Editing D'Alembert's principle (section)

==Derivations==

=== 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>

===Special case with constant mass===
Consider Newton's law for a system of particles of constant mass, <math>i</math>. The total force on each particle is<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America  |isbn=978-0-03-063366-9 |chapter=Energy Methods}}</ref>
<math display="block">\mathbf {F}_{i}^{(T)} = m_i \mathbf {a}_i,</math>
where
* <math>\mathbf {F}_{i}^{(T)}</math> are the total forces acting on the system's particles,
* <math>m_i \mathbf {a}_i</math> are the inertial forces that result from the total forces.

Moving the inertial forces to the left gives an expression that can be considered to represent quasi-static equilibrium, but which is really just a small algebraic manipulation of Newton's law:<ref name="Torby1984"/>
<math display="block">\mathbf {F}_{i}^{(T)} - m_i \mathbf {a}_i = \mathbf 0.</math>

Considering the [[virtual work]], <math>\delta W</math>, done by the total and inertial forces together through an arbitrary virtual displacement, <math>\delta \mathbf r_i</math>, of the system leads to a zero identity, since the forces involved sum to zero for each particle.<ref name="Torby1984"/>

<math display="block">\delta W = \sum_{i} \mathbf {F}_{i}^{(T)} \cdot \delta \mathbf r_i - \sum_{i} m_i \mathbf{a}_i \cdot \delta \mathbf r_i = 0</math>

The original vector equation could be recovered by recognizing that the work expression must hold for arbitrary displacements. Separating the total forces into applied forces, <math>\mathbf F_i</math>, and constraint forces, <math>\mathbf C_i</math>, yields<ref name="Torby1984"/>
<math display="block">\delta W = \sum_{i} \mathbf {F}_{i} \cdot \delta \mathbf r_i + \sum_{i} \mathbf {C}_{i} \cdot \delta \mathbf r_i - \sum_{i} m_i \mathbf{a}_i \cdot \delta \mathbf r_i = 0.</math>

If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces (which is not usually the case, so this derivation works only for special cases), the constraint forces don't do any work, <math display="inline">\sum_{i} \mathbf {C}_{i} \cdot \delta \mathbf r_i = 0</math>. Such displacements are said to be ''consistent'' with the constraints.<ref>{{Cite conference|title = Teaching Students Work and Virtual Work Method in Statics:A Guiding Strategy with Illustrative Examples |first = Ing-Chang|last = Jong|conference = 2005 American Society for Engineering Education Annual Conference & Exposition|year = 2005|url = https://peer.asee.org/teaching-students-work-and-virtual-work-method-in-statics-a-guiding-strategy-with-illustrative-examples.pdf|access-date = 2024-03-29 }}</ref> This leads to the formulation of ''d'Alembert's principle'', which states that the difference of applied forces and inertial forces for a dynamic system does no virtual work:<ref name="Torby1984"/>
<math display="block">\delta W = \sum_{i} ( \mathbf {F}_{i} - m_i \mathbf{a}_i )\cdot \delta \mathbf r_i = 0.</math>

There is also a corresponding principle for static systems called the [[virtual work#Principle of virtual work for applied forces|principle of virtual work for applied forces]].