Physics:Virtual displacement

Short description: Displacement in analytical mechanics

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) $\delta \gamma$ 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 $t,$ $\delta \gamma (t)$ is a vector tangential to the configuration space at the point $\gamma (t).$ The vectors $\delta \gamma (t)$ 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. $\gamma (\tau )=\mathbf{𝐪},$ then $\delta \gamma (\tau )=0.$

Notations

Let $M$ be the configuration space of the mechanical system, $t_0,t_1\in \mathbb{ℝ}$ be time instants, $q_0,q_1\in M,$ $C^{\mathrm{\infty }}[t_0,t_1]$ consists of smooth functions on $[t_0,t_1]$, and

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

The constraints $\gamma (t_0)=q_0,$ $\gamma (t_1)=q_1$ are here for illustration only. In practice, for each individual system, an individual set of constraints is required.

Definition

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

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

In terms of the tangent map,

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

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

Properties

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

Examples

Free particle in R³

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

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

which leads to

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

Free particles on a surface

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

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

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

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

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

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

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

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

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

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

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 $P(M)=C^{\mathrm{\infty }}([t_0,t_1],M).$ We use the standard notation $\mathfrak{𝔰}\mathfrak{𝔬}(3)$ to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map $\exp :\mathfrak{𝔰}\mathfrak{𝔬}(3)\to SO(3)$ guarantees the existence of ${ϵ}_0>0$ such that, for every path $\gamma \in P(M),$ its variation $\mathrm{\Gamma }(t,ϵ),$ and $t\in [t_0,t_1],$ there is a unique path ${\mathrm{\Theta }}^t\in C^{\mathrm{\infty }}([-{ϵ}_0,{ϵ}_0],\mathfrak{𝔰}\mathfrak{𝔬}(3))$ such that ${\mathrm{\Theta }}^t(0)=0$ and, for every $ϵ\in [-{ϵ}_0,{ϵ}_0],$ $\mathrm{\Gamma }(t,ϵ)=\gamma (t)\exp ({\mathrm{\Theta }}^t(ϵ)).$ By the definition,

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

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

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

See also

• D'Alembert principle
• Virtual work