9e71f182e146dc5c0b1 Gravitation : le problème des deux corps avec PSTricks : partie 3 Cette partie contient deux exemples : 1) le calcul, le dessin de la trajectoire des deux corps dans le repère lié au centre de masse en utilisant la résolution numérique du système d'équations différentielles par le package de Dominique Rodriguez : pst-eqdf. 2) La trajectoire de l'un des corps par rapport à l'autre et l'animation du mouvement. --- diff --git a/gravitation/2corps10.png b/gravitation/2corps10.png new file mode 100644 index 0000000..83a0781 Binary files /dev/null and b/gravitation/2corps10.png differ diff --git a/gravitation/LISTE.txt b/gravitation/LISTE.txt index 76d3aae..ac2fad1 100644 --- a/gravitation/LISTE.txt +++ b/gravitation/LISTE.txt @@ -25,6 +25,13 @@ XYM1.dat XYM2.dat XYG.dat pb2corps.dat +== Le problème des 2 corps : partie 3 +pb_2corps_doc_part3.pdf +pb_2corps_doc_part3.tex +XY1G.dat +XY2G.dat +XY12.dat +XY21.dat == Les fichiers du paquet pst-eqdf.tex pst-eqdf.sty @@ -40,3 +47,4 @@ h1.png h2.png 2corps8.png pb_2corps_animation1.png +2corps10.png diff --git a/gravitation/XY12.dat b/gravitation/XY12.dat new file mode 100644 index 0000000..865f359 --- /dev/null +++ b/gravitation/XY12.dat @@ -0,0 +1,1000 @@ +5 2 +5.01839 2.03079 +5.03657 2.06149 +5.05454 2.09211 +5.0723 2.12264 +5.08985 2.15309 +5.1072 2.18344 +5.12434 2.21372 +5.14129 2.2439 +5.15804 2.274 +5.1746 2.30402 +5.19096 2.33394 +5.20713 2.36379 +5.22312 2.39354 +5.23892 2.42322 +5.25453 2.4528 +5.26996 2.48231 +5.28522 2.51172 +5.30029 2.54105 +5.31519 2.5703 +5.32991 2.59947 +5.34446 2.62854 +5.35884 2.65754 +5.37305 2.68645 +5.38709 2.71528 +5.40097 2.74402 +5.41468 2.77268 +5.42823 2.80126 +5.44162 2.82975 +5.45485 2.85816 +5.46792 2.88649 +5.48083 2.91474 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b/gravitation/pb_2corps_doc_part3.pdf new file mode 100644 index 0000000..a6d7b97 Binary files /dev/null and b/gravitation/pb_2corps_doc_part3.pdf differ diff --git a/gravitation/pb_2corps_doc_part3.tex b/gravitation/pb_2corps_doc_part3.tex new file mode 100644 index 0000000..b849c5f --- /dev/null +++ b/gravitation/pb_2corps_doc_part3.tex @@ -0,0 +1,851 @@ + +\documentclass{article} +\usepackage[a4paper,margin=2cm]{geometry} +\usepackage[T1]{fontenc} +\usepackage[latin1]{inputenc}% +\usepackage[garamond]{mathdesign} +\usepackage{pst-eqdf,pst-node,pst-tools} +\usepackage{array,amsmath} +\usepackage{animate} +\newpsstyle{vecteurA}{arrowinset=0.05,arrowsize=0.1,linecolor={[rgb]{1 0.5 0}}} +\newpsstyle{vecteurB}{arrowinset=0.05,arrowsize=0.1,linecolor={[rgb]{0 0.5 1}}} +\newpsstyle{vecteurC}{arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{1 0 0}}} +%timeline +\begin{filecontents}{pb2corps.dat} +::0x0 +::1 +::2 +::3 +::4 +::5 +::6 +::7 +::8 +::9 +::10 +::11 +::12 +::13 +::14 +::15 +::16 +::17 +::18 +::19 +::20 +::21 +::22 +::23 +::24 +::25 +::26 +::27 +::28 +::29 +::30 +::31 +::32 +::33 +::34 +::35 +::36 +::37 +::38 +::39 +::40 +::41 +::42 +::43 +::44 +::45 +::46 +::47 +::48 +::49 +::50 +::51 +::52 +::53 +::54 +::55 +::56 +::57 +::58 +::59 +::60 +::61 +::62 +::63 +::64 +::65 +::66 +::67 +::68 +::69 +::70 +::71 +::72 +::73 +::74 +::75 +::76 +::77 +::78 +::79 +::80 +::81 +::82 +::83 +::84 +::85 +::86 +::87 +::88 +::89 +::90 +::91 +::92 +::93 +::94 +::95 +::96 +::97 +::98 +::99 +::100 +::101 +::102 +::103 +::104 +::105 +::106 +::107 +::108 +::109 +::110 +::111 +::112 +::113 +::114 +::115 +::116 +::117 +::118 +::119 +::120 +::121 +::122 +::123 +::124 +::125 +::126 +::127 +::128 +::129 +::130 +::131 +::132 +::133 +::134 +::135 +::136 +::137 +::138 +::139 +::140 +::141 +::142 +::143 +::144 +::145 +::146 +::147 +::148 +::149 +::150 +::151 +::152 +::153 +::154 +::155 +::156 +::157 +::158 +::159 +::160 +::161 +::162 +::163 +::164 +::165 +::166 +::167 +::168 +::169 +::170 +::171 +::172 +::173 +::174 +::175 +::176 +::177 +::178 +::179 +::180 +::181 +::182 +::183 +::184 +::185 +::186 +::187 +::188 +::189 +::190 +::191 +::192 +::193 +::194 +::195 +::196 +::197 +::198 +::199 +::200 +\end{filecontents} +\title{Gravitation : le problème des deux corps avec PSTricks\\ partie 3} +\date{6 juillet 2\,012} +\begin{document} +\maketitle +\section{Mouvements dans le repère du centre de masse} +D'abord un petit rappel des documents précédents : + +On considère un système de deux corps en interaction gravitationnelle $M_1$ de masse $m_1$ et $M_2$ de masse $m_2$ dans le repère galiléen \textit{inertiel} $\mathcal{R}$. Ils sont supposés ponctuels. +\begin{center} +\begin{pspicture}(-3,-1)(5,6) +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt]% +\pstVerb{/x01 -2 def /y01 1 def + /x02 5 def /y02 4 def + /xr0 x02 x01 sub def + /yr0 y02 y01 sub def + /M1 3 def + /M2 1 def + /Mt M1 M2 add def + /xG0 x01 M1 mul x02 M2 mul add Mt div def + /yG0 y01 M1 mul y02 M2 mul add Mt div def}% +\pnode(0,0){O} +\pnode(!x01 y01){M1} +\pnode(!x02 y02){M2} +\pnode(!xG0 yG0){C} +\pscircle[fillstyle=solid,fillcolor=red!50](M1){0.21} +\pscircle[fillstyle=solid,fillcolor=blue!50](M2){0.07} +\psdot(C) +\psline[linestyle=dotted](M1)(M2) +\psline[linestyle=dashed](M2)(O)(M1) +\psline{<->}(5,0)(0,0)(0,6) +\uput[dl](O){$O$} +\uput[r](0,5.8){$y$} +\uput[u](4.8,0){$x$} +\uput[l](0,5.8){$\mathcal{R}$} +\uput{0.22}[l](M1){$M_1$} +\uput{0.1}[u](M2){$M_2$} +\uput[u](C){$C$} +\pcline[offset=5pt,linestyle=none](M1)(M2) +\ncput*[nrot=:U]{$r$} +\rput(M1){\psline[style=vecteurC]{->}(!xr0 5 div yr0 5 div)\uput[u](!xr0 10 div yr0 10 div){$\overrightarrow{F}_{2/1}$}} +\rput(M2){\psline[style=vecteurC]{->}(!xr0 5 div neg yr0 5 div neg)\uput[u](!xr0 10 div neg yr0 10 div neg){$\overrightarrow{F}_{1/2}$}} +\end{pspicture} +\end{center} +On note $\overrightarrow{r}=\overrightarrow{M_1M_2}$. $M_2$ subit de la part de $M_1$ une force attractive $\overrightarrow{F}_{1/2}$ et réciproquement $M_1$ subit de la part de $M_2$ une force attractive $\overrightarrow{F}_{2/1}$ telles que : +\[ +\overrightarrow{F}_{1/2}=-\mathcal{G}\frac{m_1m_2}{r^3}\overrightarrow{r}\quad \overrightarrow{F}_{2/1}=+\mathcal{G}\frac{m_1m_2}{r^3}\overrightarrow{r} +\] +La loi de Newton appliquée à chacune des particules s'écrit : +\[ +m_1\frac{\mathrm{d}^2\overrightarrow{r}}{\mathrm{d}t^2}=\hphantom{-}\mathcal{G}\frac{m_1m_2}{r^3}\overrightarrow{r} +\] +\[ +m_2\frac{\mathrm{d}^2\overrightarrow{r}}{\mathrm{d}t^2}=-\mathcal{G}\frac{m_1m_2}{r^3}\overrightarrow{r} +\] +Sachant que $r=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}$, nous