Virtual displacement

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A virtual displacement $\delta \mathbf {r}_i$ "is an assumed infinitesimal change of system coordinates occurring while time is held constant. It is called virtual rather than real since no actual displacement can take place without the passage of time."^[1]^:263

The total differential of any set of system position vectors, $\mathbf {r}_i$ , that are functions of other variables, ${q 1, q 2,..., q m}$ , and time, $t$ may be expressed as follows:^[1]^:264

$d \mathbf{r}_i = \frac {\partial \mathbf {r}_i}{\partial t} d t + \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} d q_j$

If, instead, we want the virtual displacement (virtual differential displacement), then^[1]^:265

$\delta \mathbf{r}_i = \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j$

This equation is used in Lagrangian mechanics to relate generalized coordinates, $q j$ , to virtual work, $δ W$ , and generalized forces, $Q j$ .

In analytical mechanics the concept of a virtual displacement, related to the concept of virtual work, is meaningful only when discussing a physical system subject to constraints on its motion.^{[citation needed]} A special case of an infinitesimal displacement (usually notated $d\mathbf{r}$ ), a virtual displacement (notated $\delta \mathbf{r}$ ) refers to an infinitesimal change in the position coordinates of a system such that the constraints remain satisfied.^{[citation needed]}

For example, if a bead is constrained to move on a hoop, its position may be represented by the position coordinate $θ$ , which gives the angle at which the bead is situated. Say that the bead is at the top. Moving the bead straight upwards from its height $z$ to a height $z + d z$ would represent one possible infinitesimal displacement, but would violate the constraint. The only possible virtual displacement would be a displacement from the bead's position, $θ$ to a new position $θ + δθ$ ^{[citation needed]} (where $δθ$ could be positive or negative).

It is also worthwhile to note that virtual displacements are spatial displacements exclusively - time is fixed while they occur. When computing virtual differentials of quantities that are functions of space and time coordinates, no dependence on time is considered (formally equivalent to saying $δ t = 0$ ).

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^ ^a ^b ^c Torby, Bruce (1984). "Energy Methods", Advanced Dynamics for Engineers, HRW Series in Mechanical Engineering (in English). United States of America: CBS College Publishing. ISBN 0-03-063366-4.