Virtual displacement

A virtual displacement (or infinitesimal displacement) $\delta \mathbf{r}$ of a mechanical system is a possible change in configuration that satisfies all of the constraints imposed on the system at a fixed time $t$. It is called "virtual" because it is represents a possibility: it represent how the system could move at any given point in time while respecting constraints, not an actual movement (which would be written $d\mathbf{r}$ and would happen over a time interval $dt$). This is why time being fixed is emphasized in the definition: nothing actually happens, virtual displacement is a purely theoretical quantity and may not even equal the real displacement. It is primarily useful for analyzing the state of a system.

Mathematically, for a system with configuration space $Q$, virtual displacements at a point $P\in Q$ belong to the tangent space $T_{P}Q$ (for holonomic systems) or a subspace of it (for nonholonomic systems). Since displacements span $T_{P}Q$ in holonomic systems, we can express them as a Linear combination of a Basis of $T_{P}Q$:

where $n$ is the dimension of $Q$, i.e. the degrees of freedom of the system and $\delta q_{i}$ are virtual displacements on the axes of $T_{P}Q$. $\mathbf{r}$ is the position vector of a point (here considered to be the entire system) that is a function of the generalized coordinates: $\mathbf{r}\equiv \mathbf{r}(q_{1},\ldots,q_{N})$.