obtenons un système de quatre équations différentielles : +\[ +\left\{ +\begin{array}[m]{l} + \ddot{x_1}=\hphantom{-}\mathcal{G}\displaystyle\frac{m_2}{r^3}(x_2-x_1)\\[1em] + \ddot{y_1}=\hphantom{-}\mathcal{G}\displaystyle\frac{m_2}{r^3}(y_2-y_1)\\[1em] + \ddot{x_2}=-\mathcal{G}\displaystyle\frac{m_1}{r^3}(x_2-x_1)\\[1em] + \ddot{y_2}=-\mathcal{G}\displaystyle\frac{m_1}{r^3}(y_2-y_1) +\end{array} +\right. +\] + +% x y x' y' +% y[0] y[1] y[2] y[3] +\def\FictifAlg{% + y[2]|y[3]|% + -(M1+M2)*y[0]/(y[0]^2+y[1]^2)^1.5|% + -(M1+M2)*y[1]/(y[0]^2+y[1]^2)^1.5} +Voici comment placer sur la pile les différentes variables, ainsi que le système d'équations en notation algébrique. Il faut retenir la notation utilisée pour représenter les variables en fonction de leurs positions respectives sur la pile : +\begin{verbatim} +% 0 1 2 3 4 5 6 7 +% y[0] y[1] y[2] y[3] y[4] y[5] y[6] y[7] +% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +\def\GravAlgIIcorps{% + y[2]|y[3]|% + M2*(y[4]-y[0])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|% + M2*(y[5]-y[1])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|% + y[6]|y[7]|% + M1*(y[0]-y[4])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|% + M1*(y[1]-y[5])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5} +\end{verbatim} +Les conditions initiales, positions et vitesses respectives des deux corps, déterminent l'évolution du système. +\begin{verbatim} +\def\InitCond{x01 y01 v0x1 v0y1 x02 y02 v0x2 v0y2} +\end{verbatim} +Elles sont définies dans l'environnement \verb+\begin{pspicture}+ : +\begin{verbatim} +\pstVerb{ + /x01 2 def + /y01 2 def + /v0x1 .2 def + /v0y1 0.25 def + /x02 -3 def + /y02 0 def + /v0x2 -0.25 def + /v0y2 -0.5 def + /M1 3 def + /M2 1 def + }% +\end{verbatim} +Dans cette même variable \verb+\pstVerb+, il est utile de faire d'autres calculs, comme la période du mouvement afin de bien choisir l'intervalle de temps sur lequel on dessine les trajectoires. Les calculs s'effectuent en \textsf{postscript} et l'affichage du résultat utilise, par exemple, le package \textsf{pst-tools}. +\begin{verbatim} +\rput(-2,0){\psPrintValue[decimals=4]{periode}\hphantom{00000}s} +\end{verbatim} +On désactivera cette ligne par \% au début, une fois la période connue. Dans l'exemple proposé $T=41,0688\,\mathrm{s}$. On étudie le mouvement sur trois périodes environ. +%% 0 1 2 3 4 5 6 7 +%% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +\def\GravAlgIIcorps{% + y[2]|y[3]|% + M2*(y[4]-y[0])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|% + M2*(y[5]-y[1])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|% + y[6]|y[7]|% + M1*(y[0]-y[4])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|% + M1*(y[1]-y[5])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5} +%\newpage + +Dans $\mathcal{R}$ la position de $C$ est déterminée par la relation : +\[ +\overrightarrow{OC}=\frac{m_1\overrightarrow{OM_1}+m_2\overrightarrow{OM_2}}{m_1+m_2} +\] +Avec les conventions utilisées les coordonnées de $C$ seront calculées en postscript ainsi : +\begin{verbatim} + M1 y[0] mul M2 y[4] mul add M1 M2 add div % abscisse de C + M1 y[1] mul M2 y[5] mul add M1 M2 add div % ordonnée de C +\end{verbatim} + +Dans le repère $\mathcal{R}^*$ lié au centre de masse, les coordonnées des deux points se calculent par : + +\[ +\left\{ +\begin{array}[m]{l} +x^*_1=x_1-x_C\\ +y^*_1=x_1-y_C\\ +x^*_2=x_2-x_C\\ +y^*_2=y_2-y_C\\ +\end{array} +\right. +\] +Ce qui en postscript s'écrira : +\begin{verbatim} + y[0] % abscisse de M1 + M1 y[0] mul M2 y[4] mul add M1 M2 add div % abscisse de C + sub + y[1] % ordonnée de M1 + M1 y[1] mul M2 y[5] mul add M1 M2 add div % ordonnée de C + sub +\end{verbatim} +\begin{verbatim} + y[4] % abscisse de M2 + M1 y[0] mul M2 y[4] mul add M1 M2 add div % abscisse de C + sub + y[5] % ordonnée de M2 + M1 y[1] mul M2 y[5] mul add M1 M2 add div % ordonnée de C + sub +\end{verbatim} + +Les deux options \texttt{[plotfuncx]} et \texttt{[plotfuncy]} permettent justement de faire les calculs nécessaires à ce changement de repère. \texttt{Mt} désignant la masse totale calculée dans la variable \verb+\pstVerb+ : +\begin{verbatim} + /Mt M1 M2 add def +\end{verbatim} +\begin{verbatim} + plotfuncx=y 0 get + y 4 get M2 mul + y 0 get M1 mul add + Mt div sub , + plotfuncy=1 get + y 1 get + M1 mul + y 5 get M2 mul add + Mt div sub, +\end{verbatim} +Par défaut, c'est la variable de la fonction étudiée, ici le temps $t$, qui est notée \texttt{x}, qui est placée en abscisse. Toutes les grandeurs calculées : +\begin{verbatim} +% 0 1 2 3 4 5 6 7 +% y[0] y[1] y[2] y[3] y[4] y[5] y[6] y[7] +% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +\end{verbatim} +sont placées(empilées) dans un tableau noté \texttt{y}, c'est donc dans ce tableau qu'il faut \textit{aller à la pêche} avec l'instruction \textsf{postscript} \texttt{get}. + +Ainsi |\texttt{y 0 get}| donne : \texttt{y[0]} soit \texttt{x1}, |\texttt{y 1 get}| donne : \texttt{y[1]} soit \texttt{y1} etc. +\begin{center} +\begin{pspicture}(-6,-8)(3,3) +\pstVerb{ +/arccos { + dup + dup mul neg 1 add sqrt + exch + atan +} def + /G 1 def + /x01 2 def + /y01 2 def + /v0x1 .2 def + /v0y1 0.25 def + /x02 -3 def + /y02 0 def + /v0x2 -0.25 def + /v0y2 -0.5 def + /M1 3 def + /M2 1 def + /Mt M1 M2 add def + /mu M1 M2 mul M1 M2 add div def % masse réduite + /K G Mt mul def + /xG0 M1 x01 mul M2 x02 mul add M1 M2 add div def + /yG0 M1 y01 mul M2 y02 mul add M1 M2 add div def + /vG0x M1 v0x1 mul M2 v0x2 mul add M1 M2 add div def + /vG0y M1 v0y1 mul M2 v0y2 mul add M1 M2 add div def + /K1 M2 Mt div neg def + /K2 M1 Mt div def +% conditions initiales pour le point réduit + /xr0 x02 x01 sub def + /yr0 y02 y01 sub def + /theta_0 yr0 xr0 atan def + /r0 xr0 dup mul yr0 dup mul add sqrt def + /v0xr v0x2 v0x1 sub def + /v0yr v0y2 v0y1 sub def + /v0r_2 v0xr dup mul v0yr dup mul add def +% constante des aires + /Cste xr0 v0yr mul yr0 v0xr mul sub def % Cste des aires + /Energie 0.5 mu mul v0r_2 mul % 1/2 mv^2 + G M1 M2 mul mul r0 div sub + def + /par Cste dup mul K div def % p + /exc 2 Cste dup mul mul Energie mul mu div K dup mul div 1 add sqrt def % e + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 a_2 1 exc dup mul sub sqrt mul def % demi-petit axe + /c_2 a_2 exc mul def % distance focale + % phase + /Phi theta_0 par r0 div 1 sub exc div arccos sub def + /rP par 1 exc add div def + /rA par 1 exc sub div def +% vitesses à l'apogée et au périgée + /vA G Mt mul 2 rA div 1 a_2 div sub mul sqrt def + /vP G Mt mul 2 rP div 1 a_2 div sub mul sqrt def +% positions du périgée et de l'apogée + /xP rP Phi cos mul def + /yP rP Phi sin mul def + /xA rA Phi cos mul neg def + /yA rA Phi sin mul neg def + /xW xA xP add 2 div def + /yW yA yP add 2 div def +% periode + /periode 6.28 a_2 3 exp G div Mt div sqrt mul def + /radius {par 1 exc t Phi sub cos mul add div} def + /xE {radius t cos mul} def + /yE {radius t sin mul}def + }% +%% 0 1 2 3 4 5 6 7 +%% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +\def\InitCond{ x01 y01 v0x1 v0y1 x02 y02 v0x2 v0y2} +\psset{method=rk4} +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt](-6,-8)(3,3)% +% mouvement de M2 par rapport à M1 +%\psequadiff[plotpoints=1000,algebraic, +% plotfuncx=y dup 4 get exch 0 get sub , +% plotfuncy=dup 5 get exch 1 get sub, +% tabname=XY1]{0}{22.7}{\InitCond}{\GravAlgIIcorps} +%\listplot[unit=1,linecolor=green]{XY1 aload pop} +% mouvement de M1 par rapport à M2 +%\psequadiff[plotpoints=1000,algebraic, +% plotfuncx=y dup 0 get exch 4 get sub , +% plotfuncy=dup 1 get exch 5 get sub, +% tabname=XY2]{0}{22.7}{\InitCond}{\GravAlgIIcorps} +%\listplot[unit=1,linecolor=magenta]{XY2 aload pop} +% mouvement de M1 par rapport à G +\psequadiff[plotpoints=1000,algebraic, + plotfuncx=y 0 get + y 4 get M2 mul + y 0 get M1 mul add + Mt div sub , + plotfuncy=1 get + y 1 get + M1 mul + y 5 get M2 mul add + Mt div sub, + tabname=XY1G, + saveData,filename=XY1G.dat + ]{0}{41.07}{\InitCond}{\GravAlgIIcorps} +\listplot[unit=1,linecolor=red]{XY1G aload pop} +% mouvement de M2 par rapport à G +\psequadiff[plotpoints=1000,algebraic, + plotfuncx=y 4 get + y 4 get M2 mul + y 0 get M1 mul add + Mt div sub , + plotfuncy=5 get + y 1 get M1 mul + y 5 get M2 mul add + Mt div sub, + tabname=XY2G, + saveData,filename=XY2G.dat + ]{0}{41.07}{\InitCond}{\GravAlgIIcorps} +\listplot[unit=1,linecolor=blue]{XY2G aload pop} +% centre de masse +%\psequadiff[plotpoints=1000,algebraic, +% plotfuncx=y dup 0 get M1 mul exch +% 4 get M2 mul add +% Mt div, +% plotfuncy=dup 1 get M1 mul exch +% 5 get M2 mul add +% Mt div, +% tabname=XGYG +% ,saveData,filename=XYG.dat +% ]{0}{120}{\InitCond}{\GravAlgIIcorps} +%\listplot[unit=1,linecolor=cyan]{XGYG aload pop} +\pnode(!x01 xG0 sub y01 yG0 sub){M01} +\pnode(!x02 xG0 sub y02 yG0 sub){M02} +%\pnode(!xG0 yG0){G0} +\psdots(M01)(M02) +%\psdot[linecolor=cyan](G0) +\rput(M01){\psline[unit=5,linecolor=red]{->}(!v0x1 vG0x sub v0y1 vG0y sub)} +\rput(M02){\psline[unit=5,linecolor=blue]{->}(!v0x2 vG0x sub v0y2 vG0y sub)} +%\rput(G0){\psline[unit=5]{->}(!vG0x vG0y)} +\psline{<->}(3,0)(0,0)(0,3) +\rput(-2,2){\psPrintValue[decimals=4]{periode}\hphantom{0000000}s} +\end{pspicture} +\end{center} +%\end{document} +La partie suivante est une animation en \textit{temps réel}. + +Pour faciliter les calculs les positions successives des deux corps ont été enregistrées, avec l'option \texttt{saveData}, dans deux fichiers. Voici pour le premier : +\begin{verbatim} +\psequadiff[plotpoints=1000,algebraic, + plotfuncx=y 0 get + y 4 get M2 mul + y 0 get M1 mul add + Mt div sub , + plotfuncy=1 get + y 1 get + M1 mul + y 5 get M2 mul