Editing D'Alembert's principle (section)

==Generalization for thermodynamics==
An extension of d'Alembert's principle can be used in thermodynamics.<ref name="Gay2018"/> For instance, for an adiabatically closed [[thermodynamic system]] described by a Lagrangian depending on a single entropy ''S'' and with constant masses <math>m_i</math>, such as
<math display="block"> L(\mathbf{r}, \dot{\mathbf{r}},S,t) = \sum_i \frac{1}{2} m_i \dot { \mathbf{r} }_i^2 - V(\mathbf{r},S),</math>
it is written as follows
<math display="block">\delta \int_{t_1}^{t_2} L(\mathbf{r}, \dot{\mathbf{r}},S,t) dt + \sum_i\int_{t_1}^{t_2} \mathbf{F}_i \cdot \delta \mathbf r_i dt= 0,</math>
where the previous constraints <math display="inline">\sum_i\mathbf{C}_i \cdot \delta \mathbf{r} _i=0</math> and <math display="inline">\sum_i\mathbf{C}_i \cdot \dot \mathbf{r} _i=0</math> are generalized to involve the entropy as:
* <math>\sum_i\mathbf{C}_i \cdot \delta \mathbf{r} _i+T \delta  S=0</math>
* <math>\sum_i\mathbf{C}_i \cdot \dot \mathbf{r} _i+T \dot  S=0.</math>
Here <math>T=\partial V/\partial S</math> is the temperature of the system, <math>\mathbf{F}_i</math> are the external forces, <math>\mathbf{C}_i</math> are the internal dissipative forces. It results in the mechanical and thermal balance equations:<ref name="Gay2018"/>
<math display="block">m_i\mathbf{a}_i=- \frac{\partial V}{\partial \mathbf{r}_i}+ \mathbf{C}_i+\mathbf{F}_i, \;\;i=1,...,N \qquad \qquad T \dot  S = -\sum_i\mathbf{C}_i \cdot \dot \mathbf{r} _i.</math>
Typical applications of the principle include thermo-mechanical systems, membrane transport, and chemical reactions.

For <math>\delta S=\dot S=0</math> the classical d'Alembert principle and equations are recovered.