In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) shows how the mechanical system's trajectory can hypothetically (hence the term virtual) deviate very slightly from the actual trajectory \gamma of the system without violating the system's constraints. For every time instant t, is a vector tangential to the configuration space at the point \gamma(t). The vectors show the directions in which \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 \gamma pass through the given point \mathbf{q} at the given time \tau, i.e. then

Notations

Let M be the configuration space of the mechanical system, be time instants, consists of smooth functions on [t_0, t_1], and The constraints are here for illustration only. In practice, for each individual system, an individual set of constraints is required.

Definition

For each path and a variation of \gamma is a function such that, for every and The virtual displacement (TM being the tangent bundle of M) corresponding to the variation \Gamma assigns to every the tangent vector In terms of the tangent map, Here is the tangent map of where and

Properties

Examples

Free particle in R3

A single particle freely moving in has 3 degrees of freedom. The configuration space is and For every path and a variation of \gamma, there exists a unique such that as By the definition, which leads to

Free particles on a surface

N particles moving freely on a two-dimensional surface have 2N degree of freedom. The configuration space here is where is the radius vector of the i^\text{th} particle. It follows that and every path may be described using the radius vectors of each individual particle, i.e. This implies that, for every where Some authors express this as

Rigid body rotating around fixed point

A rigid body rotating around a fixed point with no additional constraints has 3 degrees of freedom. The configuration space here is M = SO(3), the special orthogonal group of dimension 3 (otherwise known as 3D rotation group), and We use the standard notation to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map guarantees the existence of such that, for every path its variation and there is a unique path such that and, for every By the definition, Since, for some function, as ,