Virtual displacement

One degree of freedom.

Two degrees of freedom.

Constraint force C and virtual displacement δr for a particle of mass m confined to a curve. The resultant non-constraint force is N. The components of virtual displacement are related by a constraint equation.

In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) ${\displaystyle \delta \gamma }$ shows how the mechanical system's trajectory can hypothetically (hence the term virtual) deviate very slightly from the actual trajectory ${\displaystyle \gamma }$ of the system without violating the system's constraints.^{[1][2][3]: 263} For every time instant ${\displaystyle t,}$ ${\displaystyle \delta \gamma (t)}$ is a vector tangential to the configuration space at the point ${\displaystyle \gamma (t).}$ The vectors ${\displaystyle \delta \gamma (t)}$ show the directions in which ${\displaystyle \gamma (t)}$ can "go" without breaking the constraints.

For example, the virtual displacements of the system consisting of a single particle on a two-dimensional surface fill up the entire tangent plane, assuming there are no additional constraints.

If, however, the constraints require that all the trajectories ${\displaystyle \gamma }$ pass through the given point ${\displaystyle \mathbf {q} }$ at the given time ${\displaystyle \tau ,}$ i.e. ${\displaystyle \gamma (\tau )=\mathbf {q} ,}$ then ${\displaystyle \delta \gamma (\tau )=0.}$