From a073b16d2ce5dac585504da8489bc1847beee7cc Mon Sep 17 00:00:00 2001 From: Manuel Date: Sat, 23 Jun 2012 09:26:47 +0200 Subject: [PATCH] =?utf8?q?J'ai=20compl=C3=A9t=C3=A9=20le=20document=20sur?= =?utf8?q?=20la=20mise=20en=20orbite=20d'un=20satellite=20et=20ajout=C3=A9?= =?utf8?q?=20en=20particulier=20l'hodographe=20:=20c'est=20un=20cercle=20:?= =?utf8?q?=20int=C3=A9ressant=20=C3=A0=20d=C3=A9montrer(je=20d=C3=A9couvre?= =?utf8?q?).=20J'esp=C3=A8re=20ne=20pas=20avoir=20fait=20de=20b=C3=AAtises?= =?utf8?q?,=20car=20je=20vois=20que=20le=20commit=20rajoute=20des=20fichie?= =?utf8?q?rs=20sur=20Lorentz=20existants=20d=C3=A9j=C3=A0=20=3F=20En=20pr?= =?utf8?q?=C3=A9vision=20l'animation.?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- gravitation/eqdf-grav_01.pdf | Bin 88426 -> 0 bytes gravitation/eqdf-grav_01.tex | 419 --- gravitation/g1.png | Bin 0 -> 24028 bytes gravitation/g2.png | Bin 0 -> 19847 bytes gravitation/g3.png | Bin 0 -> 12708 bytes gravitation/g4.png | Bin 0 -> 21256 bytes gravitation/gravitation_01.pdf | Bin 102885 -> 125377 bytes gravitation/gravitation_01.tex | 1070 ++++--- gravitation/gravitation_02.pdf | Bin 39057 -> 59268 bytes gravitation/gravitation_02.tex | 194 ++ gravitation/h1.png | Bin 0 -> 24177 bytes gravitation/h2.png | Bin 0 -> 31773 bytes gravitation/pst-eqdf.tex | 3 +- lorenz/LorenzXY.dat | 5002 ++++++++++++++++---------------- lorenz/LorenzXZ.dat | 5002 ++++++++++++++++---------------- lorenz/LorenzYZ.dat | 5002 ++++++++++++++++---------------- lorenz/XYZLorenz.dat | 5002 ++++++++++++++++---------------- lorenz/index.xml | 8 +- web.xml | 10 +- 19 files changed, 10806 insertions(+), 10906 deletions(-) delete mode 100644 gravitation/eqdf-grav_01.pdf delete mode 100644 gravitation/eqdf-grav_01.tex create mode 100644 gravitation/g1.png create mode 100644 gravitation/g2.png create mode 100644 gravitation/g3.png create mode 100644 gravitation/g4.png create mode 100644 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zyX%mF!;!noL%`8{^dtfS-=lXCfa2ZVUO;B{$RGlR+ROK7$ewWl0nlm>E@R>B*4OIvVql!eRBh)lB)if}g2o)p**i3_?)HNVT4K>Ao4`H(( axGm&G!V*b4gE_$bP}u%s$29ac_x}%D$$WhP diff --git a/gravitation/eqdf-grav_01.tex b/gravitation/eqdf-grav_01.tex deleted file mode 100644 index 45c744c..0000000 --- a/gravitation/eqdf-grav_01.tex +++ /dev/null @@ -1,419 +0,0 @@ -\documentclass{article} -\usepackage[a4paper,margin=2cm]{geometry} -\usepackage[latin1]{inputenc} -\usepackage{pstricks-add,pst-eucl,pst-func} -\usepackage{array,amsmath} -\newpsstyle{vecteurA}{arrowinset=0.05,arrowsize=0.125,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 0.5 0}}} -%%%%%%%%%%%%%%%%%% -\title{Interaction gravitationnelle avec PSTricks} -\date{19 juin 2\,012} -\begin{document} -\maketitle -\def\eqsatellite{% -y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)}% -\section{Mise en orbite d'un satellite} -\begin{figure}[htbp] - \begin{center} -\psset{unit=2} -\begin{pspicture}(-6,-2)(2,5) -\pstVerb{ - /GM 1 def - /theta0 -35 def - /r0 1 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.3 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def -% vitesse à l'apogée - /vA GM par div sqrt 1 exc sub mul def -% vitesse au périgée - /vP GM par div sqrt 1 exc add mul def -% coordonnées de vA - /vAx vA theta0 90 add cos mul neg def - /vAy vA theta0 90 add sin mul neg def -% coordonnées de vP - /vPx vP theta0 90 add cos mul def - /vPy vP theta0 90 add sin mul def -% Apogée - /xA par 1 exc sub div theta0 cos mul neg def - /yA par 1 exc sub div theta0 sin mul neg def -% Périgée - /xP par 1 exc add div theta0 cos mul def - /yP par 1 exc add div theta0 sin mul def -% Centre - /xO xP xA add 2 div def - /yO yP yA add 2 div def -}% -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} -\psdot[dotstyle=+](0,0) -\uput[d](0,0){O} -\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-2)(2,5) -\rput(1,4){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{0000}s}} -\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\pscircle*(!x0 y0){0.05} -\pnode(!xP yP){P} % périgée -\pnode(!xA yA){A} % Apogée -\pnode(!xO yO){O} % centre -\rput(!x0 y0){\psline[style=vecteurB]{->}(!v0x v0y)\uput[ur](!v0x v0y){$\overrightarrow{v_0}$}} -\psline[linestyle=dashed](A)(P) -\pscircle*(A){0.05} -\uput[ul](A){$A$} -\uput[dr](P){$P$} -% position du satellite à un instant quelconque -\pstVerb{/theta_i 170 def - /radius par 1 exc theta_i theta0 sub cos mul add div def - /xS radius theta_i cos mul def - /yS radius theta_i sin mul def - /ux theta_i cos 1 mul def - /uy theta_i sin 1 mul def - /xi xS ux sub def - /yi yS uy sub def - /xi2 xS ux 2 div sub def - /yi2 yS uy 2 div sub def}% -\pnode(!xi2 yi2){Mi2} -\pnode(!xi yi){Mi} -\pnode(!xS yS){S} -\pscircle*(S){0.05} -\psline[linestyle=dotted](S)(0,0) -\psline[style=vecteurA]{->}(S)(Mi) -\uput[l](S){S} -\uput[u](Mi2){$\overrightarrow{F}$} -\psarcn{->}(0,0){0.4}{0}{!theta0} -\uput{0.5}[!theta0 2 div](0,0){$\theta_0$} -\rput(A){\psline[style=vecteurB]{->}(!vAx vAy)} -%\rput(P){\psline[style=vecteurB]{->}(!vPx vPy)} -\psline[arrowinset=0.05,arrowsize=0.1]{<->}(2,0)(0,0)(0,5) -\uput[r](0,4.9){$y$}\uput[u](1.9,0){$x$} -\psdot(O)\uput[r](O){$\Omega$} -\rput{!90 theta0 add}(O){\psline[linestyle=dashed](!b_2 neg 0)(!b_2 0)} -\end{pspicture} -\caption{Mouvement d'un satellite} - \end{center} -\end{figure} -Soit (M) la masse de l'astre et (m) celle du satellite avec $m\ll M$. Le centre de masse du système \{M,m\} est confondu avec le centre de l'astre attracteur. La mise en orbite s'effectue à partir du point $M_0(x_0,y_0)$ avec une vitesse $\overrightarrow{v_0}(v_{0_x},v_{0_x})$. $\theta_0$ est l'angle que fait $\overrightarrow{Ox}$ avec $\overrightarrow{OM_0}$. Le satellite (S), supposé ponctuel, subit de la part de l'astre une force d'attraction gravitationnelle : -\[ -\overrightarrow{F}=-\mathcal{G}\frac{Mm}{r^2}\overrightarrow{u}\qquad \text{avec}\quad \overrightarrow{u}=\frac{\overrightarrow{r}}{r}\quad \text{et}\quad \overrightarrow{r}=\overrightarrow{OS} -\] -Tous les \textit{bons} livres de mécanique\footnote{Comme celui, par exemple, de José-Philippe Pérez, aux éditions Masson.} établissent les relations suivantes : -\[ -r=\frac{p}{1+\mathrm{e}\cos(\theta-\theta_0)} -\] -Paramètres et excentricité ont pour expressions respectives, avec les notations suivantes : $K=\mathcal{G}Mm$, $\mathcal{E}$ l'énergie du système et $L$ le moment cinétique. -\[ -\mathrm{e}=\sqrt{1+\frac{2\mathcal{E}L^2}{mK^2}}\qquad p=\frac{L^2}{mK} -\] -On choisit une vitesse initiale $\overrightarrow{v_0}$ perpendiculaire à $\overrightarrow{OM_0}$, dans ces conditions le moment cinétique et l'énergie, qui restent constants, valent : -\[ -L=mr_0v_0\qquad \mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 -\] -En remplaçant $L$ et $\mathcal{E}$, on obtient pour l'excentricité et le paramètre les expressions suivantes : -\[ -\mathrm{e}=\sqrt{1+\frac{1}{\mathcal{G}^2M^2}\Big(\frac{1}{2}v_0^4r_0^2-\mathcal{G}Mr_0v_0^2\Big)} -\] -\[ -p=\frac{v_0^2r_0^2}{\mathcal{G}M} -\] -On se limite au cas du mouvement elliptique, avec, en conséquence, la condition : -\[ -\mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 < 0 -\] -Le demi-grand axe $a$, le demi-petit axe $b$ sont : -\[ -a=\frac{p}{1-\mathrm{e}^2}\qquad b=\frac{p}{\sqrt{1-\mathrm{e}^2}} -\] -La période $T$ qui obéit à la troisième loi de Képler : -\[ -T^2=\frac{4\pi^2a^3}{\mathcal{G}M} -\] -La vitesse, en un point de l'ellipse, se calcule par : -\[ -v^2=\mathcal{G}M\Big( \frac{2}{r}-\frac{1}{a}\Big) -\] -Sachant que $r_p=\dfrac{p}{1+\mathrm{e}}$ et $r_A=\dfrac{p}{1-\mathrm{e}}$, on en déduit les vitesses au périgée et à l'apogée : -\[ -v_P=\sqrt{\frac{\mathcal{G}M}{p}}(1+\mathrm{e})\qquad v_A=\sqrt{\frac{\mathcal{G}M}{p}}(1-\mathrm{e}) -\] - -\section{L'étude avec PSTricks} -\subsection{La trajectoire} -\def\parametres{ - /GM 1 def % 4e14 def % GxM - /x0 6.5e6 def % position initiale - /y0 0 def - /vx0 0 def % vitesse initiale - /vy0 1e4 def -} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] -\[ -\left\{ -\begin{array}{rcl} -\ddot{x}&=&-\dfrac{GM}{r^3}x\\[1em] -\ddot{y}&=&-\dfrac{GM}{r^3}y\\ -\end{array} -\right. -\label{eq1} -\qquad r=\sqrt{x^2+y^2} -\] - -On peut dessiner la trajectoire du satellite et de ses caractéristiques de deux façons : -\begin{itemize} - \item par l'utilisation de \verb+\parametricplot+ ; - \item ou celle de \verb+\psplotDiffEqn+. -\end{itemize} -\verb+\parametricplot+ utilise l'expression exacte de l'équation de la trajectoire en coordonnées polaires : -\begin{verbatim} - \parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{% - /radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\end{verbatim} -L'excentricité, la période, demi-grand axe et demi-petit axe sont calculés par quelques lignes de code \textsf{postscript}. Il faut s'assurer que les conditions initiales choisies vérifient bien la condition d'une trajectoire elliptique, pour cela il faut que l'énergie initiale $\mathcal{E}_0<0$, sinon cela entraînera une erreur lors du passage à l'interpréteur \textsf{postscript}. -\begin{verbatim} -\pstVerb{ - /GM 1 def - /theta0 -45 def - /r0 0.5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.92 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul - v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%demi-grand axe - /a_2 par 1 exc dup mul sub div def % demi-grand axe -%demi-petit axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe -% période - /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def -}% -\end{verbatim} -\verb+\psplotDiffEqn+ utilise les équations différentielles du mouvement, en notation algébrique : -\begin{verbatim} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] -\def\eqsatellite{% -y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)} - \psplotDiffEqn[unit=2,whichabs=0,whichord=1,% - linecolor=blue,linewidth=0.1,% - method=rk4,plotpoints=1000,% - algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} -Ce qui permet, par ailleurs de vérifier la qualité du tracé par la méthode numérique, en bleu, tandis que le tracé à partir de l'expression exacte est en trait fin en rouge. -% -\begin{center} -\begin{pspicture}(-12,-2)(4,10) -\pstVerb{ - /GM 1 def - /theta0 -45 def - /r0 0.5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.92 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def -}% -\psframe*[linecolor=white](-3,-0.2)(0,0.2) -\pscircle[fillcolor=blue!50,fillstyle=solid](0,0){0.5} -\psgrid[unit=2,subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-1)(2,5) -\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} - \psplotDiffEqn[unit=2,whichabs=0,whichord=1,linecolor=blue,linewidth=0.1,method=rk4,plotpoints=1000,algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% -\parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\psdot[unit=2,dotsize=0.12](!x0 y0) -\rput(!x0 2 mul y0 2 mul){\psline[style=vecteurC]{->}(!v0x v0y)} -\end{pspicture} -\end{center} -\subsection{La vitesse} -\verb+\psplotDiffEqn+ permet de voir comment varie la vitesse sur l'ellipse : -\begin{verbatim} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] - \psplotDiffEqn[xunit=0.2,yunit=5,% - plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, - linecolor=red,method=rk4,plotpoints=1000, - algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} -\begin{center} -\begin{pspicture}(0,-1)(10,7) -\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt] -\pstVerb{/GM 1 def - /theta0 -35 def - /r0 1 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.3 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def}% -\psplotDiffEqn[xunit=0.2,yunit=5, - plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, - ,linecolor=red,method=rk4,plotpoints=1000,algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% - \multido{\i=1+1,\I=5+5}{9}{\uput[u](\i,0){\I}} -\pnode(! 36.4 5 div 0){P} -\psdot(P)\uput[d](P){Périgée} -\psline[linestyle=dashed](P)(! 36.4 5 div 7) -\pnode(! 36.4 10 div 0){A} -\psdot(A)\uput[d](A){Apogée} -\psline[linestyle=dashed](A)(! 36.4 10 div 7) -\uput[l](0,6.5){$v$} -\psline[arrowinset=0.1,arrowsize=0.2]{<->}(10,0)(0,0)(0,7) -\uput[u](10,0){$t$(s)} -\uput[ul](0,0){0} -\end{pspicture} -\end{center} -On peut obtenir les caractéristiques de la vitesse en un point quelconque, car le moment cinétique $L=mr\dot{\theta}$ étant constant, pour chaque valeur de $\theta$, on en déduit $r$ puis $\dot{\theta}$. En coordonnées polaires, la vitesse s'exprime par : -\[ -\overrightarrow{v}=\dot{r}\overrightarrow{e_r}+r\dot{\theta}\overrightarrow{e_{\theta}} -\] -$\dot{\theta}$ et $\dot{r}$ s'obtiennent par les relations suivantes : -\[ -\dot{\theta}^2=\frac{r_0v_0}{r} -\] -\[ -\dot{r}=-\frac{p(-\dot{\theta}\sin(\theta-\theta_0)}{(1+\mathrm{e}\cos(\theta-\theta_0))^2}=\frac{r^2}{p}\dot{\theta}\sin(\theta-\theta_0) -\] -La chaîne de calculs est la suivante : $\theta\Longrightarrow r\Longrightarrow \dot{\theta}\Longrightarrow \dot{r}\Longrightarrow \overrightarrow{v}$. -\section{Mouvement circulaire} - \begin{center} -\psset{unit=1} -\begin{pspicture}(-5,-5.5)(5,5.5) -\pstVerb{ - /GM 1 def - /theta0 60 def - /r0 5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 GM r0 div sqrt def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def -% vitesse à l'apogée - /vA GM par div sqrt 1 exc sub mul def -% vitesse au périgée - /vP GM par div sqrt 1 exc add mul def -% coordonnées de vA - /vAx vA theta0 90 add cos mul neg def - /vAy vA theta0 90 add sin mul neg def -% coordonnées de vP - /vPx vP theta0 90 add cos mul def - /vPy vP theta0 90 add sin mul def -}% -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.75} -\psdot[dotstyle=+](0,0) -\uput[d](0,0){O} -\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(5,5) -\rput(0,-2){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{000000}s}} -\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\pscircle*(!x0 y0){0.1} -%\pnode(!par 1 exc add div theta0 cos mul par 1 exc add div theta0 sin mul){P} % périgée -%\pnode(!par 1 exc sub div theta0 cos mul neg par 1 exc sub div theta0 sin mul neg){A} % Apogée -\rput(!x0 y0){\psline[arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{0 0.5 1}},unit=4]{->}(!v0x v0y)\uput[ur](!v0x 2 mul v0y 2 mul){$\overrightarrow{v_0}$}} -\uput[ur](!x0 y0){$M_0$} -\psline[linestyle=dashed](0,0)(!x0 y0) -% position du satellite à un instant quelconque -\pstVerb{/theta_i 170 def - /radius par 1 exc theta_i theta0 sub cos mul add div def - /xS radius theta_i cos mul def - /yS radius theta_i sin mul def - /ux theta_i cos 1 mul def - /uy theta_i sin 1 mul def - /xi xS ux sub def - /yi yS uy sub def - /xi2 xS ux 2 div sub def - /yi2 yS uy 2 div sub def}% -\pnode(!xi2 yi2){Mi2} -\pnode(!xi yi){Mi} -\pnode(!xS yS){S} -\pscircle*(S){0.1} -\psline[linestyle=dotted](S)(0,0) -\psline[style=vecteurC]{->}(S)(Mi) -\uput[l](S){S} -\uput[u](Mi2){$\overrightarrow{F}$} -\psarc{->}(0,0){1}{0}{!theta0} -\uput{1.1}[!theta0 2 div](0,0){$\theta_0$} -\psline[arrowinset=0.1,arrowsize=0.2]{<->}(5,0)(0,0)(0,5) -\psline[arrowinset=0.05,arrowsize=0.1]{<->}(5,0)(0,0)(0,5) -\uput[u](0,5){$y$}\uput[r](5,0){$x$} -\end{pspicture} - \end{center} -Il s'obtient très facilement à partir de l'étude précédente si on sait que dans ce cas : -\[ -v_0=\sqrt{\frac{\mathcal{G}M}{r_0}} -\] -\begin{verbatim} -\pstVerb{ - /GM 1 def - /theta0 60 def - /r0 5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 GM r0 div sqrt def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def }% -\end{verbatim} - -\end{document} -\begin{figure}[htbp] - \begin{center} -%\begin{pspicture}[shift=-2,showgrid=true](-3,-1.75)(2,1.5) -\begin{pspicture}(-12,-6)(4,6) -\pstVerb{\parametres}% -\psdot[dotsize=1](0,0) - \psplotDiffEqn[unit=2,whichabs=0,whichord=1,linecolor=blue,method=rk4,plotpoints=1000,algebraic]{0}{36.5}{1 0 0 1.3}{\eqsatellite}% -% \psplotDiffEqn[whichabs=0,whichord=1,linecolor=red, plotpoints=200,method=varrkiv, varsteptol=.00001,algebraic,showpoints]{0}{35}{5 0 0 %0.25}{\eqsatellite} -\psgrid[unit=2](-6,-3)(2,3) -\end{pspicture} -\caption{Mouvement d'un satellite} - \end{center} -\end{figure} - -\begin{figure}[htbp] - \begin{center} -%\begin{pspicture}[shift=-2,showgrid=true](-3,-1.75)(2,1.5) -\begin{pspicture}(-12,-6)(4,6) -\pstVerb{\parametres}% -\psset{unit=2}% -\psdot[dotsize=1](0,0) - \psplotDiffEqn[whichabs=0,whichord=1,linecolor=blue,method=rk4,plotpoints=1000,algebraic]{0}{50}{1 0 0.2 1.25}{\eqsatellite}% -\psdot(1,0) -\rput(1,0){\psline[style=vecteurA]{->}(0.2,1.25)} -\psgrid(-6,-3)(2,3) -\end{pspicture} -\caption{Mouvement d'un satellite 3} - \end{center} -\end{figure} \ No newline at end of file diff --git a/gravitation/g1.png b/gravitation/g1.png new file mode 100644 index 0000000000000000000000000000000000000000..a506f234480a4f3da1493581e5891181310fb401 GIT binary patch literal 24028 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-\usepackage{pstricks-add,pst-eqdf,pst-func} -\usepackage{array,amsmath} -\usepackage{listings} -\newpsstyle{vecteurA}{arrowinset=0.05,arrowsize=0.125,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 0.5 0}}} -%%%%%%%%%%%%%%%%%% -\title{Interaction gravitationnelle avec PSTricks} -\date{19 juin 2\,012} -\begin{document} -\maketitle -\def\eqsatellite{% -y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)}% -\section{Mise en orbite d'un satellite} -\begin{figure}[htbp] - \begin{center} -\psset{unit=2} -\begin{pspicture}(-6,-2)(2,5) -\pstVerb{ - /GM 1 def - /theta0 -35 def - /r0 1 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.3 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricitĂ© - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def -% vitesse Ă  l'apogĂ©e - /vA GM par div sqrt 1 exc sub mul def -% vitesse au pĂ©rigĂ©e - /vP GM par div sqrt 1 exc add mul def -% coordonnĂ©es de vA - /vAx vA theta0 90 add cos mul neg def - /vAy vA theta0 90 add sin mul neg def -% coordonnĂ©es de vP - /vPx vP theta0 90 add cos mul def - /vPy vP theta0 90 add sin mul def -% ApogĂ©e - /xA par 1 exc sub div theta0 cos mul neg def - /yA par 1 exc sub div theta0 sin mul neg def -% PĂ©rigĂ©e - /xP par 1 exc add div theta0 cos mul def - /yP par 1 exc add div theta0 sin mul def -% Centre - /xO xP xA add 2 div def - /yO yP yA add 2 div def -}% -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} -\psdot[dotstyle=+](0,0) -\uput[d](0,0){O} -\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-2)(2,5) -\rput(1,4){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{0000}s}} -\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\pscircle*(!x0 y0){0.05} -\pnode(!xP yP){P} % pĂ©rigĂ©e -\pnode(!xA yA){A} % ApogĂ©e -\pnode(!xO yO){O} % centre -\rput(!x0 y0){\psline[style=vecteurB]{->}(!v0x v0y)\uput[ur](!v0x v0y){$\overrightarrow{v_0}$}} -\psline[linestyle=dashed](A)(P) -\pscircle*(A){0.05} -\uput[ul](A){$A$} -\uput[dr](P){$P$} -% position du satellite Ă  un instant quelconque -\pstVerb{/theta_i 170 def - /radius par 1 exc theta_i theta0 sub cos mul add div def - /xS radius theta_i cos mul def - /yS radius theta_i sin mul def - /ux theta_i cos 1 mul def - /uy theta_i sin 1 mul def - /xi xS ux sub def - /yi yS uy sub def - /xi2 xS ux 2 div sub def - /yi2 yS uy 2 div sub def}% -\pnode(!xi2 yi2){Mi2} -\pnode(!xi yi){Mi} -\pnode(!xS yS){S} -\pscircle*(S){0.05} -\psline[linestyle=dotted](S)(0,0) -\psline[style=vecteurA]{->}(S)(Mi) -\uput[l](S){S} -\uput[u](Mi2){$\overrightarrow{F}$} -\psarcn{->}(0,0){0.4}{0}{!theta0} -\uput{0.5}[!theta0 2 div](0,0){$\theta_0$} -\rput(A){\psline[style=vecteurB]{->}(!vAx vAy)} -%\rput(P){\psline[style=vecteurB]{->}(!vPx vPy)} -\psline[arrowinset=0.05,arrowsize=0.1]{<->}(2,0)(0,0)(0,5) -\uput[r](0,4.9){$y$}\uput[u](1.9,0){$x$} -\psdot(O)\uput[r](O){$\Omega$} -\rput{!90 theta0 add}(O){\psline[linestyle=dashed](!b_2 neg 0)(!b_2 0)} -\end{pspicture} -\caption{Mouvement d'un satellite} - \end{center} -\end{figure} -Soit (M) la masse de l'astre et (m) celle du satellite avec $m\ll M$. Le centre de masse du système \{M,m\} est confondu avec le centre de l'astre attracteur. La mise en orbite s'effectue Ă  partir du point $M_0(x_0,y_0)$ avec une vitesse $\overrightarrow{v_0}(v_{0_x},v_{0_x})$. $\theta_0$ est l'angle que fait $\overrightarrow{Ox}$ avec $\overrightarrow{OM_0}$. Le satellite (S), supposĂ© ponctuel, subit de la part de l'astre une force d'attraction gravitationnelle : -\[ -\overrightarrow{F}=-\mathcal{G}\frac{Mm}{r^2}\overrightarrow{u}\qquad \text{avec}\quad \overrightarrow{u}=\frac{\overrightarrow{r}}{r}\quad \text{et}\quad \overrightarrow{r}=\overrightarrow{OS} -\] -Tous les \textit{bons} livres de mĂ©canique\footnote{Comme celui, par exemple, de JosĂ©-Philippe PĂ©rez, aux Ă©ditions Masson.} Ă©tablissent les relations suivantes : -\[ -r=\frac{p}{1+\mathrm{e}\cos(\theta-\theta_0)} -\] -Paramètres et excentricitĂ© ont pour expressions respectives, avec les notations suivantes : $K=\mathcal{G}Mm$, $\mathcal{E}$ l'Ă©nergie du système et $L$ le moment cinĂ©tique. -\[ -\mathrm{e}=\sqrt{1+\frac{2\mathcal{E}L^2}{mK^2}}\qquad p=\frac{L^2}{mK} -\] -On choisit une vitesse initiale $\overrightarrow{v_0}$ perpendiculaire Ă  $\overrightarrow{OM_0}$, dans ces conditions le moment cinĂ©tique et l'Ă©nergie, qui restent constants, valent : -\[ -L=mr_0v_0\qquad \mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 -\] -En remplaçant $L$ et $\mathcal{E}$, on obtient pour l'excentricitĂ© et le paramètre les expressions suivantes : -\[ -\mathrm{e}=\sqrt{1+\frac{1}{\mathcal{G}^2M^2}\Big(\frac{1}{2}v_0^4r_0^2-\mathcal{G}Mr_0v_0^2\Big)} -\] -\[ -p=\frac{v_0^2r_0^2}{\mathcal{G}M} -\] -On se limite au cas du mouvement elliptique, avec, en consĂ©quence, la condition : -\[ -\mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 < 0 -\] -Le demi-grand axe $a$, le demi-petit axe $b$ sont : -\[ -a=\frac{p}{1-\mathrm{e}^2}\qquad b=\frac{p}{\sqrt{1-\mathrm{e}^2}} -\] -La pĂ©riode $T$ qui obĂ©it Ă  la troisième loi de KĂ©pler : -\[ -T^2=\frac{4\pi^2a^3}{\mathcal{G}M} -\] -La vitesse, en un point de l'ellipse, se calcule par : -\[ -v^2=\mathcal{G}M\Big( \frac{2}{r}-\frac{1}{a}\Big) -\] -Sachant que $r_p=\dfrac{p}{1+\mathrm{e}}$ et $r_A=\dfrac{p}{1-\mathrm{e}}$, on en dĂ©duit les vitesses au pĂ©rigĂ©e et Ă  l'apogĂ©e : -\[ -v_P=\sqrt{\frac{\mathcal{G}M}{p}}(1+\mathrm{e})\qquad v_A=\sqrt{\frac{\mathcal{G}M}{p}}(1-\mathrm{e}) -\] - -\section{L'Ă©tude avec PSTricks} -\subsection{La trajectoire} -\def\parametres{ - /GM 1 def % 4e14 def % GxM - /x0 6.5e6 def % position initiale - /y0 0 def - /vx0 0 def % vitesse initiale - /vy0 1e4 def -} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] -\[ -\left\{ -\begin{array}{rcl} -\ddot{x}&=&-\dfrac{GM}{r^3}x\\[1em] -\ddot{y}&=&-\dfrac{GM}{r^3}y\\ -\end{array} -\right. -\label{eq1} -\qquad r=\sqrt{x^2+y^2} -\] - -On peut dessiner la trajectoire du satellite et de ses caractĂ©ristiques de deux façons : -\begin{itemize} - \item par l'utilisation de \verb+\parametricplot+ ; - \item ou celle de \verb+\psplotDiffEqn+. -\end{itemize} -\verb+\parametricplot+ utilise l'expression exacte de l'Ă©quation de la trajectoire en coordonnĂ©es polaires : -\begin{verbatim} - \parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{% - /radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\end{verbatim} -L'excentricitĂ©, la pĂ©riode, demi-grand axe et demi-petit axe sont calculĂ©s par quelques lignes de code \textsf{postscript}. Il faut s'assurer que les conditions initiales choisies vĂ©rifient bien la condition d'une trajectoire elliptique, pour cela il faut que l'Ă©nergie initiale $\mathcal{E}_0<0$, sinon cela entraĂ®nera une erreur lors du passage Ă  l'interprĂ©teur \textsf{postscript}. -\begin{verbatim} -\pstVerb{ - /GM 1 def - /theta0 -45 def - /r0 0.5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.92 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricitĂ© - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul - v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%demi-grand axe - /a_2 par 1 exc dup mul sub div def % demi-grand axe -%demi-petit axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe -% pĂ©riode - /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def -}% -\end{verbatim} -\verb+\psplotDiffEqn+ utilise les Ă©quations diffĂ©rentielles du mouvement, en notation algĂ©brique : -\begin{verbatim} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] -\def\eqsatellite{% -y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)} - \psplotDiffEqn[unit=2,whichabs=0,whichord=1,% - linecolor=blue,linewidth=0.1,% - method=rk4,plotpoints=1000,% - algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} -Ce qui permet, par ailleurs de vĂ©rifier la qualitĂ© du tracĂ© par la mĂ©thode numĂ©rique, en bleu, tandis que le tracĂ© Ă  partir de l'expression exacte est en trait fin en rouge. -% -\begin{center} -\begin{pspicture}(-12,-2)(4,10) -\pstVerb{ - /GM 1 def - /theta0 -45 def - /r0 0.5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.92 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricitĂ© - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def -}% -\psframe*[linecolor=white](-3,-0.2)(0,0.2) -\pscircle[fillcolor=blue!50,fillstyle=solid](0,0){0.5} -\psgrid[unit=2,subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-1)(2,5) -\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} - \psplotDiffEqn[unit=2,whichabs=0,whichord=1,linecolor=blue,linewidth=0.1,method=rk4,plotpoints=1000,algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% -\parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\psdot[unit=2,dotsize=0.12](!x0 y0) -\rput(!x0 2 mul y0 2 mul){\psline[style=vecteurC]{->}(!v0x v0y)} -\end{pspicture} -\end{center} -\subsection{La vitesse} -\verb+\psplotDiffEqn+ permet de voir comment varie la vitesse sur l'ellipse : -\begin{verbatim} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] - \psplotDiffEqn[xunit=0.2,yunit=5,% - plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, - linecolor=red,method=rk4,plotpoints=1000, - algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} -\begin{center} -\begin{pspicture}(0,-1)(10,7) -\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt] -\pstVerb{/GM 1 def - /theta0 -35 def - /r0 1 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.3 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def}% -\psplotDiffEqn[xunit=0.2,yunit=5, - plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, - ,linecolor=red,method=rk4,plotpoints=1000,algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% - \multido{\i=1+1,\I=5+5}{9}{\uput[u](\i,0){\I}} -\pnode(! 36.4 5 div 0){P} -\psdot(P)\uput[d](P){PĂ©rigĂ©e} -\psline[linestyle=dashed](P)(! 36.4 5 div 7) -\pnode(! 36.4 10 div 0){A} -\psdot(A)\uput[d](A){ApogĂ©e} -\psline[linestyle=dashed](A)(! 36.4 10 div 7) -\uput[l](0,6.5){$v$} -\psline[arrowinset=0.1,arrowsize=0.2]{<->}(10,0)(0,0)(0,7) -\uput[u](10,0){$t$(s)} -\uput[ul](0,0){0} -\end{pspicture} -\end{center} -On peut obtenir les caractĂ©ristiques de la vitesse en un point quelconque, car le moment cinĂ©tique $L=mr\dot{\theta}$ Ă©tant constant, pour chaque valeur de $\theta$, on en dĂ©duit $r$ puis $\dot{\theta}$. En coordonnĂ©es polaires, la vitesse s'exprime par : -\[ -\overrightarrow{v}=\dot{r}\overrightarrow{u_r}+r\dot{\theta}\overrightarrow{u_{\theta}} -\] -$\dot{\theta}$ et $\dot{r}$ s'obtiennent par les relations suivantes : -\[ -\dot{\theta}=\frac{r_0v_0}{r^2} -\] -\[ -\dot{r}=-\frac{p(-\dot{\theta}\sin(\theta-\theta_0)}{(1+\mathrm{e}\cos(\theta-\theta_0))^2}=\frac{r_0v_0}{p}\sin(\theta-\theta_0) -\] -La chaĂ®ne de calculs est la suivante : $\theta\Longrightarrow r\Longrightarrow \dot{\theta}\Longrightarrow \dot{r}\Longrightarrow \overrightarrow{v}$. - -Le package `\textsf{pst-eqdf}' comprend la commande \verb+\psequadiff+ qui est une version simplifiĂ©e de \verb+\psplotDiffEqn+, dont elle ne reprend que la mĂ©thode Runge-Kutta~4. Elle permet de sauvegarder sous forme de tableaux et/ou de fichiers toutes les variables et les dĂ©rivĂ©es de la fonction Ă©tudiĂ©e, cette possibilitĂ© est intĂ©ressante pour dĂ©terminer les caractĂ©ristiques de la vitesse au cours du temps et elle est particulièrement utile pour crĂ©er une animation. Nous allons l'utiliser pour dessiner le vecteur-vitesse Ă  quelques instants. - -On sauve successivement le tableau des positions et celui des vitesses. -\begin{verbatim} -\psequadiff[method=rk4,plotpoints=1000, - algebraic, - whichabs=0,whichord=1, - tabname=XiYi]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} -\begin{verbatim} -\psequadiff[method=rk4,plotpoints=1000, - algebraic, - whichabs=2,whichord=3, - tabname=vxvy]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} - -Pour ensuite dessiner la trajectoire et les vecteurs-vitesse. -\begin{verbatim} -%\listplot[unit=1]{vxvy aload pop} -% on dessine la vitesse un point sur 100 -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} -\multido{\i=0+100}{20}{% -\pstVerb{/vX vxvy \i\space get def - /vY vxvy \i\space 1 add get def - /xi XiYi \i\space get def - /yi XiYi \i\space 1 add get def}% -\rput(!xi yi){\psline[style=vecteurA]{->}(! vX 2 mul vY 2 mul)}} -\end{verbatim} -\begin{center} -\begin{pspicture}(-10,-10)(6,7) -\psset{unit=2}% -\pstVerb{/GM 1 def - /theta0 30 def - /r0 2 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 0.85 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricitĂ© - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def}% -\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} -\psequadiff[method=rk4, - plotpoints=1000, - algebraic, - whichabs=0, - whichord=1, - tabname=XiYi -% ,saveData,filename=XiYi.dat -]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% -\listplot{XiYi aload pop} -\psequadiff[method=rk4, - plotpoints=1000, - algebraic, - whichabs=2, - whichord=3, - tabname=vxvy -% ,saveData,filename=vxvy.dat -]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% -%\listplot[unit=1]{vxvy aload pop} -% on dessine la vitesse un point sur 100 -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} -\multido{\i=0+100}{20}{% -\pstVerb{/vX vxvy \i\space get def - /vY vxvy \i\space 1 add get def - /xi XiYi \i\space get def - /yi XiYi \i\space 1 add get def}% -\rput(!xi yi){\psline[style=vecteurA]{->}(! vX 2 mul vY 2 mul)}} -\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(3,3) -\end{pspicture} -\end{center} - - - -\section{Mouvement circulaire} - \begin{center} -\psset{unit=1} -\begin{pspicture}(-5,-5.5)(5,5.5) -\pstVerb{ - /GM 1 def - /theta0 60 def - /r0 5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 GM r0 div sqrt