From 2acea1b6946ba0bc3da40518d863899c5fefeb5a Mon Sep 17 00:00:00 2001 From: Manuel Date: Tue, 3 Jul 2012 14:48:59 +0200 Subject: [PATCH] =?utf8?q?Gravitation=20:=20le=20probl=C3=A8me=20des=202?= =?utf8?q?=20corps.=20Partie=201=20:=20Documentation=20th=C3=A9orique.?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- gravitation/2corps8.png | Bin 0 -> 20963 bytes gravitation/LISTE.txt | 6 + gravitation/pb_2corps_doc.pdf | Bin 0 -> 192685 bytes gravitation/pb_2corps_doc.tex | 1295 +++++++++++++++++++++++++++++++++ gravitation/pst-tools.sty | 8 + gravitation/pst-tools.tex | 111 +++ 6 files changed, 1420 insertions(+) create mode 100644 gravitation/2corps8.png create mode 100644 gravitation/pb_2corps_doc.pdf create mode 100644 gravitation/pb_2corps_doc.tex create mode 100644 gravitation/pst-tools.sty create mode 100644 gravitation/pst-tools.tex diff --git a/gravitation/2corps8.png b/gravitation/2corps8.png new file mode 100644 index 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+\newpsstyle{vecteurC}{arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{1 0 0}}} +%%%%%%%%%%%%%%%%%% +\title{Gravitation : le problème des deux corps avec PSTricks\\ partie 1} +\date{3 juillet 2\,012} +\begin{document} +\maketitle +\section{Présentation} + Cette première partie aborde uniquement le problème théorique et l'établissement des relations et formules indispensables à la réalisation des figures avec `PSTricks' et à l'animation avec le package `\textsf{animate}'. Elle pourra donc sembler superflue à tous ceux qui connaissent bien ce problème classique. De très bons livres traitent le \textit{problème des deux corps} de façon très claire comme celui de José-Philippe Pérez dans `\textit{Mécanique}' aux éditions Masson ou de façon très complète(et très claire aussi) comme celui publié par le \textsc{cnes} aux éditions \textsc{cepadues} : `\textit{Le mouvement du satellite}'. + + La deuxième partie détaillera la procédure suivie pour réaliser les schémas et les animations. Toutefois, on peut, dès à présent, avoir accès au code des figures dans le fichier source de ce document. +\section{Étude théorique} +\newcommand\GravAlg{% + y[2]|y[3]|% + M1*(y[4]-y[0])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|% + M1*(y[5]-y[1])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|% + y[6]|y[7]|% + M2*(y[0]-y[4])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5|% + M2*(y[1]-y[5])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5} +%% 0 1 2 3 4 5 6 7 +%% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +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 : +\[ +\overrightarrow{F}_{1/2}=-\mathcal{G}\frac{m_1m_2}{r^3}\overrightarrow{r}\quad \overrightarrow{F}_{2/1}=+\mathcal{G}\frac{m_1m_2}{r^3}\overrightarrow{r} +\] +Bien sûr : $\overrightarrow{F}_{1/2}+\overrightarrow{F}_{2/1}=\overrightarrow{0}$. Par conséquent, le centre de masse $C$ du système $\{M_1,M_2\}$ est, dans le repère $\mathcal{R}$, soit immobile, soit en mouvement de translation rectiligne uniforme suivant les conditions initiales. + +Dans $\mathcal{R}$ la position de $C$ est déterminée par la relation : +\[ +\overrightarrow{OC}=\frac{m_1\overrightarrow{OM_1}+m_2\overrightarrow{OM_2}}{m_1+m_2} +\] +\begin{equation} +\overrightarrow{v_{C/\mathcal{R}}}=\frac{m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}}{m_1+m_2}=\overrightarrow{v_0} +\label{vc} +\end{equation} +$\overrightarrow{v_0}$ étant déterminé par les valeurs initiales de $\overrightarrow{v_1}$ et $\overrightarrow{v_2}$. +L'étude peut donc se poursuivre dans le repère lié au centre de masse $\mathcal{R}^*$, lui-même galiléen. + +On pose : $\overrightarrow{r_1}=\overrightarrow{CM_1}$ et $\overrightarrow{r_2}=\overrightarrow{CM_2}$. On a toujours : $\overrightarrow{r}=\overrightarrow{M_1M_2}=\overrightarrow{r_2}-\overrightarrow{r_1}$. +\begin{center} +\begin{pspicture}(-3.5,-1)(5.5,8.25) +%\psframe(-3.5,-1)(5.5,9) +\pstVerb{/x01 -2 def /y01 1 def + /x02 4 def /y02 5 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[linecolor=gray]{<->}(5,0)(0,0)(0,6) +\rput(C){\psline[style=vecteurA]{<->}(5.5,0)(0,0)(0,6)% + \uput[l](0,5.8){$\mathcal{R^*}$} + \psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt](-3,-3)(6,6)% + \uput[r](0,5.8){$y^*$} + \uput[u](5.5,0){$x^*$}} +\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[ul](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} + +Dans le repère $\mathcal{R}^*$, la quantité de mouvement du système $\{M_1,M_2\}$ est nulle : $\overrightarrow{p}^*=(m_1+m_2)\overrightarrow{v}_C^*=\overrightarrow{0}$. +Pour chacun des deux corps, la quantité de mouvement dans $\mathcal{R}^*$ est : +\[ +\overrightarrow{p}_1^*=m_1\overrightarrow{v}_1^*=m_1(\overrightarrow{v}_{1/\mathcal{R}}-\overrightarrow{v}_{C/\mathcal{R}})=-\overrightarrow{p}^*_2 +\] +D'après (\ref{vc}) qui donne $\overrightarrow{v}_{C/\mathcal{R}}$, on obtient : +\[ +\overrightarrow{p}_1^*=m_1\left(\overrightarrow{v}_{1/\mathcal{R}}-\frac{m_1\overrightarrow{v}_{1/\mathcal{R}}+m_2\overrightarrow{v}_{2/\mathcal{R}}}{m_1+m_2}\right)=% + \frac{m_1m_2}{m_1+m_2}(\overrightarrow{v}_{1/\mathcal{R}}-\overrightarrow{v}_{2/\mathcal{R}})=-\overrightarrow{p}^*_2 +\] +Puisque, on a défini $\overrightarrow{r}$ par $\overrightarrow{r_2}-\overrightarrow{r_1}$, en posant $\mu\dfrac{m_1m_2}{m_1+m_2}$ : +\[ +\overrightarrow{p}^*_1=-\mu\dfrac{\mathrm{d}\overrightarrow{r}}{\mathrm{d}t}\qquad \overrightarrow{p}^*_2=\mu\dfrac{\mathrm{d}\overrightarrow{r}}{\mathrm{d}t} +\] +La quantité de mouvement de $M_2$ est identique à celle d'une particule fictive de masse $\mu$, appelée \textit{masse réduite}, et de vitesse $\overrightarrow{v}_{2/\mathcal{R}}-\overrightarrow{v}_{1/\mathcal{R}}$ : vitesse relative de $M_2$ par rapport à $M_1$, idem pour $M_1$. + +On considère cette particule fictive de masse $\mu$ située au