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 ${\displaystyle \delta \gamma (\tau )=0.}$

Notations

Let ${\displaystyle M}$ be the configuration space of the mechanical system, ${\displaystyle t_{0},t_{1}\in \mathbb {R} }$ be time instants, ${\displaystyle q_{0},q_{1}\in M,}$ ${\displaystyle C^{\infty }}$ consists of smooth functions on ${\displaystyle }$, and

${\displaystyle P(M)=\{\gamma \in C^{\infty }(,M)\mid \gamma (t_{0})=q_{0},\ \gamma (t_{1})=q_{1}\}.}$

The constraints ${\displaystyle \gamma (t_{0})=q_{0},}$ ${\displaystyle \gamma (t_{1})=q_{1}}$ are here for illustration only. In practice, for each individual system, an individual set of constraints is required.

Definition

For each path ${\displaystyle \gamma \in P(M)}$ and ${\displaystyle \epsilon _{0}>0,}$ a variation of ${\displaystyle \gamma }$ is a function ${\displaystyle \Gamma :\times \to M}$ such that, for every ${\displaystyle \epsilon \in -\epsilon _{0},\epsilon _{0},}$ ${\displaystyle \Gamma (\cdot ,\epsilon )\in P(M)}$ and ${\displaystyle \Gamma (t,0)=\gamma (t).}$ The virtual displacement ${\displaystyle \delta \gamma :t_{0},t_{1}\to TM}$ ${\displaystyle (TM}$ being the tangent bundle of ${\displaystyle M)}$ corresponding to the variation ${\displaystyle \Gamma }$ assigns^[1] to every ${\displaystyle t\in t_{0},t_{1}}$ the tangent vector

${\displaystyle \delta \gamma (t)={\frac {d\Gamma (t,\epsilon )}{d\epsilon }}{\Biggl |}_{\epsilon =0}\in T_{\gamma (t)}M.}$

In terms of the tangent map,

${\displaystyle \delta \gamma (t)=\Gamma _{*}^{t}\left({\frac {d}{d\epsilon }}{\Biggl |}_{\epsilon =0}\right).}$

Here ${\displaystyle {\mathrm{\Gamma <!--\; \Gamma \; -->}}_{\ast <!--\; \ast \; -->}^t:T_0}$Zdroj:https://en.wikipedia.org?pojem=Virtual_displacement
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