def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricitĂ© - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def -% vitesse Ă  l'apogĂ©e - /vA GM par div sqrt 1 exc sub mul def -% vitesse au pĂ©rigĂ©e - /vP GM par div sqrt 1 exc add mul def -% coordonnĂ©es de vA - /vAx vA theta0 90 add cos mul neg def - /vAy vA theta0 90 add sin mul neg def -% coordonnĂ©es de vP - /vPx vP theta0 90 add cos mul def - /vPy vP theta0 90 add sin mul def -}% -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.75} -\psdot[dotstyle=+](0,0) -\uput[d](0,0){O} -\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(5,5) -\rput(0,-2){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{000000}s}} -\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\pscircle*(!x0 y0){0.1} -%\pnode(!par 1 exc add div theta0 cos mul par 1 exc add div theta0 sin mul){P} % pĂ©rigĂ©e -%\pnode(!par 1 exc sub div theta0 cos mul neg par 1 exc sub div theta0 sin mul neg){A} % ApogĂ©e -\rput(!x0 y0){\psline[arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{0 0.5 1}},unit=4]{->}(!v0x v0y)\uput[ur](!v0x 2 mul v0y 2 mul){$\overrightarrow{v_0}$}} -\uput[ur](!x0 y0){$M_0$} -\psline[linestyle=dashed](0,0)(!x0 y0) -% position du satellite Ă  un instant quelconque -\pstVerb{/theta_i 170 def - /radius par 1 exc theta_i theta0 sub cos mul add div def - /xS radius theta_i cos mul def - /yS radius theta_i sin mul def - /ux theta_i cos 1 mul def - /uy theta_i sin 1 mul def - /xi xS ux sub def - /yi yS uy sub def - /xi2 xS ux 2 div sub def - /yi2 yS uy 2 div sub def}% -\pnode(!xi2 yi2){Mi2} -\pnode(!xi yi){Mi} -\pnode(!xS yS){S} -\pscircle*(S){0.1} -\psline[linestyle=dotted](S)(0,0) -\psline[style=vecteurC]{->}(S)(Mi) -\uput[l](S){S} -\uput[u](Mi2){$\overrightarrow{F}$} -\psarc{->}(0,0){1}{0}{!theta0} -\uput{1.1}[!theta0 2 div](0,0){$\theta_0$} -\psline[arrowinset=0.1,arrowsize=0.2]{<->}(5,0)(0,0)(0,5) -\psline[arrowinset=0.05,arrowsize=0.1]{<->}(5,0)(0,0)(0,5) -\uput[u](0,5){$y$}\uput[r](5,0){$x$} -\end{pspicture} - \end{center} -Il s'obtient très facilement Ă  partir de l'Ă©tude prĂ©cĂ©dente si on sait que dans ce cas : -\[ -v_0=\sqrt{\frac{\mathcal{G}M}{r_0}} -\] -\begin{verbatim} -\pstVerb{ - /GM 1 def - /theta0 60 def - /r0 5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 GM r0 div sqrt def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def }% -\end{verbatim} - -\end{document} +\documentclass{article} +\usepackage[a4paper,margin=2cm]{geometry} +\usepackage[latin1]{inputenc} +\usepackage{pstricks-add,pst-eqdf,pst-func} +\usepackage{array,amsmath} +\usepackage{listings} +\newpsstyle{vecteurA}{arrowinset=0.05,arrowsize=0.125,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 0.5 0}}} +%%%%%%%%%%%%%%%%%% +\title{Interaction gravitationnelle avec PSTricks} +\date{19 juin 2\,012} +\begin{document} +\maketitle +\def\eqsatellite{% +y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)}% +\section{Mise en orbite d'un satellite} +\begin{figure}[htbp] + \begin{center} +\psset{unit=2} +\begin{pspicture}(-6,-2)(2,5) +\pstVerb{ + /GM 1 def + /theta0 -35 def + /r0 1 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.3 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricitĂ© + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def +% vitesse Ă  l'apogĂ©e + /vA GM par div sqrt 1 exc sub mul def +% vitesse au pĂ©rigĂ©e + /vP GM par div sqrt 1 exc add mul def +% coordonnĂ©es de vA + /vAx vA theta0 90 add cos mul neg def + /vAy vA theta0 90 add sin mul neg def +% coordonnĂ©es de vP + /vPx vP theta0 90 add cos mul def + /vPy vP theta0 90 add sin mul def +% ApogĂ©e + /xA par 1 exc sub div theta0 cos mul neg def + /yA par 1 exc sub div theta0 sin mul neg def +% PĂ©rigĂ©e + /xP par 1 exc add div theta0 cos mul def + /yP par 1 exc add div theta0 sin mul def +% Centre + /xO xP xA add 2 div def + /yO yP yA add 2 div def +}% +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} +\psdot[dotstyle=+](0,0) +\uput[d](0,0){O} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-2)(2,5) +\rput(1,4){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{0000}s}} +\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def + radius t cos mul + radius t sin mul} +\pscircle*(!x0 y0){0.05} +\pnode(!xP yP){P} % pĂ©rigĂ©e +\pnode(!xA yA){A} % ApogĂ©e +\pnode(!xO yO){O} % centre +\rput(!x0 y0){\psline[style=vecteurB]{->}(!v0x v0y)\uput[ur](!v0x v0y){$\overrightarrow{v_0}$}} +\psline[linestyle=dashed](A)(P) +\pscircle*(A){0.05} +\uput[ul](A){$A$} +\uput[dr](P){$P$} +% position du satellite Ă  un instant quelconque +\pstVerb{/theta_i 170 def + /radius par 1 exc theta_i theta0 sub cos mul add div def + /xS radius theta_i cos mul def + /yS radius theta_i sin mul def + /ux theta_i cos 1 mul def + /uy theta_i sin 1 mul def + /xi xS ux sub def + /yi yS uy sub def + /xi2 xS ux 2 div sub def + /yi2 yS uy 2 div sub def}% +\pnode(!xi2 yi2){Mi2} +\pnode(!xi yi){Mi} +\pnode(!xS yS){S} +\pscircle*(S){0.05} +\psline[linestyle=dotted](S)(0,0) +\psline[style=vecteurA]{->}(S)(Mi) +\uput[l](S){S} +\uput[u](Mi2){$\overrightarrow{F}$} +\psarcn{->}(0,0){0.4}{0}{!theta0} +\uput{0.5}[!theta0 2 div](0,0){$\theta_0$} +\rput(A){\psline[style=vecteurB]{->}(!vAx vAy)} +%\rput(P){\psline[style=vecteurB]{->}(!vPx vPy)} +\psline[arrowinset=0.05,arrowsize=0.1]{<->}(2,0)(0,0)(0,5) +\uput[r](0,4.9){$y$}\uput[u](1.9,0){$x$} +\psdot(O)\uput[r](O){$\Omega$} +\rput{!90 theta0 add}(O){\psline[linestyle=dashed](!b_2 neg 0)(!b_2 0)} +\end{pspicture} +\caption{Mouvement d'un satellite} + \end{center} +\end{figure} +Soit (M) la masse de l'astre et (m) celle du satellite avec $m\ll M$. Le centre de masse du système \{M,m\} est confondu avec le centre de l'astre attracteur. La mise en orbite s'effectue Ă  partir du point $M_0(x_0,y_0)$ avec une vitesse $\overrightarrow{v_0}(v_{0_x},v_{0_x})$. $\theta_0$ est l'angle que fait $\overrightarrow{Ox}$ avec $\overrightarrow{OM_0}$. Le satellite (S), supposĂ© ponctuel, subit de la part de l'astre une force d'attraction gravitationnelle : +\[ +\overrightarrow{F}=-\mathcal{G}\frac{Mm}{r^2}\overrightarrow{u}\qquad \text{avec}\quad \overrightarrow{u}=\frac{\overrightarrow{r}}{r}\quad \text{et}\quad \overrightarrow{r}=\overrightarrow{OS} +\] +Tous les \textit{bons} livres de mĂ©canique\footnote{Comme celui, par exemple, de JosĂ©-Philippe PĂ©rez, aux Ă©ditions Masson.} Ă©tablissent les relations suivantes : +\[ +r=\frac{p}{1+\mathrm{e}\cos(\theta-\theta_0)} +\] +Paramètres et excentricitĂ© ont pour expressions respectives, avec les notations suivantes : $K=\mathcal{G}Mm$, $\mathcal{E}$ l'Ă©nergie du système et $L$ le moment cinĂ©tique. +\[ +\mathrm{e}=\sqrt{1+\frac{2\mathcal{E}L^2}{mK^2}}\qquad p=\frac{L^2}{mK} +\] +On choisit une vitesse initiale $\overrightarrow{v_0}$ perpendiculaire Ă  $\overrightarrow{OM_0}$, dans ces conditions le moment cinĂ©tique et l'Ă©nergie, qui restent constants, valent : +\[ +L=mr_0v_0\qquad \mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 +\] +En remplaçant $L$ et $\mathcal{E}$, on obtient pour l'excentricitĂ© et le paramètre les expressions suivantes : +\[ +\mathrm{e}=\sqrt{1+\frac{1}{\mathcal{G}^2M^2}\Big(\frac{1}{2}v_0^4r_0^2-\mathcal{G}Mr_0v_0^2\Big)} +\] +\[ +p=\frac{v_0^2r_0^2}{\mathcal{G}M} +\] +On se limite au cas du mouvement elliptique, avec, en consĂ©quence, la condition : +\[ +\mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 < 0 +\] +Le demi-grand axe $a$, le demi-petit axe $b$ sont : +\[ +a=\frac{p}{1-\mathrm{e}^2}\qquad b=\frac{p}{\sqrt{1-\mathrm{e}^2}} +\] +La pĂ©riode $T$ qui obĂ©it Ă  la troisième loi de KĂ©pler : +\[ +T^2=\frac{4\pi^2a^3}{\mathcal{G}M} +\] +La vitesse, en un