Editing D'Alembert's principle (section)

==Statement of the principle==
The principle states that the sum of the differences between the [[force]]s acting on a system of massive particles and the time [[derivative]]s of the [[momentum|momenta]] of the system itself projected onto any [[virtual displacement]] consistent with the constraints of the system is zero.{{clarify|date=October 2020 |reason=see Talk:D'Alembert's principle}} Thus, in mathematical notation, d'Alembert's principle is written as follows,
<math display="block">\sum_i ( \mathbf F_i - m_i \dot\mathbf{v}_i - \dot{m}_i\mathbf{v}_i)\cdot \delta \mathbf r_i = 0,</math>

where:
* <math>i</math> is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system,
* <math>\mathbf {F}_i</math> is the total applied force (excluding constraint forces) on the <math>i</math>-th particle,
* <math> m_i </math> is the mass of the <math>i</math>-th particle,
* <math>\mathbf v_i</math> is the velocity of the <math>i</math>-th particle,
* <math>\delta \mathbf r_i</math> is the virtual displacement of the <math>i</math>-th particle, consistent with the constraints.

Newton's dot notation is used to represent the derivative with respect to time. The above equation is often called d'Alembert's principle, but it was first written in this variational form by [[Joseph Louis Lagrange]].<ref>[[Arnold Sommerfeld]] (1956), ''[[Lectures on Theoretical Physics#Mechanics|Mechanics:  Lectures on Theoretical Physics]]'', Vol 1, p. 53</ref>  D'Alembert's contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish.  That is to say that the [[generalized forces]] <math>\mathbf Q_j</math> need not include constraint forces.  It is equivalent to the somewhat more cumbersome [[Gauss's principle of least constraint]].