Editing D'Alembert's principle (section)

=== General case with variable mass ===
The general statement of d'Alembert's principle mentions "the time [[derivative]]s of the [[momentum|momenta]] of the system." By Newton's second law, the first time derivative of momentum is the force. The momentum of the <math>i</math>-th mass is the product of its mass and velocity:
<math display="block">\mathbf p_i = m_i \mathbf v_i</math>
and its time derivative is
<math display="block">\dot{\mathbf{p}}_i = \dot{m}_i \mathbf{v}_i + m_i \dot{\mathbf{v}}_i.</math>

In many applications, the masses are constant and this equation reduces to
<math display="block">\dot{\mathbf{p}}_i = m_i \dot{\mathbf{v}}_i = m_i \mathbf{a}_i.</math>

However, some applications involve changing masses (for example, chains being rolled up or being unrolled) and in those cases both terms <math>\dot{m}_i \mathbf{v}_i</math> and <math>m_i \dot{\mathbf{v}}_i</math> have to remain present, giving<ref>{{Cite journal |last=Cveticanin |first=L. |date=1993-12-01 |title=Conservation Laws in Systems With Variable Mass |url=https://asmedigitalcollection.asme.org/appliedmechanics/article-abstract/60/4/954/392414/Conservation-Laws-in-Systems-With-Variable-Mass?redirectedFrom=fulltext |journal=Journal of Applied Mechanics |volume=60 |issue=4 |pages=954–958 |doi=10.1115/1.2901007 |issn=0021-8936}}</ref>
<math display="block">\sum_{i} ( \mathbf {F}_{i} - m_i \mathbf{a}_i - \dot{m}_i \mathbf{v}_i)\cdot \delta \mathbf r_i = 0.</math>
If the variable mass is ejected with a velocity <math>\mathbf{w}_i</math> the principle has an additional term:<ref>{{Cite journal |last=Guttner |first=William C. |last2=Pesce |first2=Celso P. |date=2017-06-01 |title=On Hamilton’s principle for discrete systems of variable mass and the corresponding Lagrange’s equations |url=https://link.springer.com/article/10.1007/s40430-016-0625-4 |journal=Journal of the Brazilian Society of Mechanical Sciences and Engineering |language=en |volume=39 |issue=6 |pages=1969–1976 |doi=10.1007/s40430-016-0625-4 |issn=1806-3691}}</ref>
<math display="block">\sum_{i} ( \mathbf {F}_{i} - m_i \mathbf{a}_i - \dot{m}_i (\mathbf{v}_i - \mathbf{w}_i))\cdot \delta \mathbf r_i = 0.</math>