.gitignore

@@ -6,3 +6,4 @@
 *.pdf
 *.html
 *.markdown
+.vscode

02_lois_de_newton.md

@@ -910,14 +910,135 @@ sera plus élevée, plus faible, la même? Que se passe-t-il s'il y a une force
 
 Travail pratique (Calcul de force résultante) #
 
-Générer un tableau contenant $N$ particules représentées par leur masse $m_i$
-et leur position $\vec r_i$., sachant que:
+Générer un tableau contenant $N$ particules, numérotées $p_i$, représentées par leur masse $m_i$
+et leur position $\vec r_i$, sachant que:
 
 * La position des particules est aléatoire dans le domaine $[0, 10]\times[0, 10]\m^2$.
 * La masse des particules est aléatoire dans le domaine $[0,10]\kg$.
 
-1. Calculer la force résultante dûe à l'attraction gravitationnelle sur chaque particule.
-2. Si chaque particule a de plus une vitesse $\vec v_i$, calculer la force résultante sur chaque particule si $k=10 \kg/\s$.
+1. Calculer la force résultante dûe à l'attraction gravitationnelle sur chaque particule
+
+    $$
+    \vec F_i = \sum_{j=1,j\neq i}^N\vec F_{ij},
+    $$
+    où $\vec F_{ij}$ est la force gravitationnelle entre la particule $i$ et la particule $j$, et est donnée par
+    $$
+    \vec F_{ij}=G\frac{m_i m_j}{||\vec r_{ij}||^2}\vec r_{ij},
+    $$
+    où $\vec r_{ij}$ est le vecteur reliant la particule $i$ à la particule $j$, $G=6.67\cdot 10^{-11}\ \m^3/(\kg\cdot \s^2)$
+    est la constante universelle de gravitation.
+2. Si chaque particule, $p_i$, a de plus une vitesse $\vec v_i$, calculer la force résultante sur chaque particule si $k=10 \kg/\s$ ($k$ est le coefficient de résistance de l'objet dans le fluide).
+
+---
+
+## Les équations du mouvement
+
+Les lois de Newton, nous permettent de décrire des systèmes très complexes, comme le déplacement des planètes
+dans le système solaire, ou les gaz de particules. Dans cette section nous allons voir
+comment.
+
+Pour ce faire, nous allons considérer une particule $P$ en mouvement, qui est caractérisée par une position
+$\vec x(t)$ (qui dépend donc du temps, $t$) et par une masse $m$ constante. Nous souhaitons ici décrire le mouvement
+de $P$ en fonction du temps, en d'autre termes la valeur de $x(t)$ pour toute valeur de $t$.
+Il existe différents cas de figures, tels que mouvement rectiligne uniforme, ou le mouvement réctiligne uniformément
+accéléré, que vous connaissez qui permettent de résoudre exactement le mouvement de $P$. Ici, nous souhaitons
+généraliser un petit pou tout ça, en supposant que la vitesse et l'accélération de la particule dépendent
+également du temps, $\vec v(t)$ et $\vec a(t)$. Pour les besoins de ce cours, nous allons utiliser des
+*approximations* numérique des valeurs de $x(t)$, $v(t)$, et $a(t)$.
+
+Pour simplifier, nous allons *discrétiser* le temps. Au lieu de laisser le temps prendre n'importe quelle
+valeur réelle positive, $t\in\real^+$, il ne pourra prendre que les valeurs suivantes:
+$$
+t_j=j\cdot \delta t,
+$$
+$\delta t>0$ étant le pas de discrétisation temporel et $j\in\integer^+$ (voir @fig:time_discr).
+
+![Illustration de la discrétisation temporelle.](figs/time_discr.svg){#fig:time_discr}
+
+
+### Le mouvement sans accélération
+
+Supposons que pour notre particule $P$, nous connaissons sa position initiale, $\vec x(0)$ et sa vitesse
+en tout temps, $\vec v(t)$. Nous souhaitons connaître une approximation de la position $\vec x(t_j)$.
+Pour ce faire nous commençons au point $\vec x(0)$ et calculons sa nouvelle position
+à l'aide de la vitesse $\vec v(0)$
+$$
+\vec x(\delta t)=\vec x(0)+\delta t\cdot \vec v(0).
+$$
+Plus $\delta t$ est petit, plus cette approximation sera précise.
+En utilisant la notation $t_j=j\cdot \delta t$, on obtient
+$$
+\vec x(t_1)=\vec x(t_0)+\delta t\cdot \vec v(t_0).
+$$
 