add + Mt div sub, + tabname=XY1G, + saveData,filename=XY1G.dat + ]{0}{41.07}{\InitCond}{\GravAlgIIcorps} +\end{verbatim} +Cette option est ensuite désactivée. Pour l'animation, les fichiers sont lus et les données placées dans un tableau. +\begin{verbatim} +\pstVerb{/XY1 [(XYM1.dat) run] def + /XY2 [(XYM2.dat) run] def +\end{verbatim} +On pourra, éventuellement, tracer un segment joignant les deux corps en activant la ligne : +\begin{verbatim} + %\psline[linestyle=dashed,linecolor=white](!X1 Y1)(!X2 Y2) +\end{verbatim} + +\newpage +\section{L'animation} +\begin{center} +\pstVerb{/XY1 [(XY1G.dat) run] def + /XY2 [(XY2G.dat) run] def + }% +\def\nFrames{200}% 200 images +\begin{animateinline}[controls,loop,timeline=pb2corps.dat,% + begin={\begin{pspicture}(-6,-8)(3,3)}, + end={\end{pspicture}}]{20}% 20 images/s +\psframe*[linecolor={[cmyk]{1 1 0 0.7}}](-6,-8)(3,3) +\psgrid[subgriddiv=0,gridcolor=white,griddots=10,gridlabels=0pt]% +\listplot[linecolor=gray]{XY1 aload pop} +\listplot[linecolor=gray]{XY2 aload pop} +\psdot[linecolor=yellow,dotstyle=+](0,0) +\newframe +\multiframe{\nFrames}{i=0+10}{% 1 point sur 10 +\pstVerb{/X1 XY1 \i\space get def + /Y1 XY1 \i\space 1 add get def + /X2 XY2 \i\space get def + /Y2 XY2 \i\space 1 add get def + }% +%\psdot(!X1 Y1) +\pscircle*[linecolor=red](!X1 Y1){0.3} +\pscircle*[linecolor=blue!50](!X2 Y2){0.1} +%\psline[linestyle=dashed,linecolor=white](!X1 Y1)(!X2 Y2) +} +\end{animateinline} +\end{center} +\section{Mouvement d'un corps par rapport à l'autre} +Étudions le mouvement de $M_2$ par rapport à $M_1$. Dans un repère lié à $M_1$ les coordonnées de $M_2$ s'expriment par : +\[ +\left\{ +\begin{array}[m]{l} +x_{2/1}=x_2-x_1\\ +y_{2/1}=y_2-y_1 +\end{array} +\right. +\] +\begin{verbatim} +% 0 1 2 3 4 5 6 7 +% y[0] y[1] y[2] y[3] y[4] y[5] y[6] y[7] +% x1 y1 x'1 y'1 x2 y2 x'2 y'2 + y[4] y[0] sub % x2 - x1 + y[5] y[1] sub % y2 - y1 +\end{verbatim} + +\begin{verbatim} +% mouvement de M2 par rapport à M1 +\psequadiff[plotpoints=1000,algebraic, + plotfuncx=y dup 4 get exch 0 get sub , + plotfuncy=dup 5 get exch 1 get sub, + tabname=XY21, + saveData,filename=XY21.dat + ]{0}{41.07}{\InitCond}{\GravAlgIIcorps} +\end{verbatim} +\begin{center} +\begin{pspicture}(-7,-9)(3,3) +\pstVerb{ +/arccos { + dup + dup mul neg 1 add sqrt + exch + atan +} def + /G 1 def + /x01 2 def + /y01 2 def + /v0x1 .2 def + /v0y1 0.25 def + /x02 -3 def + /y02 0 def + /v0x2 -0.25 def + /v0y2 -0.5 def + /M1 3 def + /M2 1 def + /Mt M1 M2 add def + /mu M1 M2 mul M1 M2 add div def % masse réduite + /K G Mt mul def + /xG0 M1 x01 mul M2 x02 mul add M1 M2 add div def + /yG0 M1 y01 mul M2 y02 mul add M1 M2 add div def + /vG0x M1 v0x1 mul M2 v0x2 mul add M1 M2 add div def + /vG0y M1 v0y1 mul M2 v0y2 mul add M1 M2 add div def + /K1 M2 Mt div neg def + /K2 M1 Mt div def +% conditions initiales pour le point réduit + /xr0 x02 x01 sub def + /yr0 y02 y01 sub def + /theta_0 yr0 xr0 atan def + /r0 xr0 dup mul yr0 dup mul add sqrt def + /v0xr v0x2 v0x1 sub def + /v0yr v0y2 v0y1 sub def + /v0r_2 v0xr dup mul v0yr dup mul add def +% constante des aires + /Cste xr0 v0yr mul yr0 v0xr mul sub def % Cste des aires + /Energie 0.5 mu mul v0r_2 mul % 1/2 mv^2 + G M1 M2 mul mul r0 