point $M$ tel que : $\overrightarrow{CM}=\overrightarrow{r}$. + +La relation fondamentale de la dynamique appliquée à chacun des deux corps donne : +\[ +\left\{ +\begin{array}[m]{l} + \dfrac{\mathrm{d}\overrightarrow{p}^*_1}{\mathrm{d}t}=\overrightarrow{F}_{2/1}\\[1em] +\dfrac{\mathrm{d}\overrightarrow{p}^*_2}{\mathrm{d}t}=\overrightarrow{F}_{1/2} +\end{array} +\right. +\] +En remplaçant $\overrightarrow{p}^*_1$ (ou $\overrightarrow{p}^*_2$) par son expression en fonction de $\mu$, on a : +\begin{equation} + \mu\dfrac{\mathrm{d}^2\overrightarrow{r}}{\mathrm{d}t^2}=\overrightarrow{F}_{1/2}=-\mathcal{G}\frac{m_1m_2}{r^3}\overrightarrow{r} +\label{RFD} +\end{equation} +Nous désignons cette force par $\overrightarrow{F}$. Remarquons qu'elle peut s'écrire, en notant la masse totale $(m_1+m_2)$ du système $\{M_1,M_2\}$ par $m_t$~: +\begin{equation} +\overrightarrow{F}=-\mathcal{G}\frac{\mu m_t}{r^3}\overrightarrow{r} +\label{RF} +\end{equation} +En conclusion, cette particule fictive $M$, de masse $\mu$ positionnée en $\overrightarrow{CM}=\overrightarrow{r}$ est attirée par un corps fictif de masse égale à la masse totale du système $(m_1+m_2)$, placé en $C$. Sa position et sa vitesse initiales sont déterminées par les positions et vitesses initiales de $M_1$ et $M_2$. +\[ +\overrightarrow{r}_0=\overrightarrow{r}_{2_0}-\overrightarrow{r}_{1_0} \quad \mbox{et} \quad \overrightarrow{v}^*_0=\overrightarrow{v}_{0/R}=\overrightarrow{v}_{2_0}^*-\overrightarrow{v}_{1_0}^*= +\overrightarrow{v}_{{0_2}/R}-\overrightarrow{v}_{{0_1}/R} +\] +Si nous connaissons le mouvement de $M$ nous pourrons en déduire les mouvements respectifs de $M_1$ et $M_2$, puisque d'après la définition du centre de masse : +\begin{equation} +\overrightarrow{r_1}=-\frac{m_2}{m_1+m_2}\overrightarrow{r}\quad\mbox{et}\quad \overrightarrow{r_2}=\frac{m_1}{m_1+m_2}\overrightarrow{r} +\label{r1r2} +\end{equation} +Les trajectoires de $M_1$ et $M_2$ se déduiront de celle de $M$ par une homothétie de centre C. +\begin{center} +\begin{pspicture}(-3.5,-1)(6.5,8.25) +%\psframe(-3.5,-1)(5.5,9) +\pstVerb{/x01 -2 def /y01 1 def + /x02 4 def /y02 5 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} +\pscircle[hatchcolor=gray,fillstyle=vlines,hatchwidth=0.02,hatchsep=0.05](C){0.25} +\psdot(C) +\psline[linestyle=dotted](M1)(M2) +\psline[linestyle=dashed](M2)(O)(M1) +\psline[linecolor=gray]{<->}(5,0)(0,0)(0,6) +\rput(C){\psline[style=vecteurA]{<->}(5.5,0)(0,0)(0,6)% + \uput[l](0,5.8){$\mathcal{R^*}$} + \psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt](-3,-3)(7,6)% + \uput[r](0,5.8){$y^*$} + \uput[u](5.5,0){$x^*$} + \pnode(!xr0 yr0){M} + \pscircle*(M){0.05} + \pcline[offset=0.3,linewidth=0.01,arrowsize=0.1]{|->}(0,0)(M) + \ncput*[nrot=:U]{$\overrightarrow{r}$} + \psline[linestyle=dotted](M)\uput[r](M){$M(\mu)$} + \rput(M){\psline[style=vecteurC]{->}(!xr0 5 div neg yr0 5 div neg)\uput[dr](!xr0 10 div neg yr0 10 div neg){$\overrightarrow{F}_{1/2}$}}} +\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}[dr](M2){$M_2$} +\uput{0.3}[d](C){$C$} +%\pcline[offset=10pt,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} +Pour étudier, dans $\mathcal{R}^*$, le mouvement de ce point matériel fictif $M$ de masse $(\mu)$ soumis à la force centrale $\overrightarrow{F}$, il est avantageux de passer en coordonnées polaires. +\begin{center} +\begin{pspicture}(-3.5,-1)(6.5,8.25) +%\psframe(-3.5,-1)(5.5,9) +\pstVerb{/x01 -2 def /y01 1 def + /x02 4 def /y02 5 def + /xr0 x02 x01 sub def + /yr0 y02 y01 sub def + /theta_0 yr0 xr0 atan 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} +\pscircle[hatchcolor=gray,fillstyle=vlines,hatchwidth=0.02,hatchsep=0.05](C){0.25} +\psdot(C) +\psline[linestyle=dotted](M1)(M2) +\psline[linestyle=dashed](M2)(O)(M1) +\psline[linecolor=gray]{<->}(5,0)(0,0)(0,6) +\rput(C){\psline[style=vecteurA]{<->}(5.5,0)(0,0)(0,6)% + \uput[l](0,5.8){$\mathcal{R^*}$} + \psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt](-3,-3)(7,6)% + \uput[r](0,5.8){$y^*$} + \uput[u](5.5,0){$x^*$} + \pnode(!xr0 yr0){M} + \pscircle*(M){0.05} + \pcline[offset=0.3,linewidth=0.01,arrowsize=0.1]{|->}(0,0)(M) + \ncput*[nrot=:U]{$\overrightarrow{r}$} + \psline[linestyle=dotted](M)\uput[r](M){$M(\mu)$} + \rput(M){\psline[style=vecteurC]{->}(!xr0 5 div neg yr0 5 div neg)\uput[dr](!xr0 10 div neg yr0 10 div neg){$\overrightarrow{F}$}}} +\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}[dr](M2){$M_2$} +\uput{0.3}[d](C){$C$} +\rput(C){\psline[arrowinset=0.1,arrowsize=0.15]{->}(!theta_0 90 add cos theta_0 90 add sin) + \psline[arrowinset=0.1,arrowsize=0.15]{->}(!theta_0 cos theta_0 sin) + \uput{.1}[u](!theta_0 90 add cos theta_0 90 add sin){$\overrightarrow{u}_\theta$} + \uput[d](!theta_0 cos theta_0 sin){$\overrightarrow{u}_r$} + \psarc{->}(0,0){1.5}{0}{!theta_0} + \uput{1.6}[!theta_0 2 div](0,0){$\theta$}} +\end{pspicture} +\end{center} +Pour la suite, retenons que dans $\mathcal{R}*$ : +\[ +\overrightarrow{u}_r= +\left| +\begin{array}[m]{l} +\cos\theta\\ +\sin\theta +\end{array} +\right. +\quad +\overrightarrow{u}_\theta= +\left| +\begin{array}[m]{l} +-\sin\theta\\ +\cos\theta +\end{array} +\right. +\] +\[ +\frac{\mathrm{d}\overrightarrow{u}_r}{\mathrm{d}t}%= +%\left| +%\begin{array}[m]{l} +%-\dot{\theta}\sin\theta\\ +%\dot{\theta}\cos\theta +%\end{array} +%\right. +=\dot{\theta}\overrightarrow{u}_\theta +\quad +\frac{\mathrm{d}\overrightarrow{u}_\theta}{\mathrm{d}t}= +%\left| +%\begin{array}[m]{l} +%-\dot{\theta}\cos\theta\\ +%-\dot{\theta}\sin\theta +%\end{array} +%\right. +-\dot{\theta}\overrightarrow{u}_r +\] +Dans ce repère, la position de $M$ et sa vitesse ont pour expressions respectives : +\[ +\overrightarrow{r}=r \overrightarrow{u}_r\quad ;\quad \overrightarrow{v}=\frac{\mathrm{d}\overrightarrow{r}}{\mathrm{d}t}=\frac{\mathrm{d}r}{\mathrm{d}t}\overrightarrow{u}_r + r\frac{\mathrm{d}\overrightarrow{u}_r}{\mathrm{d}t}=\frac{\mathrm{d}r}{\mathrm{d}t}\overrightarrow{u}_r + r\dot{\theta}\overrightarrow{u}_\theta +\] +l'accélération s'écrit : +\[ +\overrightarrow{\gamma}=\frac{\mathrm{d}\overrightarrow{v}}{\mathrm{d}t}=\ddot{r}\overrightarrow{u}_r + + \dot{r}\dot{\theta}\overrightarrow{u}_\theta+ + \dot{r}\dot{\theta}\overrightarrow{u}_\theta + + r\ddot{\theta}\overrightarrow{u}_\theta -r\dot{\theta}^2\overrightarrow{u}_r +\] +\[ +=(\ddot{r} -r\dot{\theta}^2)\overrightarrow{u}_r + + (2\dot{r}\dot{\theta}+r\ddot{\theta})\overrightarrow{u}_\theta +\] +En résumé : +\[ +\overrightarrow{v}= +\left| +\begin{array}{l} +\dot{r}\\ +r\dot{\theta} +\end{array} +\right. +\quad \quad +\overrightarrow{\gamma}= +\left| +\begin{array}{l} +\ddot{r} -r\dot{\theta}^2\\ +\dfrac{1}{r}\dfrac{\mathrm{d}}{\mathrm{d}t}(r^2\dot{\theta}) +\end{array} +\right. +\] +L'équation (\ref{RFD}) nous donne, en divisant par $\mu$, l'accélération : +\[ + \dfrac{\mathrm{d}^2\overrightarrow{r}}{\mathrm{d}t^2}=-\mathcal{G}\frac{m_t}{r^2}\overrightarrow{u}_r +\] +Et puisque l'accélération radiale est nulle : +\[ +\dfrac{1}{r}\dfrac{\mathrm{d}}{\mathrm{d}t}(r^2\dot{\theta})=0 +\] +D'où : +\[ +r^2\dot{\theta}=\mathrm{C^{ste}} +\] +Il est intéressant ici de calculer le moment cinétique de $M$ et d'exprimer la loi des aires. + +Le moment cinétique : +\[ +\overrightarrow{\sigma}=\overrightarrow{r} \wedge \mu\overrightarrow{v} +\] +$\overrightarrow{F}$ étant dirigée vers $C$, son moment par rapport à $C$ est nul. Le moment cinétique est donc constant. Il en découle que le mouvement est plan, puisque le vecteur-position $\overrightarrow{r}$ reste toujours perpendiculaire à un vecteur constant. Ce plan est le plan perpendiculaire à $\sigma_0$ contenant $C$. + +On peut calculer le moment cinétique $\overrightarrow{\sigma}$, à partir des coordonnées polaires : +\[ +\overrightarrow{\sigma}=\mu +\left| +\begin{array}{l} +r\\ +0\\ +0 +\end{array} +\right. +\wedge +\left| +\begin{array}{l} +\dot{r}\\ +r\dot{\theta}\\ +0 +\end{array} +\right. += +\left| +\begin{array}{l} +0\\ +0\\ +r^2\dot{\theta} +\end{array} +\right. +=\mu r^2\dot{\theta}\overrightarrow{u}_z +\] +La constante des aires(voir plus loin pourquoi elle s'appelle ainsi) est $\mathrm{C}=\dfrac{\sigma}{\mu}=r^2\dot{\theta}$, c'est aussi la valeur du moment cinétique par unité de masse. + +On peut aussi calculer le moment cinétique en coordonnées cartésiennes. Les conditions initiales étant fixées, $\overrightarrow{v}_0$ est donné par ses coordonnées, ainsi que $\overrightarrow{r}_0$. +\[ +\overrightarrow{\sigma}=\mu +\left| +\begin{array}{l} +x_0\\ +y_0\\ +0 +\end{array} +\right. +\wedge +\left| +\begin{array}{l} +v_{x_0}\\ +v_{y_0}\\ +0 +\end{array} +\right. += +\left| +\begin{array}{l} +0\\ +0\\ +x_0 v_{y_0} - y_0 v_{x_0} +\end{array} +\right. +=\mu(x_0 v_{y_0} - y_0 v_{x_0})\overrightarrow{u}_z +\] +Les livres d'astronomie\footnote{page 43 dans \textit{Le mouvement du satellite} : conférences et exercices de mécanique spatiale. Cepadues-éditions 1983.} préfèrent définir la vitesse par un autre paramètre $\gamma$, \textit{angle de l'horizontale locale }$\overrightarrow{u}_z\wedge \overrightarrow{u}_r$ (c'est-à-dire $\overrightarrow{u}_\theta$) \textit{avec la vitesse}, en se plaçant toujours en coordonnées polaires. +\begin{center} +\begin{pspicture}(-3.5,-1)(6.5,8.25) +%\psframe(-3.5,-1)(5.5,9) +\pstVerb{/x01 -2 def /y01 1 def + /x02 4 def /y02 5 def + /xr0 x02 x01 sub def + /yr0 y02 y01 sub def + /theta_0 yr0 xr0 atan 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} +\pscircle[hatchcolor=gray,fillstyle=vlines,hatchwidth=0.02,hatchsep=0.05](C){0.25} +\psdot(C) +%\psline[linestyle=dotted](M1)(M2) +%\psline[linestyle=dashed](M2)(O)(M1) +\psline[linecolor=gray]{<->}(5,0)(0,0)(0,6) +\rput(C){\psline[style=vecteurA]{<->}(5.5,0)(0,0)(0,6)% + \uput[l](0,5.8){$\mathcal{R^*}$} + \psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt](-3,-3)(7,6)% + \uput[r](0,5.8){$y^*$} + \uput[u](5.5,0){$x^*$} + \pnode(!xr0 yr0){M} + \pscircle*(M){0.05} +% \pcline[offset=0.3,linewidth=0.01,arrowsize=0.1]{|->}(0,0)(M) +% \ncput*[nrot=:U]{$\overrightarrow{r}$} + \psline[linestyle=dotted](M)\uput[r](M){$M(\mu)$} + \rput(M){\psline[style=vecteurC]{->}(!xr0 5 div neg yr0 5 div neg)\uput[dr](!xr0 10 div neg yr0 10 div neg){$\overrightarrow{F}$} + \psline[linestyle=dotted](!theta_0 90 add cos 2 mul theta_0 90 add sin 2 mul) + \psline[arrowinset=0.1,arrowsize=0.15]{->}(!theta_0 90 add cos theta_0 90 add sin) + \psline[arrowinset=0.1,arrowsize=0.15,linecolor=blue]{->}(!theta_0 45 add cos 2 mul theta_0 45 add sin 2 mul) + \uput[r](!theta_0 45 add cos 1.75 mul theta_0 45 add sin 1.55 mul){$\overrightarrow{v}$} + \psarc{->}(0,0){1.2}{!theta_0 45 add}{!theta_0 90 add} + \uput{1.3}[!theta_0 67.5 add](0,0){$\gamma$}} + } +\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}[dr](M2){$M_2$} +\uput{0.3}[d](C){$C$} +\rput(C){\psline[arrowinset=0.1,arrowsize=0.15]{->}(!theta_0 90 add cos theta_0 90 add sin) + \psline[arrowinset=0.1,arrowsize=0.15]{->}(!theta_0 cos theta_0 sin) + \uput{.1}[u](!theta_0 90 add cos theta_0 90 add sin){$\overrightarrow{u}_\theta$} + \uput[d](!theta_0 cos theta_0 sin){$\overrightarrow{u}_r$} + \psarc{->}(0,0){1.5}{0}{!theta_0} + \uput{1.6}[!theta_0 2 div](0,0){$\theta$}} +\end{pspicture} +\end{center} +Dans ce cas, on peut exprimer la constante des aires par : +$ +\mathrm{C}=rv\cos\gamma +$ +\begin{center} +\begin{pspicture}(-2,-2)(7,5) +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt] +\pstVerb{/par 2 def + /exc 0.7 def + /Phi 30 def + /radius {par 1 exc t Phi sub cos mul add div} def + /xE {radius t cos mul neg} def + /yE {radius t sin mul neg} def}% +\parametricplot[plotpoints=360]{0}{360}{ + xE + yE}% +\pscustom[fillstyle=vlines,hatchwidth=0.02,hatchcolor=gray]{% + \psline(0,0)(! /t 