Editing Virtual work (section)

== Mathematical treatment ==
Consider a particle ''P'' that moves from a point ''A'' to a point ''B'' along a trajectory {{math|'''r'''(''t'')}}, while a force {{math|'''F'''('''r'''(''t''))}} is applied to it. The work done by the force {{math|'''F'''}} is given by the integral
<math display="block"> W = \int_{\mathbf{r}(t_0)=A}^{\mathbf{r}(t_1)=B} \mathbf{F} \cdot d\mathbf{r} = \int_{t_0}^{t_1} \mathbf{F} \cdot \frac{d\mathbf{r}}{dt}~dt = \int_{t_0}^{t_1}\mathbf{F} \cdot \mathbf{v} ~ dt,</math>
where {{math|''d'''''r'''}} is the differential element along the curve that is the trajectory of ''P'', and {{math|'''v'''}} is its velocity.  It is important to notice that the value of the work {{math|''W''}} depends on the trajectory {{math|'''r'''(''t'')}}.

Now consider particle ''P'' that moves from point ''A'' to point ''B'' again, but this time it moves along the nearby trajectory that differs from {{math|'''r'''(''t'')}} by the variation {{math|1=''δ'''''r'''(''t'') = ''ε'''''h'''(''t'')}}, where {{math|''ε''}} is a scaling constant that can be made as small as desired and {{math|'''h'''(''t'')}} is an arbitrary function that satisfies {{math|1='''h'''(''t''<sub>0</sub>) = '''h'''(''t''<sub>1</sub>) = 0}}. Suppose the force {{math|'''F'''('''r'''(''t'') + ''ε'''''h'''(''t''))}} is the same as {{math|'''F'''('''r'''(''t''))}}. The work done by the force is given by the integral
<math display="block">\bar{W} = \int_{\mathbf{r}(t_0)=A}^{\mathbf{r}(t_1)=B} \mathbf{F} \cdot d(\mathbf{r}+\varepsilon \mathbf{h}) = \int_{t_0}^{t_1} \mathbf{F} \cdot \frac{d(\mathbf{r}(t) + \varepsilon\mathbf{h}(t))}{dt}~ dt = \int_{t_0}^{t_1}\mathbf{F} \cdot (\mathbf{v} + \varepsilon \dot{\mathbf{h}}) ~ dt .</math>
The variation of the work {{math|''δW''}} associated with this nearby path, known as the ''virtual work'', can be computed to be
<math display="block"> \delta W = \bar{W}-W = \int_{t_0}^{t_1} (\mathbf{F} \cdot \varepsilon \dot{\mathbf{h}}) ~dt.</math>

If there are no constraints on the motion of ''P'', then 3 parameters are needed to completely describe ''P''{{'}}s position at any time {{math|''t''}}. If there are {{math|''k''}} ({{math|''k'' ≤ 3}}) constraint forces, then {{math|1=''n'' = (3 − ''k'')}} parameters are needed. Hence, we can define {{math|''n''}} generalized coordinates {{math|''q''<sub>''i''</sub> (''t'')}} ({{math|1=''i'' = 1,...,''n''}}), and express {{math|'''r'''(''t'')}} and {{math|1=''δ'''''r''' = ''ε'''''h'''(''t'')}} in terms of the generalized coordinates. That is,
<math display="block">\mathbf{r}(t) = \mathbf{r}(q_1,q_2,\dots,q_n;t),</math>
<math display="block">\mathbf{h}(t) = \mathbf{h}(q_1,q_2,\dots,q_n;t).</math>
Then, the derivative of the variation {{math|1=''δ'''''r''' = ''ε'''''h'''(''t'')}} is given by
<math display="block"> \frac{d}{dt} \delta \mathbf{r} = \frac{d}{dt} \varepsilon\mathbf{h} = \sum_{i=1}^n \frac{\partial \mathbf{h}}{\partial q_i} \varepsilon \dot{q}_i,</math>
then we have  
<math display="block"> \delta W = \int_{t_0}^{t_1} \left(\sum_{i=1}^n \mathbf{F} \cdot \frac{\partial\mathbf{h}}{\partial q_i} \varepsilon \dot{q}_i\right) dt = \sum_{i=1}^n \left(\int_{t_0}^{t_1} \mathbf{F} \cdot \frac{\partial\mathbf{h}}{\partial q_i} \varepsilon\dot{q}_i ~dt\right).</math>
 
The requirement that the virtual work be zero for an arbitrary variation {{math|1=''δ'''''r'''(''t'') = ''ε'''''h'''(''t'')}} is equivalent to the set of requirements
<math display="block"> Q_i = \mathbf{F} \cdot \frac{\partial \mathbf{h}}{\partial q_i} = 0, \quad i=1, \ldots, n.</math>
The terms {{math|''Q<sub>i</sub>''}} are called the ''generalized forces'' associated with the virtual displacement {{math|''δ'''''r'''}}.