Editing Virtual work (section)

==Overview==
If a force acts on a particle as it moves from point <math>A</math> to point <math>B</math>, then, for each possible trajectory that the particle may take, it is possible to compute the total work done by the force along the path.   The ''principle of virtual work'', which is the form of the principle of least action applied to these systems, states that the path actually followed by the particle is the one for which the difference between the work along this path and other nearby paths is zero (to the first order). The formal procedure for computing the difference of functions evaluated on nearby paths is a generalization of the derivative known from differential calculus, and is termed ''the calculus of variations''.

Consider a point particle that moves along a path which is described by a function <math>\mathbf{r}(t)</math> from point <math>A</math>, where <math>\mathbf{r}(t=t_0)</math>, to point <math>B</math>, where <math>\mathbf{r}(t=t_1)</math>. It is possible that the particle moves from <math>A</math> to <math>B</math> along a nearby path described by <math>\mathbf{r}(t) + \delta \mathbf{r}(t)</math>, where <math>\delta \mathbf{r}(t)</math> is called the variation of <math>\mathbf{r}(t)</math>. The variation <math>\delta \mathbf{r}(t)</math> satisfies the requirement <math>\delta \mathbf{r}(t_0) = \delta \mathbf{r}(t_1) = 0</math>. The scalar components of the variation  <math>\delta r_1(t)</math>, <math>\delta r_2(t)</math> and <math>\delta r_3(t)</math> are called virtual displacements. This can be generalized to an arbitrary mechanical system defined by the [[generalized coordinates]] <math>q_i</math>, <math>i = 1,2,...,n</math>.  In which case, the variation of the trajectory <math>q_i(t)</math> is defined by the virtual displacements <math>\delta q_i</math>, <math>i = 1,2,...,n</math>.

Virtual work is the total work done by the applied forces and the inertial forces of a mechanical system as it moves through a set of virtual displacements. When considering forces applied to a body in static equilibrium, the principle of least action requires the virtual work of these forces to be zero.
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'''Virtual work''' on a [[physical system|system]] is the [[mechanical work|work]] resulting from either virtual forces acting through a real [[Displacement (vector)|displacement]] or real [[forces]] acting through a [[virtual displacement]]. In this discussion, the term ''displacement'' may refer to a translation or a rotation, and the term ''force'' to a force or a moment. When the virtual quantities are [[independent variable]]s, they are also ''arbitrary''. Being arbitrary is an essential characteristic that enables one to draw important conclusions from mathematical relations. For example, in the matrix relation

:<math>\mathbf{R}^{*T} \mathbf{r} = \mathbf{R}^{*T} \mathbf{B}^{T} \mathbf{q}</math>,

if <math>\mathbf{R}^{*}</math> is an arbitrary vector, then one can conclude that <math> \mathbf{r} = \mathbf{B}^{T} \mathbf{q} </math>. In this way, the arbitrary quantities disappear from the final useful results.
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