200 def xE yE) +\parametricplot[plotpoints=360]{200}{210}{ + xE + yE}% + \psline(! /t 210 def xE yE)(0,0)} +\parametricplot[plotpoints=360,arrows=->]{200}{210}{ + /radius {2.25 1 exc t Phi sub cos mul add div} def + xE + yE}% +\psarc{->}(0,0){3}{20}{30} +\uput{3.2}[25](0,0){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white,framesep=0pt]{d$\theta$}} +\rput(5,2.3){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white,framesep=0pt]{d$\mathcal{A}$}} +%\psline[linearc=0.25]{->}(2.2,1.8)(3.5,1.8)(3.5,1.5) +\pcline[offset=-0.3,linestyle=none](0,0)(! /t 200 def xE yE) +\ncput*[nrot=:U]{$r$} +\uput[l](0,4.8){$\mathcal{R^*}$} +\uput[r](0,4.8){$y^*$} +\uput[u](7,0){$x^*$} +\uput[dl](0,0){$C$} +\psline[style=vecteurA]{<->}(7,0)(0,0)(0,5) +\end{pspicture} +\end{center} +Géométriquement lorsque $\theta$ varie de $\mathrm{d}\theta$, la surface balayée par le rayon vecteur vaut : +\[ +\mathrm{d}\mathcal{A}=\frac{1}{2}r^2\mathrm{d}\theta +\] +On définit ainsi la vitesse \textit{aérolaire}, qui est l'aire balayée par unité de temps, en fonction de C : +\[ +\frac{\mathrm{d}A}{\mathrm{d}t}=\frac{\mathrm{C}}{2} +\] +On retrouve ainsi la deuxième loi de Képler. + +Pour déterminer l'équation de la trajectoire, on utilise la méthode de Binet, qui consiste en un changement de variable en posant $u=\dfrac{1}{r}$. +\[ +\dfrac{\mathrm{d}r}{\mathrm{d}t}=-\dfrac{1}{u^2}\dfrac{\mathrm{d}u}{\mathrm{d}\theta}\dfrac{\mathrm{d}\theta}{\mathrm{d}t} +\] +Nous avons la constante des aires $\mathrm{C}=r^2\dot{\theta}=\dfrac{\dot{\theta}}{u^2}$, par conséquent : +\[ +\dfrac{\mathrm{d}r}{\mathrm{d}t}=-\mathrm{C}\dfrac{\mathrm{d}u}{\mathrm{d}\theta} +\] +\[ +\dfrac{\mathrm{d}^2r}{\mathrm{d}t^2}=-\mathrm{C}\dfrac{\mathrm{d}^2u}{\mathrm{d}\theta^2}\dfrac{\mathrm{d}\theta}{\mathrm{d}t} +\] +On remplace $\dot{\theta}$ par $\mathrm{C}u^2$ : +\[ +\dfrac{\mathrm{d}^2r}{\mathrm{d}t^2}=-\mathrm{C^2}u^2\dfrac{\mathrm{d}^2u}{\mathrm{d}\theta^2} +\] +Nous avons vu qu'en appliquant la loi de Newton, la composante de l'accélération $\gamma$ suivant $\sqrt[n]{u}_r$ vaut : +\[ +\ddot{r} -r\dot{\theta}^2=-\mathcal{G}\frac{m_t}{r^2} +\] +Ce qui s'écrit avec la variable $u$ : +\[ +-\mathrm{C^2}u^2\dfrac{\mathrm{d}^2u}{\mathrm{d}\theta^2}-\mathrm{C^2}u^3=-\mathcal{G}m_tu^2 +\] +Ce qui, après simplification, donne : +\begin{equation} +\psframebox[linestyle=none,fillstyle=solid,fillcolor={[rgb]{1 1 0.5}}]{% +\dfrac{\mathrm{d}^2u}{\mathrm{d}\theta^2}+u=\dfrac{\mathcal{G}m_t}{\mathrm{C^2}}} +\label{equ} +\end{equation} +La solution générale de cette équation peut s'écrire : +\[ +u=A\cos(\theta-\varphi)+\dfrac{\mathcal{G}m_t}{\mathrm{C^2}} +\] +$A$ et $\varphi$ sont fixés par les conditions initiales. Le rayon-vecteur a pour expression : +\[ +r=\frac{1}{A\cos(\theta-\varphi)+\dfrac{\mathcal{G}m_t}{\mathrm{C^2}}} +\] +Il s'écrit encore ainsi : +\[ +r=\dfrac{\dfrac{\mathrm{C^2}}{\mathcal{G}m_t}}{A\dfrac{\mathrm{C^2}}{\mathcal{G}m_t}\cos(\theta-\varphi)+1} +\] +Si on pose $p=\dfrac{\mathrm{C^2}}{\mathcal{G}m_t}$ et $\mathrm{e}=Ap$, on reconnaît l'équation d'une conique dont l'un des foyers est $C$, de paramètre $p$ et d'excentricité $\mathrm{e}$. +\begin{equation} +\psframebox[linestyle=none,fillstyle=solid,fillcolor={[rgb]{1 1 0.5}}]{% +r=\dfrac{p}{1+e\cos(\theta-\varphi)}} +\label{equr} +\end{equation} +Déterminons les constantes en fonction des conditions initiales. + +Nous avons vu que la constante des aires C pouvait se calculer de deux façons suivant le choix fait pour définir la vitesse initiale, soit $ +\mathrm{C}=r_0v_0\cos\gamma_0$, soit $C=x_0 v_{y_0} - y_0 v_{x_0}$. Par conséquent $p$ est bien déterminé. + +Lorsque $\theta=\varphi$ alors $r=\dfrac{p}{1+\mathrm{e}}$, ce qui est la plus petite valeur possible de $r$. On est donc au périgée P de la conique, on note, en ce point, le rayon-vecteur par $\overrightarrow{r}_\mathrm{P}$. En ce point la vitesse $\overrightarrow{v}_\mathrm{P}$ est perpendiculaire à $\overrightarrow{r}_\mathrm{P}$ et la constante des aires peut s'écrire : $\mathrm{C}=r_\mathrm{P}v_\mathrm{P}$. + +L'angle $\varphi$ est l'inclinaison du grand axe de la conique avec $Ox$, plus précisément l'angle du rayon-vecteur au périgée, $\overrightarrow{r}_P$ avec $Ox$. + +L'énergie mécanique se conserve au cours du mouvement, elle est la somme de l'énergie cinétique et de l'énergie potentielle. En fonction des conditions initiales elle s'écrit(on rappelle que $\mu$ est la masse(\textit{réduite})) de la particule fictive $M$ étudiée : +\[ +\mathcal{E}=\frac{1}{2}\mu v_0^2-\mathcal{G}\frac{\mu(m_1+m_2)}{r_0}\quad \text{ou}\quad \frac{1}{2}\mu v_0^2-\mathcal{G}\frac{m_1m_2}{r_0} +\] +Au périgée, elle s'exprimera par : +\[ +\mathcal{E}=\frac{1}{2}\mu v_\mathrm{P}^2-\mathcal{G}\frac{m_1m_2}{r_\mathrm{P}} +\] +Pour exprimer l'énergie dans le cas général, calculons la vitesse en fonction de $u$. Rappelons que : +\[ +\overrightarrow{v}= +\left| +\begin{array}{l} +\dot{r}\\ +r\dot{\theta} +\end{array} +\right. +\] +Calculons séparément $\dot{r}$ et $r\dot{\theta}$ : +\[ +\dot{r}=\dfrac{\mathrm{d}r}{\mathrm{d}u}\dfrac{\mathrm{d}u}{\mathrm{d}\theta}\dfrac{\mathrm{d}\theta}{\mathrm{d}t}= + -\dfrac{1}{u^2}\dfrac{\mathrm{d}u}{\mathrm{d}\theta}\dot{\theta}= + -\mathrm{C}\dfrac{\mathrm{d}u}{\mathrm{d}\theta} +\] +\[ +r\dot{\theta}=\dfrac{1}{u}\mathrm{C}u^2=\mathrm{C}u +\] +\[ +v^2=\mathrm{C^2}\left[\left(\frac{\mathrm{d}u}{\mathrm{d}\theta}\right)^2+u^2\right] +\] +\[ +v^2=\mathrm{C^2}\left[ + \frac{\mathrm{e^2}}{p^2}\sin^2(\theta-\varphi)+\frac{1}{p^2}\left[1+2\mathrm{e}\cos(\theta-\varphi)+\mathrm{e^2}\cos^2(\theta-\varphi)\right] + \right] +\] +\[ +v^2=\frac{\mathrm{C^2}}{p^2}\left[1+2\mathrm{e}\cos(\theta-\varphi)+\mathrm{e^2} \right] +\] +L'énergie cinétique s'exprime par : +\[ +E_C=\frac{1}{2}\mu\frac{\mathrm{C^2}}{p^2}\left[ 