Update textbook.fr.md

12.temporary_ins/40.classical-mechanics/40.n4/70.lagrangian-mechanics/07.coherence/textbook.fr.md

@@ -198,12 +198,6 @@ $`\;=\displaystyle\int_{t_1}^{t_2}
  +\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg)`$
  $`\,-\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg]\,dt`$
  
-$`\;=\displaystyle\int_{t_1}^{t_2}
-\bigg[
- \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
- +\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg)`$
- $`\,-\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg]\,dt`$
- 
 $`\;=\displaystyle\int_{t_1}^{t_2}
 \bigg[
  \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
@@ -218,13 +212,31 @@ $`\;=\displaystyle\int_{t_1}^{t_2}
 $`+\displaystyle\int_{t_1}^{t_2}
 \dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i`$
 
-comme s'impose $`\delta x_i(t_1)=\delta x_i(t_2)=0`$
+comme s'impose $`\delta x_i(t_1)=\delta x_i(t_2)=0`$, alors 
+$`displaystyle\int_{t_1}^{t_2}
+\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i=0`$
 
-$`\;=\displaystyle\int_{t_1}^{t_2}
+$`\delta \mathcal{S}=\displaystyle\int_{t_1}^{t_2}
 \bigg[
  \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
 -\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg]\,dt`$
 
+$`\quad=\displaystyle\int_{t_1}^{t_2}
+\bigg[
+ \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
+-\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\bigg]\,\delta x_i\,dt`$
+
+Stationnarité de l'action impose $`\delta \mathcal{S}=0`$
+
+d'où l'équation d'Euler-Lagrange :
+
+$`\dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
+-\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}`$
+
+
+
+
+