Update textbook.fr.md

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

@@ -186,7 +186,7 @@ $`\dfrac{dv_i}{dt}=\dfrac{d\,\delta x_i}{dt}`$
 $`\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot\dfrac{d\,\delta x_i}{dt}`$
 $`\quad=\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg)
 -\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\bigg)\cdot \delta x_i`$
-$`\quad=\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i
+$`\quad=\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`$
 
 
@@ -195,8 +195,39 @@ $`\delta \mathcal{S}`$
 $`\;=\displaystyle\int_{t_1}^{t_2}
 \bigg[
  \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
- +\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg)
- -\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg]\,dt`$
+ +\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg)`$
+ $`\,-\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg]\,dt`$
+ 
+$`\;=\displaystyle\int_{t_1}^{t_2}
+\bigg[
+ \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
+ +\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg)`$
+ $`\,-\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg]\,dt`$
+ 
+$`\;=\displaystyle\int_{t_1}^{t_2}
+\bigg[
+ \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
+-\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg]\,dt`$
+$`+\displaystyle\int_{t_1}^{t_2}
+\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg)\,dt`$
+
+$`\;=\displaystyle\int_{t_1}^{t_2}
+\bigg[
+ \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
+-\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg]\,dt`$
+$`+\displaystyle\int_{t_1}^{t_2}
+\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i`$
+
+comme s'impose $`\delta x_i(t_1)=\delta x_i(t_2)=0`$
+
+$`\;=\displaystyle\int_{t_1}^{t_2}
+\bigg[
+ \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
+-\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg]\,dt`$
+
+
+ 
+