1+2\mathrm{e}\cos(\theta-\varphi)+\mathrm{e^2} \right] +\] +On se souvient que $p=\dfrac{\mathrm{C^2}}{\mathcal{G}m_t}$, il vient : +\[ +E_C=\frac{1}{2}\frac{\mathcal{G}\mu m_t}{p}\left[ 1+2\mathrm{e}\cos(\theta-\varphi)+\mathrm{e^2} \right] +\] +Pour l'énergie potentielle nous avons : +\[ +E_p=-\frac{\mathcal{G}\mu m_t}{r}=-\frac{\mathcal{G}\mu m_t}{p}\left[1+e\cos(\theta-\varphi)\right] +\] +et pour l'énergie mécanique, nous obtenons après simplification : +\[ +\mathcal{E}=E_C+E_p=\frac{\mathcal{G}\mu m_t}{2p}(\mathrm{e^2}-1) +\] +On en déduit l'excentricité : +\[ +\mathrm{e^2}=\frac{2p\mathcal{E}}{\mathcal{G}\mu m_t}+1\Longrightarrow \mathrm{e}=\sqrt{\frac{2p\mathcal{E}}{\mathcal{G}\mu m_t}+1} +\] +On peut l'exprimer en fonction de la constante des aires en posant $K=\mathcal{G}m_t$ et en remplaçant $p$ par $\dfrac{\mathrm{C^2}}{K}$ : +\begin{equation} +\psframebox[linestyle=none,fillstyle=solid,fillcolor={[rgb]{1 1 0.5}}]{% +\mathrm{e}=\sqrt{\frac{2\mathrm{C^2}\mathcal{E}}{\mu K^2}+1}} +\label{exc} +\end{equation} +Il reste à déterminer $\varphi$, c'est le problème le plus simple à résoudre. +\[ +r_0=\frac{p}{1+\mathrm{e}\cos(\theta_0-\varphi)} +\] +On en déduit : +\begin{equation} +\psframebox[linestyle=none,fillstyle=solid,fillcolor={[rgb]{1 1 0.5}}]{% +\varphi=\theta_0-\arccos\left[\left(\frac{p}{r_0}-1\right)\frac{1}{\mathrm{e}}\right]} +\label{phi} +\end{equation} +On distingue trois cas : +\begin{itemize} +\item + $\mathrm{e}<1$ ou $\mathcal{E}<0$ : ellipse +\item + $\mathrm{e}=1$ ou $\mathcal{E}=0$ : parabole +\item + $\mathrm{e}=>$ ou $\mathcal{E}>0$ : hyperbole +\end{itemize} +Et on s'intéresse maintenant au mouvement elliptique. Les conditions initiales $(\overrightarrow{r}_0,\overrightarrow{v}_0)$ doivent vérifier : +\[ +\mathcal{E}=\frac{1}{2}\mu v_0^2-\mathcal{G} \frac{m_1 m_2}{r_0}<0 +\] +\begin{center} +\begin{pspicture}(-2,-3)(5,7) +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt] +\pstVerb{ +/arccos { + dup + dup mul neg 1 add sqrt + exch + atan +} def + /G 1 def + /x01 2 def + /y01 2 def + /v0x1 .05 def + /v0y1 0.25 def + /x02 -2 def + /y02 -0.5 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 +% conditions initiales pour le point réduit + /xr0 x01 x02 sub def + /yr0 y01 y02 sub def + /theta_0 yr0 xr0 atan def + /r0 xr0 dup mul yr0 dup mul add sqrt def + /v0xr v0x1 v0x2 sub def + /v0yr v0y1 v0y2 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 + }% +\parametricplot[plotpoints=360]{0}{360}{xE yE}% +\uput[l](0,6.8){$\mathcal{R^*}$} +\uput[r](0,6.8){$y^*$} +\uput[u](5,0){$x^*$} +\uput[l](0,0){$C$} +\psdot(0,0) +\psline[style=vecteurA]{<->}(5,0)(0,0)(0,7) +\pnode(!xr0 yr0){M0}\psdot(M0) +\rput(M0){\psline[unit=2,style=vecteurA]{->}(!v0xr v0yr)\uput[r](0,0){$M_0$}\uput[r](!v0xr 2 mul v0yr 2 mul){$\overrightarrow{v}_0$}} +\psline(M0) +\pcline[offset=0.25,linestyle=none](0,0)(M0) +\ncput*[nrot=:U]{$r_0$} +\psarc[style=vecteurB]{->}(0,0){1.5}{0}{!theta_0} +\uput{1.6}[!theta_0 2 div](0,0){$\theta_0$} +\pnode(!rA Phi cos mul neg rA Phi sin mul neg){A} +\psline[linestyle=dotted](A) +\uput[ur](A){$A$} +\pnode(!rP Phi cos mul rP Phi sin mul){P} +\psline[linestyle=dotted](P) +\uput[dl](P){$P$} +\rput(-2,0){\psPrintValue[decimals=4]{exc}} +\pnode(!xW yW){W}\psdot(W)\uput[r](W){$\Omega$} +\rput{!Phi}(W){\pnode(!0 b_2){B1}\pnode(!0 b_2 neg){B2}\psline[linestyle=dotted](B1)(B2)} +\uput[ul](B2){$B$} +\pcline[offset=-0.25,linestyle=none](W)(B2) +\ncput*{$b$} +\pcline[offset=0.25,linestyle=none](W)(A) +\ncput*{$a$} +\pcline[offset=0.25,linestyle=none](0,0)(W) +\ncput*{$c$} +\psline[linecolor=blue](W)(B2) +\psline[linecolor=red](W)(A) +\psline[linecolor=green](W)(0,0) +\rput{! 180 Phi add}(W){\psdot(!c_2 0)} +\psarcn{->}(0,0){0.5}{0}{!Phi}\uput{0.55}[!Phi 2 div](0,0){$\varphi$} +\rput{!Phi}(P){\pnode(! 0 vP){vP}\psline[style=vecteurA]{->}(vP)} +\uput[r](vP){$\overrightarrow{v}_P$} +\rput{!Phi}(A){\pnode(! 0 vA neg){vA}\psline[style=vecteurA]{->}(vA)} +\uput[u](vA){$\overrightarrow{v}_A$} +\end{pspicture} +\end{center} +L'un des foyers de l'ellipse est le centre attracteur $C$. L'autre est le symétrique de $C$ par rapport à $\Omega$. + +Au périgée, le rayon-vecteur vaut : $r_P=\dfrac{p}{1+\mathrm{e}}$, à l'apogée $r_A=\dfrac{p}{1-\mathrm{e}}$. + +Le demi-grand axe est égal à : $a=\dfrac{r_P+r_A}{2}=\dfrac{p}{1-\mathrm{e^2}}$. + +La distance focale est égale à : $c=a-r_P=\dfrac{p\mathrm{e}}{1-\mathrm{e^2}}=a\mathrm{e}$. + +Pour calculer le demi-petit axe de l'ellipse $b$, on peut rappeler que l'ellipse est lieu des points dont la somme des distances aux foyers est égale à $2a$. Au point $B$ cette somme est égale à $2a$, par conséquent le théorème de Pythagore appliqué dans le triangle rectangle $C\Omega B$ donne la valeur du demi-axe : $b=\sqrt{a^2-c^2}=a\sqrt{1-\mathrm{e^2}}$. + +Rappelons que $\mathrm{e^2}=\dfrac{2p\mathcal{E}}{\mathcal{G}\mu m_t}+1$. On peut exprimer $\mathcal{E}$ en fonction de $\mathrm{e}$ : + +\[ +\mathcal{E}=\frac{\mathcal{G}\mu m_t}{2p}(\mathrm{e^2}-1)=-\frac{\mathcal{G}\mu m_t}{2a}\quad\text{ou}\quad\mathcal{E}=-\frac{\mathcal{G}m_1m_2}{2a} +\] +Pour le système $\{M_1,M_2\}$, son énergie s'exprime très simplement en fonction du grand axe de l'ellipse $2a$. + +Calculons la vitesse tout au long de l'ellipse : + +\[ +\mathcal{E}=\frac{1}{2}\mu v^2-\frac{\mathcal{G}\mu m_t}{r}=-\frac{\mathcal{G}\mu m_t}{2a} +\] +\[ +v^2=\mathcal{G}m_t\left(\frac{2}{r}-\frac{1}{a}\right) +\] +On en déduit les vitesses au périgée et à l'apogée : +\[ +v_P=\sqrt{\frac{\mathcal{G}m_t}{p}}(1+\mathrm{e})\qquad v_A=\sqrt{\frac{\mathcal{G}m_t}{p}}(1-\mathrm{e}) +\] +La vitesse est maximale au périgée et minimale à l'apogée. + +La période de révolution se détermine à partir de la vitesse \textit{aérolaire}. Cette vitesse vaut : +\[ +\mathcal{V}=\frac{\mathrm{C}}{2}=\frac{\sqrt{p\mathcal{G}m_t}}{2} +\] +Sachant que surface