One degree of freedom.

Two degrees of freedom.

Constraint forceC and virtual displacementδr for a particle of massm confined to a curve. The resultant non-constraint force isN. The components of virtual displacement are related by a constraint equation.

Inanalytical mechanics, a branch ofapplied mathematics andphysics, avirtual displacement (orinfinitesimal variation)${\displaystyle \delta \gamma }$ shows how the mechanical system's trajectory canhypothetically (hence the termvirtual) 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 vectortangential to theconfiguration 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${\displaystyle \delta \gamma (\tau )=0.}$

Let${\displaystyle M}$ be theconfiguration 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^{\mathrm{\infty }}[t_0,t_1]}$ consists ofsmooth functions on${\displaystyle [t_0,t_1]}$, and

$${\displaystyle P(M)=\{\gamma \in C^{\mathrm{\infty }}([t_0,t_1],M)\mid \gamma (t_0)=q_0,\gamma (t_1)=q_1\}.}$$

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.

For each path${\displaystyle \gamma \in P(M)}$ and${\displaystyle {ϵ}_0>0,}$ avariation of${\displaystyle \gamma }$ is a smooth function${\displaystyle \mathrm{\Gamma }:[t_0,t_1]\times [-{ϵ}_0,{ϵ}_0]\to M}$ such that, for every${\displaystyle ϵ\in [-{ϵ}_0,{ϵ}_0],}$${\displaystyle \mathrm{\Gamma }(\cdot ,ϵ)\in P(M)}$ and${\displaystyle \mathrm{\Gamma }(t,0)=\gamma (t).}$ Thevirtual displacement${\displaystyle \delta \gamma :[t_0,t_1]\to TM}$${\displaystyle (TM}$ being thetangent bundle of${\displaystyle M)}$ corresponding to the variation${\displaystyle \mathrm{\Gamma }}$ assigns^[1] to every${\displaystyle t\in [t_0,t_1]}$ thetangent vector

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

In terms of thetangent map,

$${\displaystyle \delta \gamma (t)={\mathrm{\Gamma }}_{\ast }^t\left({\frac{d}{dϵ}|}_{ϵ=0}\right).}$$

Here${\displaystyle {\mathrm{\Gamma }}_{\ast }^t:T_0[-ϵ,ϵ]\to T_{\mathrm{\Gamma }(t,0)}M=T_{\gamma (t)}M}$ is the tangent map of${\displaystyle {\mathrm{\Gamma }}^t:[-ϵ,ϵ]\to M,}$ where${\displaystyle {\mathrm{\Gamma }}^t(ϵ)=\mathrm{\Gamma }(t,ϵ),}$ and${\displaystyle \frac{d}{dϵ}{|}_{ϵ=0}\in T_0[-ϵ,ϵ].}$

• Coordinate representation. If${\displaystyle \{q_i{\}}_{i=1}^n}$ are the coordinates in an arbitrary chart on${\displaystyle M}$ and${\displaystyle n=\dim M,}$ then$${\displaystyle \delta \gamma (t)=\sum _{i=1}^n\frac{d[q_i(\mathrm{\Gamma }(t,ϵ))]}{dϵ}{|}_{ϵ=0}\cdot \frac{d}{d{q}_i}{|}_{\gamma (t)}.}$$
• If, for some time instant${\displaystyle \tau }$ and every${\displaystyle \gamma \in P(M),}$${\displaystyle \gamma (\tau )=\text{const},}$ then, for every${\displaystyle \gamma \in P(M),}$${\displaystyle \delta \gamma (\tau )=0.}$
• If${\displaystyle \gamma ,\frac{d\gamma }{dt}\in P(M),}$ then${\displaystyle \delta \frac{d\gamma }{dt}=\frac{d}{dt}\delta \gamma .}$

A single particle freely moving in${\displaystyle {\mathbb{R}}^3}$ has 3 degrees of freedom. The configuration space is${\displaystyle M={\mathbb{R}}^3,}$ and${\displaystyle P(M)=C^{\mathrm{\infty }}([t_0,t_1],M).}$ For every path${\displaystyle \gamma \in P(M)}$ and a variation${\displaystyle \mathrm{\Gamma }(t,ϵ)}$ of${\displaystyle \gamma ,}$ there exists a unique${\displaystyle \sigma \in T_0{\mathbb{R}}^3}$ such that${\displaystyle \mathrm{\Gamma }(t,ϵ)=\gamma (t)+\sigma (t)ϵ+o(ϵ),}$ as${\displaystyle ϵ\to 0.}$By the definition,

$${\displaystyle \delta \gamma (t)={\left(\frac{d}{dϵ}(\gamma (t)+\sigma (t)ϵ+o(ϵ))\right)|}_{ϵ=0}}$$

which leads to

$${\displaystyle \delta \gamma (t)=\sigma (t)\in T_{\gamma (t)}{\mathbb{R}}^3.}$$

