# Virtual displacement

For the particle trajectory ${\displaystyle x(t)}$ and its virtual trajectory ${\displaystyle x'(t)}$, at position ${\displaystyle x_{1}}$, time ${\displaystyle t_{1}}$, the virtual displacement is ${\displaystyle \delta x}$ 。 The starting and ending positions for both trajectories are at ${\displaystyle x_{0}}$ and ${\displaystyle x_{2}}$ respectively.
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 δri "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 Also, 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 δt = 0).

In modern terminology virtual displacement is a tangent vector to the manifold representing the constraints at a fixed time. Unlike regular displacement which arises from differentiating with respect to time parameter t along the path of the motion (thus pointing in the direction of the motion), virtual displacement arises from differentiating with respect to the parameter ε enumerating paths of the motion varied in a manner consistent with the constraints (thus pointing at a fixed time in the direction tangent to the constraining manifold). The symbol δ is traditionally used to denote the corresponding derivative

${\displaystyle \delta =\left.{\frac {\partial }{\partial \epsilon }}\right|_{\epsilon =0}\,.}$

## Comparison between virtual and actual displacements

The total differential of any set of system position vectors, ri, that are functions of other variables

${\displaystyle \lbrace q_{1},q_{2},...,q_{m}\rbrace \,}$

and time t may be expressed as follows:[1]:264

${\displaystyle d\mathbf {r} _{i}={\frac {\partial \mathbf {r} _{i}}{\partial t}}dt+\sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}}dq_{j}\,}$

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

${\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}}\delta q_{j}\,}$

This equation is used in Lagrangian mechanics to relate generalized coordinates, qj, to virtual work, δW, and generalized forces, Qj.

## Virtual work

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 dr), a virtual displacement (denoted δr) 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 + dz 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 (where δθ could be positive or negative).