de l'ellipse est $S=\pi a b$ : +\[ +T=\frac{2\pi a b}{\sqrt{p\mathcal{G}m_t}} +\] +On remplace $b$ par $a\sqrt{(1-\mathrm{e^2})}$ et $p$ par $a(1-\mathrm{e^2})$ : +\[ +T=\frac{2\pi a^2 \sqrt{(1-\mathrm{e^2})}}{\sqrt{a(1-\mathrm{e^2})\mathcal{G}m_t}}=\frac{2\pi a^2}{\sqrt{a\mathcal{G}m_t}} +\] +En élevant les deux membres au carré on retrouve la troisième loi de Képler : +\begin{equation} +\psframebox[linestyle=none,fillstyle=solid,fillcolor={[rgb]{1 1 0.5}}]{% +\frac{T^2}{a^3}=\frac{4\pi^2}{\mathcal{G}m_t}} +\label{Kepler3} +\end{equation} +\section{Le mouvement des 2 corps dans le repère du centre de masse} +Nous avons vu (\ref{r1r2}), que les trajectoires de $M_1$ et $M_2$ se déduisent de celle de $M$ par une homothétie de centre~C. +\[ +\overrightarrow{r_1}=-\frac{m_2}{m_1+m_2}\overrightarrow{r}\quad\mbox{et}\quad \overrightarrow{r_2}=\frac{m_1}{m_1+m_2}\overrightarrow{r} +\] +Les trajectoires sont donc deux ellipses dont les caractéristiques, paramètre, demi-grand axe et demi-petit axe, se déduisent de l'ellipse de la particule de masse réduite dans le même rapport. + +\begin{itemize} + \item Paramètre : $p_1=p\dfrac{m_2}{m_1+m_2}$. + \item rayon-vecteur à l'apogée : $r_{A_1}=r_A\dfrac{m_2}{m_1+m_2}$ + \item rayon-vecteur au périgée : $r_{P_1}=r_P\dfrac{m_2}{m_1+m_2}$ + \item demi-grand axe : $a_1=\dfrac{r_{A_1}+r_{P_1}}{2}=\dfrac{m_2}{m_1+m_2}\dfrac{r_{A}+r_{P}}{2}=a\dfrac{m_2}{m_1+m_2}$ + \item excentricité, on vérifie facilement qu'elle est inchangée : $\mathrm{e_1}=\dfrac{r_{A_1}-r_{P_1}}{r_{A_1}+r_{P_1}}=\mathrm{e}$ +\end{itemize} +pour la deuxième : +\begin{itemize} + \item Paramètre : $p_2=p\dfrac{m_1}{m_1+m_2}$. + \item rayon-vecteur à l'apogée : $r_{A_2}=r_A\dfrac{m_1}{m_1+m_2}$ + \item rayon-vecteur au périgée : $r_{P_2}=r_P\dfrac{m_1}{m_1+m_2}$ + \item demi-grand axe : $a_1=\dfrac{r_{A_2}+r_{P_2}}{2}=\dfrac{m_1}{m_1+m_2}\dfrac{r_{A}+r_{P}}{2}=a\dfrac{m_1}{m_1+m_2}$ + \item excentricité : $\mathrm{e_2}=\dfrac{r_{A_2}-r_{P_2}}{r_{A_2}+r_{P_2}}=\mathrm{e}$ +\end{itemize} +Leurs équations respectives s'écrivent : +\[ +r_1=\dfrac{p_1}{1+\mathrm{e}\cos(\theta-\varphi)} \quad r_2=\dfrac{p_2}{1+\mathrm{e}\cos(\theta-\varphi)} +\] +Dans l'exemple suivant $\{m_1=3,m_2=1\}$. Les vitesses initiales dans $\mathcal{R}^*$ sont indiquées par une flèche. La trajectoire de $M_1$ est en bleu, celle de $M_2$ en rouge et celle du point fictif est en pointillés. +\begin{center} +\begin{pspicture}(-8,-10)(3,3) +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt] +\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 + }% +\parametricplot[plotpoints=360,linestyle=dotted]{0}{360}{xE yE}% +\parametricplot[plotpoints=360,linecolor=blue]{0}{360}{xE K1 mul yE K1 mul}% +\parametricplot[plotpoints=360,linecolor=red]{0}{360}{xE K2 mul yE K2 mul}% +\uput[l](0,2.8){$\mathcal{R^*}$} +\uput[r](0,2.8){$y^*$} +\uput[u](3,0){$x^*$} +\uput[l](0,0){$C$} +\psdot(0,0) +\psline[style=vecteurA]{<->}(3,0)(0,0)(0,3) +\pnode(!xr0 yr0){M0}\psdot(M0) +\rput(M0){\psline[unit=2,style=vecteurA]{->}(!v0xr v0yr)\uput[l](0,0){$M_0$}\uput[l](!v0xr 2 mul v0yr 2 mul){$\overrightarrow{v}_0$}} +\pcline[offset=0.25,linestyle=none](M0)(0,0) +\ncput*[nrot=:U]{$r_0$} +\psarcn[style=vecteurB]{->}(0,0){0.75}{0}{!theta_0} +\uput{0.75}[!theta_0 2 div 180 add](0,0){$\theta_0$} +\pnode(!rA Phi cos mul neg rA Phi sin mul neg){A} +\psline[linestyle=dotted](A) +%\uput[ur](A){$A$} +\pnode(!rP Phi cos mul rP Phi sin mul){P} +\psline[linestyle=dotted](P) +%\uput[dl](P){$P$} +%\rput(-2,0){\psPrintValue[decimals=4]{exc}} +\pnode(!xW yW){W}\psdot(W)\uput[r](W){$\Omega$} +%\psline[linecolor=blue](W)(B2) +%\psline[linecolor=red](W)(A) +%\psline[linecolor=green](W)(0,0) +%\rput{! 180 Phi add}(W){\psdot(!c_2 0)} +%\psarc{->}(0,0){0.5}{0}{!Phi}\uput{0.55}[!Phi 2 div](0,0){$\varphi$} +%\rput{!Phi}(P){\pnode(! 0 vP){vP}\psline[style=vecteurA]{->}(vP)} +%\uput[r](vP){$\overrightarrow{v}_P$} +%\rput{!Phi}(A){\pnode(! 0 vA neg){vA}\psline[style=vecteurA]{->}(vA)} +%\uput[u](vA){$\overrightarrow{v}_A$} +\pnode(!x01 xG0 sub y01 yG0 sub){M01}\psdot(M01)\uput[r](M01){$M_{0_1}$} +\pnode(!x02 xG0 sub y02 yG0 sub){M02}\psdot(M02)\uput[dr](M02){$M_{0_2}$} +\psline[linestyle=dashed](M0)(M02)(M01) +\rput(M01){\psline[unit=2,linecolor=blue]{->}(!v0x1 vG0x sub v0y1 vG0y sub)} +\rput(M02){\psline[unit=2,linecolor=red]{->}(!v0x2 vG0x sub v0y2 vG0y sub)} +\end{pspicture} +\end{center} +\section{Exemples à développer, réalisés avec pst-eqdf} +\[ +\mu\frac{\mathrm{d}^2\overrightarrow{r}}{\mathrm{d}t^2}=-G\frac{m_1m_2}{r^3}\overrightarrow{r} +\] +\[ +\frac{\mathrm{d}^2\overrightarrow{r}}{\mathrm{d}t^2}=-G\frac{(m_1+m_2)}{r^3}\overrightarrow{r} +\] + +\[ +\left\{ +\begin{array}[m]{l} + \ddot{x}=-G\displaystyle\frac{m_1+m_2}{r^3}x\\[1em] + \ddot{y}=-G\displaystyle\frac{m_1+m_2}{r^3}y +\end{array} +\right. +\] +% 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} +\begin{center} +\begin{pspicture}(-7,-4)(7,8) +\pstVerb{/G 1 def + /x01 1 def + /y01 2 def + /v0x1 .1 def + /v0y1 0.1 def + /x02 -1 def + /y02 -2 def + /v0x2 -2 def + /v0y2 0 def + /M1 2 def + /M2 20 def + /mu M1 M2 mul M1 M2 add div def % masse réduite + /xG M1 x01 mul M2 x02 mul add M1 M2 add div def + /yG M1 y01 mul M2 y02 mul add M1 M2 add div def + /x0 x01 x02 sub def + /y0 y01 y02 sub def + /v0x v0x1 v0x2 sub def + /v0y v0y1 v0y2 sub def + }% +%% 0 1 2 3 4 5 6 7 +%% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +\def\InitCondred{ x0 y0 v0x v0y} +\def\InitCond{ x01 y01 v0x1 v0y1 x02 y02 v0x2 v0y2} +\psset{method=rk4} +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=8pt]% +\psequadiff[whichabs=0,whichord=1, + plotpoints=1000,algebraic, + tabname=X1Y1]{0}{50}{\InitCond}{\GravAlg} +\listplot[unit=1,linecolor=red]{X1Y1 aload pop} +\psequadiff[whichabs=4,whichord=5, + plotpoints=1000,algebraic, + tabname=X2Y2]{0}{50}{\InitCond}{\GravAlg} +\listplot[unit=1,linecolor=blue]{X2Y2 