 ---
 
+Remarque ($\delta t\rightarrow 0$) #
+
+On peut réécrire l'équation ci-dessus comme
+$$
+\vec v(0)=\frac{\vec x(\delta t)-\vec x(0)}{\delta t}.
+$$
+En prenant la limite $\delta t\rightarrow 0$, on obtient
+$$
+\vec v(0)=\vec x'(0),
+$$
+Soit la relation bien connue que la dérivée de la position donne la vitesse.
+Ou encore de façon similaire
+$$
+\vec v(t_0)=\vec x'(t_0),
+$$
+
+---
+
+Nous pouvons ainsi calculer la position pour $t_2$[^7]
+$$
+\vec x(t_2)=\vec x(t_1)+\delta t\cdot \vec v(t_1),
+$$
+et ainsi de suite pour n'importe quelle valeur de $t_j$
+$$
+\vec x(t_j)=\vec x(t_{j-1})+\delta t\cdot \vec v(t_{j-1}).
+$$
+De cette équation, nous pouvons également déduire que
+$$
+\vec v(t_{j-1})=\frac{\vec x(t_j)-\vec x(t_{j-1})}{\delta t},
+$${#eq:vt0}
+ou encore 
+$$
+\vec v(t_{j})=\frac{\vec x(t_j+1)-\vec x(t_{j})}{\delta t}.
+$${#eq:vt1}
+
+Nous pouvons ainsi approximer le mouvement de la particule $P$
+pour tout instant $t_j$. A l'inverse en connaissant la position de la
+particule à tous les $t_j$, nous pouvons calculer sa vitesse.
+
+### Le mouvement avec accélération
+
+De façon similaire à ce que nous avons fait pour le lien entre la position et
+la vitesse, nous pouvons faite le lien entre l'accélération et la vitesse.
+En connaissant la vitesse de la particule $P$ à tout instant $\vec v(t_j)$,
+nous pouvons approximer l'accélération $\vec a(t_j)$ comme
+$$
+\vec a(t_j) = \frac{\vec v(t_j)-v(t_{j-1})}{\delta t}.
+$$
+En utilisant l'@eq:vt0 et l'@eq:vt1, il vient
+$$
+\vec a(t_j) = \frac{\frac{\vec x(t_j+1)-\vec x(t_{j})}{\delta t}-\frac{\vec x(t_j)-\vec x(t_{j-1})}{\delta t}}{\delta t}=\frac{\vec x(t_{j+1})-2\vec x(t_j)+\vec x(t_{j-1})}{\delta t^2}.
+$$
+En isolant $\vec x(t_{j+1})$ il vient
+$$
+\vec x(t_{j+1})=2\vec x(t_j)-\vec x(t_{j-1})+\vec a(t_j)\delta t^2.
+$$
+Cette formule est correcte pour $j\geq 1$. Pour $j=0$, on a
+$$
+\vec x(t_{1})=\vec x(t_0)+\delta t\vec v(t_0)+\vec a(t_0)\delta t^2.
+$$
+Il est donc nécessaire de connaître la vitesse initiale et la position
+initiale de la particule $P$ pour pouvoir calculer son évolution,
+ainsi que son accélération en tout temps $t_j$. Afin de calculer
+l'accélération de la particule $P$, on utilise souvent les 
+lois de Newton. Ainsi,
+$$
+\vec a(t_j)=\frac{\vec F(t_j)}{m}.
+$$
+

10_footer.md

@@ -16,3 +16,4 @@ les jointures des articulation $\mu_k=0.01$.
 [^6]: A titre d'exemple, pour un contact bois sur bois, on a $\mu_s=0.4$, pour du caoutchouc sur du béton $\mu_s=1$, et pour 
 les jointures des articulation $\mu_s=0.01$.
 
+[^7]: Cette équation peut aussi s'écrire $\vec x(2\delta t)=\vec x(\delta t)+\delta t\cdot \vec v(\delta t)$.

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