Update textbook.fr.md

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

@@ -176,7 +176,8 @@ $`\;=\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot\dfrac{d\,\delta x_i}{d
 
 $`u_i=\dfrac{\partial \mathcal{L}}{\partial\dpt{x_i}}`$
 
-$`\dfrac{du_i}{dt}=\dfrac{d}{dt}\bigg(\,\dfrac{\partial \mathcal{L}}{\partial\dpt{x}_i}\bigg)`$
+$`\dfrac{du_i}{dt}=\dfrac{d}{dt}\bigg(\,\dfrac{\partial \mathcal{L}}{\partial\dpt{x}_i}\bigg)
+=\dfrac{d}{dt}\dfrac{\partial \mathcal{L}}{\partial\dpt{x}_i}`$
 
 $`v_i=\delta x_i`$
 
@@ -185,15 +186,17 @@ $`\dfrac{dv_i}{dt}=\dfrac{d\,\delta x_i}{dt}`$
 $`\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot\dfrac{d\,\delta x_i}{dt}`$
 $`\quad=\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i\bigg)
 -\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\bigg)\cdot \delta x_i`$
+$`\quad=\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i
+-\dfrac{d}{dt}\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot \delta x_i`$
 
 
 
 $`\delta \mathcal{S}`$$`\;=\displaystyle\int_{t_1}^{t_2}
-\bigg(
+\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
- --\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\bigg)\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
+ \bigg]\;dt`$