aload pop} +\psequadiff[whichabs=0,whichord=1, + plotpoints=1000,algebraic, + tabname=XrYr]{0}{50}{\InitCondred}{\FictifAlg} +\listplot[unit=1,linecolor=green]{XrYr aload pop} +\psdots(!x01 y01)(!x02 y02) +\psdot[linecolor=red](!xG yG) +\psline{<->}(7,0)(0,0)(0,8) +\end{pspicture} +\end{center} +% +\newpage +% 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} +% on se place au centre de masse +\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} +%% 0 1 2 3 4 5 6 7 +%% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +\newpage +\begin{center} +\def\InitCondred{ xr0 yr0 v0xr v0yr} +\begin{pspicture}(-7,-4)(7,8) +\pstVerb{/G 1 def + /x01 2 def + /y01 2 def + /v0x1 .05 def + /v0y1 0.25 def + /x02 -2 def + /y02 -0.35 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 + /xG M1 x01 mul M2 x02 mul add M1 M2 add div def + /yG M1 y01 mul M2 y02 mul add M1 M2 add div def + /x0 x01 x02 sub def + /y0 y01 y02 sub def + /v0x v0x1 v0x2 sub def + /v0y v0y1 v0y2 sub def + /mu M1 M2 mul M1 M2 add div def % masse réduite + /xG M1 x01 mul M2 x02 mul add M1 M2 add div def + /yG M1 y01 mul M2 y02 mul add M1 M2 add div def + /vxG M1 v0x1 mul M2 v0x2 mul add M1 M2 add div def + /vyG M1 v0y1 mul M2 v0y2 mul add M1 M2 add div def +% conditions initiales pour le point réduit + /xr0 x01 x02 sub def + /yr0 y01 y02 sub def + /r0 xr0 dup mul yr0 dup mul add sqrt def + /v0xr v0x1 v0x2 sub def + /v0yr v0y1 v0y2 sub def + /Lc xr0 v0yr mul yr0 v0xr mul sub mu mul def % moment cinetique + /K xr0 v0yr mul yr0 v0xr mul sub def + /Energie 0.5 mu mul v0xr dup mul v0yr dup mul add mul % 1/2 mv^2 +% Lc dup mul 2 div mu div r0 dup mul div + G M1 M2 mul mul r0 div sub + def +% /par xr0 v0yr mul yr0 v0xr mul sub dup mul G Mt mul div def % paramètre de l'ellipse +%%%%%%%%%%%%%% + % /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /a_2 G M1 M2 add mul mu mul Energie div 2 div neg def % + % excentricité + /exc 1 K 2 exp a_2 G mul Mt mul div sub sqrt def +% paramètre +% /par2 a_2 1 exc dup mul sub mul def +% periode + /periode 6.28 a_2 3 exp G div Mt div sqrt mul def + }% +%% 0 1 2 3 4 5 6 7 +%% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +\def\InitCondred{ x0 y0 v0x v0y} +\def\InitCond{ x01 y01 v0x1 v0y1 x02 y02 v0x2 v0y2} +\psset{method=rk4} +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=8pt]% +\psequadiff[whichabs=0,whichord=1, + plotpoints=1000,algebraic, + tabname=X1Y1]{0}{100}{\InitCond}{\GravAlgIIcorps} +\listplot[unit=1,linecolor=red]{X1Y1 aload pop} +\psequadiff[whichabs=4,whichord=5, + plotpoints=1000,algebraic, + tabname=X2Y2]{0}{100}{\InitCond}{\GravAlgIIcorps} +\listplot[unit=1,linecolor=blue]{X2Y2 aload pop} +% 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=XY3]{0}{22.7}{\InitCond}{\GravAlgIIcorps} +%\listplot[unit=1]{XY3 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=XY4]{0}{22.7}{\InitCond}{\GravAlgIIcorps} +%\listplot[unit=1,linecolor=red]{XY4 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]{0}{100}{\InitCond}{\GravAlgIIcorps} +\listplot[unit=1,linecolor=cyan]{XGYG aload pop} +\psdots(!x01 y01)(!x02 y02) +\psdot[linecolor=red](!xG yG) +\psline{<->}(7,0)(0,0)(0,8) +\rput(-2,0){\psPrintValue[decimals=4]{periode}\hphantom{00000}s} +\end{pspicture} +\end{center} +% +\newpage +\begin{center} +\def\InitCondred{ xr0 yr0 v0xr v0yr} +\begin{pspicture}(-7,-6)(7,8) +\pstVerb{ +/arccos { + dup + dup mul neg 1 add sqrt + exch + atan +} def + /G 1 def + /x01 2 def + /y01 2 def + /v0x1 .05 def + /v0y1 0.25 def + /x02 -2 def + /y02 -0.5 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 + /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 + /x0 x01 x02 sub def + /y0 y01 y02 sub def + /theta_0 y0 x0 atan def + /v0x v0x1 v0x2 sub def + /v0y v0y1 v0y2 sub def + /mu M1 M2 mul M1 M2 add div def % masse réduite + /xG M1 x01 mul M2 x02 mul add M1 M2 add div def + /yG M1 y01 mul M2 y02 mul add M1 M2 add div def + /vxG M1 v0x1 mul M2 v0x2 mul add M1 M2 add div def + /vyG M1 v0y1 mul M2 v0y2 mul add M1 M2 add div def +% conditions initiales pour le point réduit + /xr0 x01 x02 sub def + /yr0 y01 y02 sub def + /r0 xr0 dup mul yr0 dup mul add sqrt def + /v0xr v0x1 v0x2 sub def + /v0yr v0y1 v0y2 sub def + /v0r_2 v0xr dup mul v0yr dup mul add def +% v0r 2 Mt mul r0 div G mul ge { a faire} if + /Lc xr0 v0yr mul yr0 v0xr mul sub mu mul def % moment cinetique + /K xr0 v0yr mul yr0 v0xr mul sub def + /Energie 0.5 mu mul v0xr dup mul v0yr dup mul add mul % 1/2 mv^2 +% Lc dup mul 2 div mu div r0 dup mul div + G M1 M2 mul mul r0 div sub + def +% /par xr0 v0yr mul yr0 v0xr mul sub dup mul G Mt mul div def % paramètre de l'ellipse +%%%%%%%%%%%%%% + % /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /a_2 G M1 M2 add mul mu mul Energie div 2 div neg def % + % excentricité + /exc 1 K 2 exp a_2 G mul Mt mul div sub sqrt def +% paramètre + /par2 a_2 1 exc dup mul sub mul def +% periode + /periode 6.28 a_2 3 exp G div Mt div sqrt mul def +% phase + /Phi theta_0 par2 r0 div 1 sub exc div arccos sub def + }% +%% 0 1 2 3 4 5 6 7 +%% x1 y1 x'1 y'1 x2 y2 x'2 y'2 +\def\InitCondred{ x0 y0 v0x v0y} +\def\InitCond{ x01 y01 v0x1 v0y1 x02 y02 v0x2 v0y2} +\psset{method=rk4} +%\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=8pt]% +% 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=XY3]{0}{23.6}{\InitCond}{\GravAlgIIcorps} +\listplot[unit=1,linecolor=red]{XY3 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=XY4]{0}{23.6}{\InitCond}{\GravAlgIIcorps} +\listplot[unit=1,linecolor=blue]{XY4 aload pop} +\psline{<->}(7,0)(0,0)(0,8) +\rput(-2,2){\psPrintValue[decimals=4]{periode}\hphantom{0000000}s} +%\parametricplot[plotpoints=360,linestyle=dotted]{0}{360}{% +% /radius par2 1 exc t theta_0 sub cos mul add div def +% radius t cos mul +% radius t sin mul}% +\psdots(!x01 xG0 sub