point de l'ellipse, se calcule par : +\[ +v^2=\mathcal{G}M\Big( \frac{2}{r}-\frac{1}{a}\Big) +\] +Sachant que $r_p=\dfrac{p}{1+\mathrm{e}}$ et $r_A=\dfrac{p}{1-\mathrm{e}}$, on en dĂ©duit les vitesses au pĂ©rigĂ©e et Ă  l'apogĂ©e : +\[ +v_P=\sqrt{\frac{\mathcal{G}M}{p}}(1+\mathrm{e})\qquad v_A=\sqrt{\frac{\mathcal{G}M}{p}}(1-\mathrm{e}) +\] + +\section{L'Ă©tude avec PSTricks} +\subsection{La trajectoire} +\def\parametres{ + /GM 1 def % 4e14 def % GxM + /x0 6.5e6 def % position initiale + /y0 0 def + /vx0 0 def % vitesse initiale + /vy0 1e4 def +} +% x0 y0 x'0 y'0 +% y[0] y[1] y[2] y[3] +\[ +\left\{ +\begin{array}{rcl} +\ddot{x}&=&-\dfrac{GM}{r^3}x\\[1em] +\ddot{y}&=&-\dfrac{GM}{r^3}y\\ +\end{array} +\right. +\label{eq1} +\qquad r=\sqrt{x^2+y^2} +\] + +On peut dessiner la trajectoire du satellite et de ses caractĂ©ristiques de deux façons : +\begin{itemize} + \item par l'utilisation de \verb+\parametricplot+ ; + \item ou celle de \verb+\psplotDiffEqn+. +\end{itemize} +\verb+\parametricplot+ utilise l'expression exacte de l'Ă©quation de la trajectoire en coordonnĂ©es polaires : +\begin{verbatim} + \parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{% + /radius par 1 exc t theta0 sub cos mul add div def + radius t cos mul + radius t sin mul} +\end{verbatim} +L'excentricitĂ©, la pĂ©riode, demi-grand axe et demi-petit axe sont calculĂ©s par quelques lignes de code \textsf{postscript}. Il faut s'assurer que les conditions initiales choisies vĂ©rifient bien la condition d'une trajectoire elliptique, pour cela il faut que l'Ă©nergie initiale $\mathcal{E}_0<0$, sinon cela entraĂ®nera une erreur lors du passage Ă  l'interprĂ©teur \textsf{postscript}. +\begin{verbatim} +\pstVerb{ + /GM 1 def + /theta0 -45 def + /r0 0.5 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.92 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricitĂ© + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul + v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%demi-grand axe + /a_2 par 1 exc dup mul sub div def % demi-grand axe +%demi-petit axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe +% pĂ©riode + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def +}% +\end{verbatim} +\verb+\psplotDiffEqn+ utilise les Ă©quations diffĂ©rentielles du mouvement, en notation algĂ©brique : +\begin{verbatim} +% x0 y0 x'0 y'0 +% y[0] y[1] y[2] y[3] +\def\eqsatellite{% +y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)} + \psplotDiffEqn[unit=2,whichabs=0,whichord=1,% + linecolor=blue,linewidth=0.1,% + method=rk4,plotpoints=1000,% + algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% +\end{verbatim} +Ce qui permet, par ailleurs de vĂ©rifier la qualitĂ© du tracĂ© par la mĂ©thode numĂ©rique, en bleu, tandis que le tracĂ© Ă  partir de l'expression exacte est en trait fin en rouge. +% +\begin{center} +\begin{pspicture}(-12,-2)(4,10) +\pstVerb{ + /GM 1 def + /theta0 -45 def + /r0 0.5 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.92 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricitĂ© + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def +}% +\psframe*[linecolor=white](-3,-0.2)(0,0.2) +\pscircle[fillcolor=blue!50,fillstyle=solid](0,0){0.5} +\psgrid[unit=2,subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-1)(2,5) +\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} + \psplotDiffEqn[unit=2,whichabs=0,whichord=1,linecolor=blue,linewidth=0.1,method=rk4,plotpoints=1000,algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% +\parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def + radius t cos mul + radius t sin mul} +\psdot[unit=2,dotsize=0.12](!x0 y0) +\rput(!x0 2 mul y0 2 mul){\psline[style=vecteurC]{->}(!v0x v0y)} +\end{pspicture} +\end{center} +\subsection{La vitesse} +\verb+\psplotDiffEqn+ permet de voir comment varie la vitesse sur l'ellipse : +\begin{verbatim} +% x0 y0 x'0 y'0 +% y[0] y[1] y[2] y[3] + \psplotDiffEqn[xunit=0.2,yunit=5,% + plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, + linecolor=red,method=rk4,plotpoints=1000, + algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% +\end{verbatim} +\begin{center} +\begin{pspicture}(0,-1)(10,7) +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt] +\pstVerb{/GM 1 def + /theta0 -35 def + /r0 1 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.3 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def}% +\psplotDiffEqn[xunit=0.2,yunit=5, + plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, + linecolor=red,method=rk4,plotpoints=1000,algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% + \multido{\i=1+1,\I=5+5}{9}{\uput[u](\i,0){\I}} +\pnode(! 36.4 5 div 0){P} +\psdot(P)\uput[d](P){PĂ©rigĂ©e} +\psline[linestyle=dashed](P)(! 36.4 5 div 7) +\pnode(! 36.4 10 div 0){A} +\psdot(A)\uput[d](A){ApogĂ©e} +\psline[linestyle=dashed](A)(! 36.4 10 div 7) +\uput[l](0,6.5){$v$} +\psline[arrowinset=0.1,arrowsize=0.2]{<->}(10,0)(0,0)(0,7) +\uput[u](10,0){$t$(s)} +\uput[ul](0,0){0} +\end{pspicture} +\end{center} +On peut obtenir les caractĂ©ristiques de la vitesse en un point quelconque, car le moment cinĂ©tique $L=mr\dot{\theta}$ Ă©tant constant, pour chaque valeur de $\theta$, on en dĂ©duit $r$ puis $\dot{\theta}$. En coordonnĂ©es polaires, la vitesse s'exprime par : +\[ +\overrightarrow{v}=\dot{r}\overrightarrow{u_r}+r\dot{\theta}\overrightarrow{u_{\theta}} +\] +$\dot{\theta}$ et $\dot{r}$ s'obtiennent par les relations suivantes : +\[ +\dot{\theta}=\frac{r_0v_0}{r^2} +\] +\[ +\dot{r}=-\frac{p(-\dot{\theta}\sin(\theta-\theta_0)}{(1+\mathrm{e}\cos(\theta-\theta_0))^2}=\frac{r_0v_0}{p}\sin(\theta-\theta_0) +\] +La chaĂ®ne de calculs est la suivante : $\theta\Longrightarrow r\Longrightarrow \dot{\theta}\Longrightarrow \dot{r}\Longrightarrow \overrightarrow{v}$. + +Le package `\textsf{pst-eqdf}' comprend la commande \verb+\psequadiff+ qui est une version simplifiĂ©e de \verb+\psplotDiffEqn+, dont elle ne reprend que la mĂ©thode Runge-Kutta~4. Elle permet de sauvegarder sous forme de tableaux et/ou de fichiers toutes les variables et les dĂ©rivĂ©es de la fonction Ă©tudiĂ©e, cette possibilitĂ© est intĂ©ressante pour dĂ©terminer les caractĂ©ristiques de la vitesse au cours du temps et elle est particulièrement utile pour crĂ©er une animation. Nous allons l'utiliser pour dessiner le vecteur-vitesse Ă  quelques instants. + +On sauve successivement le tableau des positions et celui des vitesses. +\begin{verbatim} +\psequadiff[method=rk4,plotpoints=1000, + algebraic, + whichabs=0,whichord=1, + tabname=XiYi]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +\end{verbatim} +\begin{verbatim} +\psequadiff[method=rk4,plotpoints=1000, + algebraic, + whichabs=2,whichord=3, + tabname=vxvy]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +\end{verbatim} + +Pour ensuite dessiner la trajectoire et les vecteurs-vitesse. +\begin{verbatim} +%\listplot[unit=1]{vxvy aload pop} +% on dessine la vitesse un point sur 100 +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} +\multido{\i=0+100}{20}{% +\pstVerb{/vX vxvy \i\space get def + /vY vxvy \i\space 1 add get def + /xi XiYi \i\space get def + /yi XiYi \i\space 1 add get def}% +\rput(!xi yi){\psline[style=vecteurA]{->}(! vX 2 mul vY 2 mul)}} +\end{verbatim} +\begin{center} +\begin{pspicture}(-10,-10)(6,7) +\psset{unit=2}% +\pstVerb{/GM 1 def + /theta0 30 def + /r0 2 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 0.85 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricitĂ© + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def}% +\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=0, + whichord=1, + tabname=XiYi +% ,saveData,filename=XiYi.dat +]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot{XiYi aload pop} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=2, + whichord=3, + tabname=vxvy +% ,saveData,filename=vxvy.dat +]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +%\listplot[unit=1]{vxvy aload pop} +% on dessine la vitesse un point sur 100 +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} +\multido{\i=0+100}{20}{% +\pstVerb{/vX vxvy \i\space get def + /vY vxvy \i\space 1 add get def + /xi XiYi \i\space get def + /yi XiYi \i\space 1 add get def}% +\rput(!xi yi){\psline[style=vecteurA]{->}(! vX 2 mul vY 2 mul)}} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(3,3) +\end{pspicture} +\end{center} +% +\section{Mouvement circulaire} + \begin{center} +\psset{unit=1} +\begin{pspicture}(-5,-5.5)(5,5.5) +\pstVerb{ + /GM 1 def + /theta0 60 def + /r0 5 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 GM r0 div sqrt def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricitĂ© + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def +% vitesse Ă  l'apogĂ©e + /vA GM par div sqrt 1 exc sub mul def +% vitesse au pĂ©rigĂ©e + /vP GM par div sqrt 1 exc add mul def +% coordonnĂ©es de vA + /vAx vA theta0 90 add cos mul neg def + /vAy vA theta0 90 add sin mul neg def +% coordonnĂ©es de vP + /vPx vP theta0 90 add cos mul def + /vPy vP theta0 90 add sin mul def +}% +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.75} +\psdot[dotstyle=+](0,0) +\uput[d](0,0){O} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(5,5) +\rput(0,-2){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{000000}s}} +\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def + radius t cos mul + radius t sin mul} +\pscircle*(!x0 y0){0.1} +%\pnode(!par 1 exc add div theta0 cos mul par 1 exc add div theta0 sin mul){P} % pĂ©rigĂ©e +%\pnode(!par 1 exc sub div theta0 cos mul neg par 1 exc sub div theta0 sin mul neg){A} % ApogĂ©e +\rput(!x0 y0){\psline[arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{0 0.5 1}},unit=4]{->}(!v0x v0y)\uput[ur](!v0x 2 mul v0y 2 mul){$\overrightarrow{v_0}$}} +\uput[ur](!x0 y0){$M_0$} +\psline[linestyle=dashed](0,0)(!x0 y0) +% position du satellite Ă  un instant quelconque +\pstVerb{/theta_i 170 def + /radius par 1 exc theta_i theta0 sub cos mul add div def + /xS radius theta_i cos mul def + /yS radius theta_i sin mul def + /ux theta_i cos 1 mul def + /uy theta_i sin 1 mul def + /xi xS ux sub def + /yi yS uy sub def + /xi2 xS ux 2 div sub def + /yi2 yS uy 2 div sub def}% +\pnode(!xi2 yi2){Mi2} +\pnode(!xi yi){Mi} +\pnode(!xS yS){S} +\pscircle*(S){0.1} +\psline[linestyle=dotted](S)(0,0) +\psline[style=vecteurC]{->}(S)(Mi) +\uput[l](S){S} +\uput[u](Mi2){$\overrightarrow{F}$} +\psarc{->}(0,0){1}{0}{!theta0} +\uput{1.1}[!theta0 2 div](0,0){$\theta_0$} +\psline[arrowinset=0.1,arrowsize=0.2]{<->}(5,0)(0,0)(0,5) +\psline[arrowinset=0.05,arrowsize=0.1]{<->}(5,0)(0,0)(0,5) +\uput[u](0,5){$y$}\uput[r](5,0){$x$} +\end{pspicture} + \end{center} +Il s'obtient très facilement Ă  partir de l'Ă©tude prĂ©cĂ©dente si on sait que dans ce cas : +\[ +v_0=\sqrt{\frac{\mathcal{G}M}{r_0}} +\] +\begin{verbatim} +\pstVerb{ + /GM 1 def + /theta0 60 def + /r0 5 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 GM r0 div sqrt def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def }% +\end{verbatim} +\section{L'hodographe du mouvement du satellite} +On rappelle qu'en coordonnĂ©es polaires le vecteur-vitesse s'Ă©crit : +\[ +\overrightarrow{v}=\dot{r}\overrightarrow{u_r}+r\dot{\theta}\overrightarrow{u_{\theta}} +\] +\[ +\dot{\theta}=\frac{r_0v_0}{r^2} +\] +\[ +\dot{r}=-\frac{p(-\mathrm{e}\dot{\theta}\sin(\theta-\theta_0)}{(1+\mathrm{e}\cos(\theta-\theta_0))^2}=\frac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0) +\] +On peut exprimer $v$ uniquement en fonction de $\theta$ : +\[ +\overrightarrow{v}=\frac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0)\overrightarrow{u_r}+\frac{r_0v_0}{p}\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\overrightarrow{u_\theta} +\] +Ses composantes dans la base $(\overrightarrow{u_r},u_\theta)$ sont : +\[ +\left\{ +\begin{array}{rcl} +v_r&=&\dfrac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0)\\[1em] +v_\theta&=&\dfrac{r_0v_0}{p}\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big) +\end{array} +\right. +\label{eqv1} +\] +On repasse aux coordonnĂ©es cartĂ©siennes par une rotation d'angle $(-\theta)$. +\[ +\left\{ +\begin{array}{rcl} +\dot{x}&=&v_r\cos\theta-v_\theta\sin\theta\\[1em] +\dot{y}&=&v_r\sin\theta+v_\theta\cos\theta +\end{array} +\right. +\label{eqv2} +\] +\[ +\left\{ +\begin{array}{rcl} +\dfrac{p}{r_0v_0}\dot{x}&=&\mathrm{e}\sin(\theta-\theta_0)\cos\theta-\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\sin\theta\\[1em] +\dfrac{p}{r_0v_0}\dot{y}&=&\mathrm{e}\sin(\theta-\theta_0)\sin\theta+\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\cos\theta +\end{array} +\right. +\label{eqv3} +\] +En utilisant les relations trigonomĂ©triques de soustraction : +\[ +\sin\alpha\cos\beta-\cos\alpha\sin\beta=\sin(\alpha-\beta)\qquad \cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\alpha-\beta) +\] +On obtient : +\[ +\left\{ +\begin{array}{rcl} +\dfrac{p}{r_0v_0}\dot{x}&=&-\mathrm{e}\sin\theta_0-\sin\theta\\[1em] +\dfrac{p}{r_0v_0}\dot{y}&=&\hphantom{-}\mathrm{e}\cos\theta_0+\cos\theta +\end{array} +\right. +\label{eqv4} +\] +\[ +\left\{ +\begin{array}{rcl} +\dot{x}+\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0&=&-\dfrac{r_0v_0}{p}\sin\theta\\[1em] +\dot{y}-\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0&=&\dfrac{r_0v_0}{p}\cos\theta +\end{array} +\right. +\label{eqv5} +\] +L'Ă©quation de l'hodographe : +\[ +\left(\dot{x}+\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0\right)^2+\left(\dot{y}-\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0\right)^2=\left(\dfrac{r_0v_0}{p}\right)^2 +\] +est celle d'un cercle centrĂ© en $\Big(\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0,\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0\Big)$, de rayon $R=\dfrac{r_0v_0}{p}$. Ce qu'on vĂ©rifie sur le graphe suivant : en rouge l'hodographe a Ă©tĂ© obtenu Ă  partir de l'Ă©quation diffĂ©rentielle et en noir avec un trait plus fin par l'expression exacte de l'Ă©quation du cercle. +\begin{center} +\begin{pspicture}(-12,-2)(4,8) +\psset{unit=2}% +\uput[r](0,4){$y$}\uput[u](2,0){$x$} +\pstVerb{/GM 1 def + /theta0 -35 def + /r0 1 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.3 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricitĂ© + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def +% rayon de l'hodographe + /rH r0 v0 mul par div def +% les coordonnĂ©es du centre + /xCH rH exc mul theta0 sin mul neg def + /yCH rH exc mul theta0 cos mul def}% +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=0, + whichord=1, + tabname=XiYi, +% saveData,filename=XiYi.dat +]{0}{36.5}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot{XiYi aload pop} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=2, + whichord=3, + tabname=vxvy, +% saveData,filename=vxvy.dat +]{0}{36.5}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot[unit=1,linecolor=red,linewidth=0.075]{vxvy aload pop} +% on dessine la vitesse un point sur 50 +\multido{\i=0+50}{40}{% +\pstVerb{ /vX vxvy \i\space get def + /vY vxvy \i\space 1 add get def + /xi XiYi \i\space get def + /yi XiYi \i\space 1 add get def}% +\rput(!xi yi){\psline[unit=1,linecolor=blue]{->}(!vX vY)}} +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.2} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-2)(2,4) +\psline[arrowinset=0.05,arrowsize=0.1]{<->}(2,0)(0,0)(0,4) +\pscircle(!xCH yCH){!rH} +\psdot[dotstyle=+](!xCH yCH) +\end{pspicture} +\end{center} +\end{document} diff --git a/gravitation/gravitation_02.pdf b/gravitation/gravitation_02.pdf index 8b08656d0740bd19cb8809e502eeed42fc78b449..ccab03209bb0ac43c9e1b74371d65dc591a592b3 100644 GIT binary patch literal 59268 zcma%?Q*b2=u&rZb$4+)^+cqb5c5Isy+qP{xnQ&s;w#}J2|EW4}_vJoyS3h@m^|xv@ zrHX_UBP$aJ0_E6sLNNj-kOk;yVuK(cz^q{IVCiNB_znKxMj`UJ%?y9Pp9rB=4m?nQNK{#l zx>?WR^ROGEHvcQQQ!