Editing Generalized coordinates (section)

==Generalized coordinates and virtual work==
The ''principle of [[virtual work]]'' states that if a system is in static equilibrium, the virtual work of the applied forces is zero for all virtual movements of the system from this state, that is, {{math|1=δ''W'' = 0}} for any variation {{math|δ'''r'''}}.<ref name="Torby1984">{{harvnb|Torby|1984|chapter="Energy Methods"}}</ref>  When formulated in terms of generalized coordinates, this is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is {{math|1=''F''{{sub|''i''}} = 0}}.

Let the forces on the system be {{math|1='''F'''{{sub|''j''}} (''j'' = 1, 2, …, ''m'')}} be applied to points with Cartesian coordinates {{math|1='''r'''{{sub|''j''}} (''j'' = 1, 2, …, ''m'')}}, then the virtual work generated by a virtual displacement from the equilibrium position is given by
:<math>\delta W = \sum_{j=1}^m \mathbf{F}_j\cdot \delta\mathbf{r}_j.</math>
where {{math|1=δ'''r'''{{sub|''j''}} (''j'' = 1, 2, …, ''m'')}} denote the virtual displacements of each point in the body.

Now assume that each {{math|δ'''r'''{{sub|''j''}}}} depends on the generalized coordinates {{math|1=''q{{sub|i}}'' (''i'' = 1, 2, …, ''n'')}} then
:<math> \delta \mathbf{r}_j = \frac{\partial \mathbf{r}_j}{\partial q_1} \delta{q}_1 + \ldots + \frac{\partial \mathbf{r}_j}{\partial q_n} \delta{q}_n,</math>
and
:<math> \delta W = \left(\sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{r}_j}{\partial q_1}\right) \delta{q}_1 + \ldots + \left(\sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{r}_j}{\partial q_n}\right) \delta{q}_n. </math>

The {{mvar|n}} terms
:<math> F_i = \sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{r}_j}{\partial q_i},\quad i=1,\ldots, n,</math>
are the generalized forces acting on the system.   Kane<ref>T. R. Kane and D. A. Levinson, Dynamics: theory and applications, McGraw-Hill, New York, 1985</ref> shows that these generalized forces can also be formulated in terms of the ratio of time derivatives,
:<math> F_i = \sum_{j=1}^m \mathbf{F}_j\cdot \frac{\partial \mathbf{v}_j}{\partial \dot{q}_i},\quad i=1,\ldots, n,</math>
where {{math|'''v'''{{sub|''j''}}}} is the velocity of the point of application of the force {{math|'''F'''{{sub|''j''}}}}.

In order for the virtual work to be zero for an arbitrary virtual displacement, each of the generalized forces must be zero, that is
:<math> \delta W = 0 \quad \Rightarrow \quad F_i =0, i=1,\ldots, n.</math>