y01 yG0 sub)(!x02 xG0 sub y02 yG0 sub) +\psline(!x01 xG0 sub y01 yG0 sub)(!x02 xG0 sub y02 yG0 sub) +% mouvement de M (masse réduite) 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 + Mt mul M1 div, + plotfuncy=5 get + y 1 get M1 mul + y 5 get M2 mul add + Mt div sub + Mt mul M1 div, + tabname=XYM]{0}{23.6}{\InitCond}{\GravAlgIIcorps}% +\listplot[unit=1,linecolor=gray!80,linewidth=0.1]{XYM aload pop} +\rput(!x01 xG0 sub y01 yG0 sub){\psline[unit=5,linecolor=red!50]{->}(!v0x1 vxG sub v0y1 vyG sub)} +\rput(!x02 xG0 sub y02 yG0 sub){\psline[unit=5,linecolor=blue!50]{->}(!v0x2 vxG sub v0y2 vyG sub)} +% point "réduit" +\pstVerb{/xr0 x01 xG0 sub Mt mul M2 div neg def + /yr0 y01 yG0 sub Mt mul M2 div neg def}% +\psdot[linecolor=gray](!xr0 yr0) +\psline(!x01 xG0 sub y01 yG0 sub)(!x02 xG0 sub y02 yG0 sub)(!xr0 yr0) +\uput[d](0,0){$C$} +\uput[r](!x01 xG0 sub y01 yG0 sub){$M_1$} +\uput[dr](!x02 xG0 sub y02 yG0 sub){$M_2$} +\uput[l](!xr0 yr0){$M$} +\rput(!xG0 neg yG0 neg){\psline{<->}(7,0)(0,0)(0,8)\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=8pt]} +\parametricplot[plotpoints=360,linecolor=white]{0}{360}{ + /radius par2 1 exc t Phi sub cos mul add div def + radius t cos mul neg + radius t sin mul neg}% +\end{pspicture} +\end{center} +\end{document} \ No newline at end of file diff --git a/gravitation/pst-tools.sty b/gravitation/pst-tools.sty new file mode 100644 index 0000000..d4e98a6 --- /dev/null +++ b/gravitation/pst-tools.sty @@ -0,0 +1,8 @@ +\RequirePackage{pstricks} +\ProvidesPackage{pst-tools}[2012/01/01 package wrapper for + pst-tools.tex (hv)] +\input{pst-tools.tex} +\ProvidesFile{pst-tools.tex} + [\filedate\space v\fileversion\space `PST-tools' (hv)] +\endinput +%% $Id: pst-tools.sty 355 2010-06-21 10:02:44Z herbert $ diff --git a/gravitation/pst-tools.tex b/gravitation/pst-tools.tex new file mode 100644 index 0000000..a6e4cad --- /dev/null +++ b/gravitation/pst-tools.tex @@ -0,0 +1,111 @@ +%% $Id: pst-tools.tex 599 2011-11-03 19:38:28Z herbert $ +%% +%% This is file `pst-tools.tex', +%% +%% IMPORTANT NOTICE: +%% +%% Package `pst-tools.tex' +%% +%% Herbert Voss +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN archives +%% in directory macros/latex/base/lppl.txt. +%% +%% DESCRIPTION: +%% `pst-tools' is a PSTricks package for helper functions +%% +%% +\csname PSTtoolsLoaded\endcsname +\let\PSTtoolsLoaded\endinput + +\ifx\PSTricksLoaded\endinput\else\input pstricks.tex\fi +\ifx\PSTXKeyLoaded\endinput\else \input pst-xkey.tex \fi +% +\edef\PstAtCode{\the\catcode`\@} \catcode`\@=11\relax +% interface to the `xkeyval' package +\pst@addfams{pst-tools} +% +\def\fileversion{0.1} +\def\filedate{2012/01/01} +\message{`PST-tools' v\fileversion, \filedate\space (hv)} +% +\define@boolkey[psset]{pst-tools}[Pst@]{dot}[true]{} +\define@key[psset]{pst-tools}{xShift}[0]{\def\psk@xShift{#1}} +\define@key[psset]{pst-tools}{PSfont}[Times-Roman]{\def\psk@PSfont{/#1 }} +\define@key[psset]{pst-tools}{valuewidth}[10]{\pst@getint{#1}\psk@valuewidth } +\define@key[psset]{pst-tools}{fontscale}[10]{\pst@checknum{#1}\psk@fontscale } +\define@key[psset]{pst-tools}{decimals}[-1]{\pst@getint{#1}\psk@decimals } +\psset[pst-tools]{PSfont=Times-Roman,fontscale=10,valuewidth=10,decimals=-1,xShift=0,dot} +% +\def\psPrintValue{\pst@object{psPrintValue}} +\def\psPrintValue@i#1{\expandafter\psPrintValue@ii#1,,\@nil} +\def\psPrintValue@ii#1,#2,#3\@nil{% #1,#2 only for algebraic code + \begin@SpecialObj + \addto@pscode{ + gsave \psk@PSfont findfont \psk@fontscale scalefont setfont + \ifPst@algebraic + /x #1 def + /Func (#2) tx@AlgToPs begin AlgToPs end cvx def + Func + \else #1 \fi + \psk@decimals -1 gt { 10 \psk@decimals exp dup 3 1 roll mul cvi exch div } if + \psk@valuewidth string cvs %/Output exch def % save output + \ifPst@dot dot2comma \fi % do we have to change dot to comma + \psk@xShift\space 0 moveto %Output + show grestore + }% + \end@SpecialObj% +} + +\define@boolkey[psset]{pst-tools}[Pst@]{round}[true]{}% +\define@boolkey[psset]{pst-tools}[Pst@]{science}[true]{% + \ifPst@science\def\psk@Scin{true }\else\def\psk@Scin{false }\fi} +\psset[pst-tools]{science=false,round=false} +\def\psPrintValueNew{\pst@object{psPrintValueNew}} +\def\psPrintValueNew@i#1{\expandafter\psPrintValueNew@ii#1,,\@nil} +\def\psPrintValueNew@ii#1,#2,#3\@nil{% #1,#2 only for algebraic code + \begin@SpecialObj + \addto@pscode{ % thanks to Buddy Ledger + /mfont { \psk@PSfont findfont \psk@fontscale scalefont setfont } bind def + /mfontexp { \psk@PSfont findfont \psk@fontscale 1.2 div scalefont setfont } bind def + /s1 { /Symbol findfont \psk@fontscale scalefont setfont } bind def + \ifPst@algebraic + /x #1 def + /Func (#2) tx@AlgToPs begin AlgToPs end cvx def + Func + \else #1 \fi + /value ED + \psk@Scin { + value 0 ne { value log floor cvi /expon ED }{ /expon 0 def } ifelse + value 10 expon exp div + \psk@decimals -1 gt { 10 \psk@decimals exp dup 3 1 roll mul + \ifPst@round round \else cvi \fi exch div } if + \psk@decimals 0 eq { cvi } if /numb ED + expon \psk@valuewidth string cvs /expon exch def + numb \psk@valuewidth string cvs + \ifPst@dot dot2comma \fi % do we have to change dot to comma + /Output exch def + /txspc \psk@fontscale 4 div def + \psk@xShift\space 0 moveto mfont Output show + txspc 0 rmoveto s1 (\string\264) show + txspc 0 rmoveto mfont (10) show + txspc 2 div txspc 1.5 mul rmoveto mfontexp expon show } + { value + \psk@decimals -1 gt { 10 \psk@decimals exp dup 3 1 roll mul + \ifPst@round round \else cvi \fi exch div } if + \psk@decimals 0 eq { cvi } if %inserted to handle decimals=0 + \psk@valuewidth string cvs + \ifPst@dot dot2comma \fi % do we have to change dot to comma + \psk@xShift\space 0 moveto mfont %Output + show + } ifelse + }% + \end@SpecialObj% +} +% +\catcode`\@=\PstAtCode\relax +% +%% END: pst-tools.tex +\endinput +% -- 2.20.1