Update textbook.fr.md

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

@@ -151,9 +151,6 @@ L'action est stationnaire par rapport à de petites variations ($`1^{er}`$ ordre
 
 $`\mathcal{S}=\displaystyle\int_{t_1}^{t_2}\mathcal{L}\big(x_i\,,\,\dpt{x}_i\big) dt`$
 
-$`\delta \mathcal{S}=\displaystyle\int_{t_1}^{t_2}\bigg( \dfrac{\partial\mathcal{L}}{\partial x_i} \delta x_i
-+\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i} \delta \dpt{x}_i\bigg)\,dt`$
-
 $`\delta \mathcal{S}=\displaystyle\int_{t_1}^{t_2}\bigg( \dfrac{\partial\mathcal{L}}{\partial x_i}\cdot \delta x_i
 +\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i} \cdot\delta \dpt{x}_i\bigg)\,dt`$
 
@@ -177,6 +174,8 @@ $`\displaystyle\;=\int_{\alpha_1}^{\alpha_2}\dfrac{d (uv)}{d\alpha}\,d\alpha
 $`\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i} \cdot\delta \dpt{x}_i`$
 $`\;=\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot\dfrac{d x_i}{dt}`$
 
+$`u=\dfrac{\partial \mathcal{L}}{\partial\dpt{q}}
+
 $`\quad=\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\cdot x_i\bigg)
 -\dfrac{d}{dt}\bigg(\,\dfrac{\partial\mathcal{L}}{\partial \dpt{x}_i}\bigg)\cdot x_i`$