Update textbook.fr.md

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

@@ -154,6 +154,9 @@ $`\mathcal{S}=\displaystyle\int_{t_1}^{t_2}\mathcal{L}\big(x_i\,,\,\dpt{x}_i\big
 $`\delta \mathcal{S}=\displaystyle\int_{t_1}^{t_2}\bigg( \dfrac{\partial\mathcal{L}}{\partial x_i} \delta x_i
 +\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i} \delta \dpt{x}_i\bigg)\,dt`$
 
+$`\delta \mathcal{S}=\displaystyle\int_{t_1}^{t_2}\bigg( \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
++\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i} \cdot\delta \dpt{x}_i\bigg)\,dt`$
+
 <details markdown=1>
 <summary>intégration par partie du second terme de l'intégrande
 </summary>
@@ -169,10 +172,15 @@ $`\displaystyle\;=\int_{\alpha_1}^{\alpha_2}\dfrac{d (uv)}{d\alpha}\,d\alpha
 -\int_{\alpha_1}^{\alpha_2}\dfrac{d u}{d\alpha}\cdot v\,d\alpha`$
 </details>
 
-$`\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i} \delta \dpt{x}_i`$
-$`\;=\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\dfrac{d x_i}{dt}`$
+<br>
+
+$`\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i} \ cdot\delta \dpt{x}_i`$
+$`\;=\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot\dfrac{d x_i}{dt}`$
+
+$`\quad=\dfrac{d}{dt}\bigg({\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot x_i\bigg)
+-\dfrac{d}{dt}\bigg({\partial\mathcal{L}}{\partial \dpt{x}_i}\bigg)\,x_i`$
+
 
-$`\quad=\dfrac{\partial\mathcal{L}}\,x_i-{\partial \dpt{x}_i}\dfrac{d x_i}{dt}`$