${\displaystyle N}$ particles moving freely on a two-dimensional surface${\displaystyle S\subset {\mathbb{R}}^3}$ have${\displaystyle 2N}$ degree of freedom. The configuration space here is

$${\displaystyle M=\{({\mathbf{r}}_1,\dots ,{\mathbf{r}}_N)\in {\mathbb{R}}^{3N}\mid {\mathbf{r}}_i\in {\mathbb{R}}^3;{\mathbf{r}}_i\ne {\mathbf{r}}_j\text{if}i\ne j\},}$$

where${\displaystyle {\mathbf{r}}_i\in {\mathbb{R}}^3}$ is the radius vector of the${\displaystyle i^{\text{th}}}$ particle. It follows that

$${\displaystyle T_{({\mathbf{r}}_1,\dots ,{\mathbf{r}}_N)}M=T_{{\mathbf{r}}_1}S\oplus \dots \oplus T_{{\mathbf{r}}_N}S,}$$

and every path${\displaystyle \gamma \in P(M)}$ may be described using the radius vectors${\displaystyle {\mathbf{r}}_i}$ of each individual particle, i.e.

$${\displaystyle \gamma (t)=({\mathbf{r}}_1(t),\dots ,{\mathbf{r}}_N(t)).}$$

This implies that, for every${\displaystyle \delta \gamma (t)\in T_{({\mathbf{r}}_1(t),\dots ,{\mathbf{r}}_N(t))}M,}$

$${\displaystyle \delta \gamma (t)=\delta {\mathbf{r}}_1(t)\oplus \dots \oplus \delta {\mathbf{r}}_N(t),}$$

where${\displaystyle \delta {\mathbf{r}}_i(t)\in T_{{\mathbf{r}}_i(t)}S.}$ Some authors express this as

$${\displaystyle \delta \gamma =(\delta {\mathbf{r}}_1,\dots ,\delta {\mathbf{r}}_N).}$$

Arigid body rotating around a fixed point with no additional constraints has 3 degrees of freedom. The configuration space here is${\displaystyle M=SO(3),}$ thespecial orthogonal group of dimension 3 (otherwise known as3D rotation group), and${\displaystyle P(M)=C^{\mathrm{\infty }}([t_0,t_1],M).}$ We use the standard notation${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$ to refer to the three-dimensional linear space of allskew-symmetric three-dimensional matrices. Theexponential map${\displaystyle \exp :\mathfrak{s}\mathfrak{o}(3)\to SO(3)}$ guarantees the existence of${\displaystyle {ϵ}_0>0}$ such that, for every path${\displaystyle \gamma \in P(M),}$ its variation${\displaystyle \mathrm{\Gamma }(t,ϵ),}$ and${\displaystyle t\in [t_0,t_1],}$ there is a unique path${\displaystyle {\mathrm{\Theta }}^t\in C^{\mathrm{\infty }}([-{ϵ}_0,{ϵ}_0],\mathfrak{s}\mathfrak{o}(3))}$ such that${\displaystyle {\mathrm{\Theta }}^t(0)=0}$ and, for every${\displaystyle ϵ\in [-{ϵ}_0,{ϵ}_0],}$${\displaystyle \mathrm{\Gamma }(t,ϵ)=\gamma (t)\exp ({\mathrm{\Theta }}^t(ϵ)).}$ By the definition,

$${\displaystyle \delta \gamma (t)={\left(\frac{d}{dϵ}(\gamma (t)\exp ({\mathrm{\Theta }}^t(ϵ)))\right)|}_{ϵ=0}=\gamma (t){\frac{d{\mathrm{\Theta }}^t(ϵ)}{dϵ}|}_{ϵ=0}.}$$

Since, for some function${\displaystyle \sigma :[t_0,t_1]\to \mathfrak{s}\mathfrak{o}(3),}$${\displaystyle {\mathrm{\Theta }}^t(ϵ)=ϵ\sigma (t)+o(ϵ)}$, as${\displaystyle ϵ\to 0}$,

$${\displaystyle \delta \gamma (t)=\gamma (t)\sigma (t)\in T_{\gamma (t)}\mathrm{S}\mathrm{O}(3).}$$

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