Editing D'Alembert's principle (section)

==Formulation using the Lagrangian==

D'Alembert's principle can be rewritten in terms of the Lagrangian <math>L=T-V</math> of the system as a generalized version of [[Hamilton's principle]] for the case of point particles, as follows,
<math display="block">\delta \int_{t_1}^{t_2} L(\mathbf{r}, \dot{\mathbf{r}},t) dt + \sum_i\int_{t_1}^{t_2} \mathbf{F}_i \cdot \delta \mathbf r_i dt= 0,</math>
where:
* <math> \mathbf{r}=(\mathbf{r}_1,..., \mathbf{r} _N)</math> 

* <math> \mathbf{F}_i </math> are the applied forces

* <math> \delta \mathbf{r}_i</math> is the virtual displacement of the <math>i</math>-th particle, consistent with the constraints <math>\sum_i\mathbf{C}_i \cdot \delta \mathbf{r} _i=0</math>
* the critical curve satisfies the constraints <math>\sum_i\mathbf{C}_i \cdot \dot \mathbf{r} _i=0</math>
With the Lagrangian
<math display="block"> L(\mathbf{r}, \dot{\mathbf{r}},t) = \sum_i \frac{1}{2} m_i \dot { \mathbf{r} }_i^2,</math>
the previous statement of d'Alembert principle is recovered.