@2zC9QjtzawbasMe&hxuE~~>8?E2%cMOlIe7o;@4-XN%AW@N zhwBO~`oB>Bg!TFiyQB^a?)m)#zKgw_;Aad5?a4beM^E19i31LywfYx|8&$aw0=$|& z%l*_>nisNl)C|=*9eZ0R)8^=R2ReSI3Tw9A-)l}h%04z5>&h4i#Uwc1`<`$fooM)d z*}T(vd}8vuBOmo&sq-1`wlcT1ogiLMAY5F?aEnhB(3280)CR{F5}oa)C?ED@U@7Fu z-b{8h%AMlhBI7mcHKZgz|3TNgl!8PH5L=`v$t}F{-l|u17MJRdH%JAX_)#2 zR>H#qqiem7=n{SP8>=mxyS&p60tua`HD&cubtp767^~|M)^7@W8s&5r-?0so1vqQi zEQK4i=M4E2@teV8@W}MKGbfTJa6u{)$j15B%*ujg9awv#{#SFQ`$0?xZ5&Dsjj3^m zW0IC+9BSz;Ru9BKE$4MdD6l%v(C7jTX2mCNXec;TCu&_dvCWP0RM_-jO$H~QimC5X zHRI6`ig)DSaAOd7CHH(53!ua`9)EO{HQ=QY*WpGTLMl}mSU 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+\usepackage[a4paper,margin=2cm]{geometry} +\usepackage[latin1]{inputenc} +\usepackage{pst-eqdf,pst-func} +\usepackage{array,amsmath} +\newpsstyle{vecteurA}{arrowinset=0.05,arrowsize=0.125,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 0.5 0}}} +%%%%%%%%%%%%%%%%%% +\title{L'hodographe du mouvement d'un satellite} +\date{23 juin 2\,012} +\begin{document} +\maketitle +\def\eqsatellite{% +y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)}% +On rappelle qu'en coordonnĂ©es polaires le vecteur-vitesse s'Ă©crit : +\[ +\overrightarrow{v}=\dot{r}\overrightarrow{u_r}+r\dot{\theta}\overrightarrow{u_{\theta}} +\] +\[ +\dot{\theta}=\frac{r_0v_0}{r^2} +\] +\[ +\dot{r}=-\frac{p(-\mathrm{e}\dot{\theta}\sin(\theta-\theta_0)}{(1+\mathrm{e}\cos(\theta-\theta_0))^2}=\frac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0) +\] +On peut exprimer $v$ uniquement en fonction de $\theta$ : +\[ +\overrightarrow{v}=\frac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0)\overrightarrow{u_r}+\frac{r_0v_0}{p}\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\overrightarrow{u_\theta} +\] +Ses composantes dans la base $(\overrightarrow{u_r},u_\theta)$ sont : +\[ +\left\{ +\begin{array}{rcl} +v_r&=&\dfrac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0)\\[1em] +v_\theta&=&\dfrac{r_0v_0}{p}\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big) +\end{array} +\right. +\label{eqv1} +\] +On repasse aux coordonnĂ©es cartĂ©siennes par une rotation d'angle $(-\theta)$. +\[ +\left\{ +\begin{array}{rcl} +\dot{x}&=&v_r\cos\theta-v_\theta\sin\theta\\[1em] +\dot{y}&=&v_r\sin\theta+v_\theta\cos\theta +\end{array} +\right. +\label{eqv2} +\] +\[ +\left\{ +\begin{array}{rcl} +\dfrac{p}{r_0v_0}\dot{x}&=&\mathrm{e}\sin(\theta-\theta_0)\cos\theta-\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\sin\theta\\[1em] +\dfrac{p}{r_0v_0}\dot{y}&=&\mathrm{e}\sin(\theta-\theta_0)\sin\theta+\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\cos\theta +\end{array} +\right. +\label{eqv3} +\] +En utilisant les relations trigonomĂ©triques de soustraction : +\[ +\sin\alpha\cos\beta-\cos\alpha\sin\beta=\sin(\alpha-\beta)\qquad \cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\alpha-\beta) +\] +On obtient : +\[ +\left\{ +\begin{array}{rcl} +\dfrac{p}{r_0v_0}\dot{x}&=&-\mathrm{e}\sin\theta_0-\sin\theta\\[1em] +\dfrac{p}{r_0v_0}\dot{y}&=&\hphantom{-}\mathrm{e}\cos\theta_0+\cos\theta +\end{array} +\right. +\label{eqv4} +\] +\[ +\left\{ +\begin{array}{rcl} +\dot{x}+\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0&=&-\dfrac{r_0v_0}{p}\sin\theta\\[1em] +\dot{y}-\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0&=&\dfrac{r_0v_0}{p}\cos\theta +\end{array} +\right. +\label{eqv5} +\] +L'Ă©quation de l'hodographe : +\[ +\left(\dot{x}+\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0\right)^2+\left(\dot{y}-\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0\right)^2=\left(\dfrac{r_0v_0}{p}\right)^2 +\] +est celle d'un cercle centrĂ© en $\Big(\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0,\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0\Big)$, de rayon $R=\dfrac{r_0v_0}{p}$. Ce qu'on vĂ©rifie sur le graphe suivant : en rouge l'hodographe a Ă©tĂ© obtenu Ă  partir de l'Ă©quation diffĂ©rentielle et en noir avec un trait plus fin par l'expression exacte de l'Ă©quation du cercle. +\begin{center} +\begin{pspicture}(-10,-2)(4,8) +\psset{unit=2}% +\uput[r](0,4){$y$}\uput[u](2,0){$x$} +\pstVerb{/GM 1 def + /theta0 -35 def + /r0 1 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.3 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricitĂ© + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def +% rayon de l'hodographe + /rH r0 v0 mul par div def +% les coordonnĂ©es du centre + /xCH rH exc mul theta0 sin mul neg def + /yCH rH exc mul theta0 cos mul def}% +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=0, + whichord=1, + tabname=XiYi, + saveData,filename=XiYi.dat]{0}{36.5}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot{XiYi aload pop} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=2, + whichord=3, + tabname=vxvy, + saveData,filename=vxvy.dat]{0}{36.5}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot[unit=1,linecolor=red,linewidth=0.075]{vxvy aload pop} +% on dessine la vitesse un point sur 50 +\multido{\i=0+50}{40}{% +\pstVerb{ /vX vxvy \i\space get def + /vY vxvy \i\space 1 add get def + /xi XiYi \i\space get def + /yi XiYi \i\space 1 add get def}% +\rput(!xi yi){\psline[unit=1,linecolor=blue]{->}(!vX vY)}} +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.2} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-2)(2,4) +\psline[arrowinset=0.05,arrowsize=0.1]{<->}(2,0)(0,0)(0,4) +\pscircle(!xCH yCH){!rH} +\psdot[dotstyle=+](!xCH yCH) +\end{pspicture} +\end{center} +\newpage +\begin{center} +\begin{pspicture}(-10,-10)(6,6) +\psset{unit=2}% +\uput[r](0,3){$y$}\uput[u](3,0){$x$} +\pstVerb{/GM 1 def + /theta0 30 def + /r0 2 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 0.85 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricitĂ© + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def}% +\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=0, + whichord=1, + tabname=XiYi, +% saveData,filename=XiYi.dat +]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot{XiYi aload pop} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=2, + whichord=3, + tabname=vxvy, +% saveData,filename=vxvy.dat +]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot[unit=2,linecolor=red]{vxvy aload pop} +% on dessine la vitesse un point sur 100 +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} +\multido{\i=0+100}{20}{% +\pstVerb{/vX vxvy \i\space get def + /vY vxvy \i\space 1 add get def + /xi XiYi \i\space get def + /yi XiYi \i\space 1 add get def}% +\rput(!xi yi){\psline[style=vecteurA]{->}(! vX 2 mul vY 2 mul)}} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(3,3) +\psline[arrowinset=0.05,arrowsize=0.1]{<->}(3,0)(0,0)(0,3) +\end{pspicture} +\end{center} +\end{document} diff --git a/gravitation/h1.png b/gravitation/h1.png new file mode 100644 index 0000000000000000000000000000000000000000..7d3b31b2c327a1dc8ee30c47173620485296e5cb GIT binary patch literal 24177 zcmY(rWmFtp(=|MUThIgzHdunYTkrsZ1b26LcPF^J2Mg}*?jg9lyF0v{>;9e}-&qS- 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Luque - Paquet pst-eqdf - PSTricks  Niveau supĂ©rieur - + + M. Luque + Paquet pst-eqdf + PSTricks  Niveau supĂ©rieur + \ No newline at end of file -- 2.20.1