div sub + def + /par Cste dup mul K div def % p + /exc 2 Cste dup mul mul Energie mul mu div K dup mul div 1 add sqrt def % e + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 a_2 1 exc dup mul sub sqrt mul def % demi-petit axe + /c_2 a_2 exc mul def % distance focale + % phase + /Phi theta_0 par r0 div 1 sub exc div arccos sub def + /rP par 1 exc add div def + /rA par 1 exc sub div def +% vitesses à l'apogée et au périgée + /vA G Mt mul 2 rA div 1 a_2 div sub mul sqrt def + /vP G Mt mul 2 rP div 1 a_2 div sub mul sqrt def +% positions du périgée et de l'apogée + /xP rP Phi cos mul def + /yP rP Phi sin mul def + /xA rA Phi cos mul neg def + /yA rA Phi sin mul neg def + /xW xA xP add 2 div def + /yW yA yP add 2 div def +% periode + /periode 6.28 a_2 3 exp G div Mt div sqrt mul def + /radius {par 1 exc t Phi sub cos mul add div} def + /xE {radius t cos mul} def + /yE {radius t sin mul}def + }% +%% 0 1 2 3 4 5 6 7 +%% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +\def\InitCond{ x01 y01 v0x1 v0y1 x02 y02 v0x2 v0y2} +\psset{method=rk4} +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt](-7,-9)(3,3)% +% mouvement de M2 par rapport à M1 +\psequadiff[plotpoints=1000,algebraic, + plotfuncx=y dup 4 get exch 0 get sub , + plotfuncy=dup 5 get exch 1 get sub, + tabname=XY21, + saveData,filename=XY21.dat + ]{0}{41.07}{\InitCond}{\GravAlgIIcorps} +\listplot[unit=1,linecolor=green]{XY21 aload pop} +% mouvement de M1 par rapport à M2 +%\psequadiff[plotpoints=1000,algebraic, +% plotfuncx=y dup 0 get exch 4 get sub , +% plotfuncy=dup 1 get exch 5 get sub, +% tabname=XY2]{0}{22.7}{\InitCond}{\GravAlgIIcorps} +%\listplot[unit=1,linecolor=magenta]{XY2 aload pop} +\pnode(!x02 x01 sub y02 y01 sub){M021} +\pscircle*[linecolor=red](0,0){0.3} +\pscircle*[linecolor=blue!50](M021){0.1} +\uput[r](M021){$(t_0)$} +\psdot(M021)\uput[l](M021){$M_{2/1}$} +\rput(M021){\psline[unit=5,linecolor=blue]{->}(!v0x2 v0x1 sub v0y2 v0y1 sub)} +\uput{0.3}[d](0,0){$M_1(0)$} +\psline{<->}(3,0)(0,0)(0,3) +\rput(-2,2){\psPrintValue[decimals=4]{periode}\hphantom{0000000}s} +\end{pspicture} +\end{center} +Le mouvement de $M_1$ par rapport à $M_2$ : +\begin{verbatim} +% mouvement de M1 par rapport à M1 +\psequadiff[plotpoints=1000,algebraic, + plotfuncx=y dup 0 get exch 4 get sub , + plotfuncy=dup 1 get exch 5 get sub, + tabname=XY12, + saveData,filename=XY12.dat + ]{0}{41.07}{\InitCond}{\GravAlgIIcorps} +\end{verbatim} +\begin{center} +\begin{pspicture}(-3,-2)(8,8) +\pstVerb{ +/arccos { + dup + dup mul neg 1 add sqrt + exch + atan +} def + /G 1 def + /x01 2 def + /y01 2 def + /v0x1 .2 def + /v0y1 0.25 def + /x02 -3 def + /y02 0 def + /v0x2 -0.25 def + /v0y2 -0.5 def + /M1 3 def + /M2 1 def + /Mt M1 M2 add def + /mu M1 M2 mul M1 M2 add div def % masse réduite + /K G Mt mul def + /xG0 M1 x01 mul M2 x02 mul add M1 M2 add div def + /yG0 M1 y01 mul M2 y02 mul add M1 M2 add div def + /vG0x M1 v0x1 mul M2 v0x2 mul add M1 M2 add div def + /vG0y M1 v0y1 mul M2 v0y2 mul add M1 M2 add div def + /K1 M2 Mt div neg def + /K2 M1 Mt div def +% conditions initiales pour le point réduit + /xr0 x02 x01 sub def + /yr0 y02 y01 sub def + /theta_0 yr0 xr0 atan def + /r0 xr0 dup mul yr0 dup mul add sqrt def + /v0xr v0x2 v0x1 sub def + /v0yr v0y2 v0y1 sub def + /v0r_2 v0xr dup mul v0yr dup mul add def +% constante des aires + /Cste xr0 v0yr mul yr0 v0xr mul sub def % Cste des aires + /Energie 0.5 mu mul v0r_2 mul % 1/2 mv^2 + G M1 M2 mul mul r0 div sub + def + /par Cste dup mul K div def % p + /exc 2 Cste dup mul mul Energie mul mu div K dup mul div 1 add sqrt def % e + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 a_2 1 exc dup mul sub sqrt mul def % demi-petit axe + /c_2 a_2 exc mul def % distance focale + % phase + /Phi theta_0 par r0 div 1 sub exc div arccos sub def + /rP par 1 exc add div def + /rA par 1 exc sub div def +% vitesses à l'apogée et au périgée + /vA G Mt mul 2 rA div 1 a_2 div sub mul sqrt def + /vP G Mt mul 2 rP div 1 a_2 div sub mul sqrt def +% positions du périgée et de l'apogée + /xP rP Phi cos mul def + /yP rP Phi sin mul def + /xA rA Phi cos mul neg def + /yA rA Phi sin mul neg def + /xW xA xP add 2 div def + /yW yA yP add 2 div def +% periode + /periode 6.28 a_2 3 exp G div Mt div sqrt mul def + /radius {par 1 exc t Phi sub cos mul add div} def + /xE {radius t cos mul} def + /yE {radius t sin mul}def + }% +%% 0 1 2 3 4 5 6 7 +%% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +\def\InitCond{ x01 y01 v0x1 v0y1 x02 y02 v0x2 v0y2} +\psset{method=rk4} +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt](-3,-2)(8,9)% +% mouvement de M2 par rapport à M1 +%\psequadiff[plotpoints=1000,algebraic, +% plotfuncx=y dup 4 get exch 0 get sub , +% plotfuncy=dup 5 get exch 1 get sub, +% tabname=XY21, +% saveData,filename=XY21.dat +% ]{0}{41.07}{\InitCond}{\GravAlgIIcorps} +%\listplot[unit=1,linecolor=green]{XY21 aload pop} +% mouvement de M1 par rapport à M2 +\psequadiff[plotpoints=1000,algebraic, + plotfuncx=y dup 0 get exch 4 get sub , + plotfuncy=dup 1 get exch 5 get sub, + tabname=XY12, + saveData,filename=XY12.dat + ]{0}{41.07}{\InitCond}{\GravAlgIIcorps} +\listplot[unit=1,linecolor=magenta]{XY12 aload pop} +\pnode(!x01 x02 sub y01 y02 sub){M012} +\pscircle*[linecolor=red](M012){0.3} +\pscircle*[linecolor=blue!50](0,0){0.1} +\uput{0.3}[r](M012){$(t_0)$} +\uput[d](0,0){$M_2(0)$} +\psdot(M012)\uput{0.3}[l](M012){$M_{1/2}$} +\rput(M012){\psline[unit=5,linecolor=red]{->}(!v0x1 v0x2 sub v0y1 v0y2 sub)} +%\rput(G0){\psline[unit=5]{->}(!vG0x vG0y)} +\psline{<->}(3,0)(0,0)(0,3) +\rput(-2,2){\psPrintValue[decimals=4]{periode}\hphantom{0000000}s} +\end{pspicture} +\end{center} +\section{Animation du mouvement de $M_1$ par rapport à $M_2$} +\begin{center} +\pstVerb{/XY12 [(XY12.dat) run] def + }% +\def\nFrames{199}% 199 images au lieu de 200 ? +\begin{animateinline}[controls,loop,timeline=pb2corps.dat,% + begin={\begin{pspicture}(-3,-2)(8,10)}, + end={\end{pspicture}}]{20}% 20 images/s +\psframe*[linecolor={[cmyk]{1 1 0 0.7}}](-3,-2)(8,10) +\psgrid[subgriddiv=0,gridcolor=white,griddots=10,gridlabels=0pt]% +\listplot[linecolor=gray]{XY12 aload pop} +\pscircle*[linecolor=blue!50](0,0){0.1} +\newframe +\multiframe{\nFrames}{i=0+10}{% 1 point sur 10 +\pstVerb{/X1 XY12 \i\space get def + /Y1 XY12 \i\space 1 add get def + }% +\pscircle*[linecolor=red](!X1 Y1){0.3} +} +\end{animateinline} +\end{center} +\end{document} \ No newline at end of file