virtual displacement

Let $M$ be the configuration space of the mechanical system, $t\_0,t\_1\; \backslash in\; \backslash mathbb\{R\}$ be time instants, $q\_0,q\_1\; \backslash in\; M,$ $C^\backslash infty[t\_0,\; t\_1]$ consists of smooth functions on $[t\_0,\; t\_1]$, and

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

The constraints $\backslash gamma(t\_0)=q\_0,$ $\backslash gamma(t\_1)=q\_1$ are here for illustration only. In practice, for each individual system, an individual set of constraints is required.

For each path $\backslash gamma\; \backslash in\; P(M)$ and $\backslash epsilon\_0\; >\; 0,$ a variation of $\backslash gamma$ is a smooth function $\backslash Gamma\; :\; [t\_0,t\_1]\; \backslash times\; [-\backslash epsilon\_0,\backslash epsilon\_0]\; \backslash to\; M$ such that, for every $\backslash epsilon\; \backslash in\; [-\backslash epsilon\_0,\backslash epsilon\_0],$ $\backslash Gamma(\backslash cdot,\backslash epsilon)\; \backslash in\; P(M)$ and $\backslash Gamma(t,0)\; =\; \backslash gamma(t).$ The virtual displacement $\backslash delta\; \backslash gamma\; :\; [t\_0,t\_1]\; \backslash to\; TM$ $(TM$ being the tangent bundle of $M)$ corresponding to the variation $\backslash Gamma$ assigns to every $t\; \backslash in\; [t\_0,t\_1]$ the tangent vector

$$\backslash delta\; \backslash gamma(t)\; =\; \backslash left.\backslash frac\{d\backslash Gamma(t,\backslash epsilon)\}\{d\backslash epsilon\}\backslash right|\_\{\backslash epsilon=0\}\; \backslash in\; T\_\{\backslash gamma(t)\}M.$$

In terms of the tangent map,

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

Here $\backslash Gamma^t\_*:\; T\_0[-\backslash epsilon,\backslash epsilon]\; \backslash to\; T\_\{\backslash Gamma(t,0)\}M\; =\; T\_\{\backslash gamma(t)\}M$ is the tangent map of $\backslash Gamma^t\; :\; [-\backslash epsilon,\backslash epsilon]\; \backslash to\; M,$ where $\backslash Gamma^t(\backslash epsilon)\; =\; \backslash Gamma(t,\backslash epsilon),$ and $\backslash textstyle\; \backslash frac\{d\}\{d\backslash epsilon\}\backslash Bigl|\_\{\backslash epsilon\; =\; 0\}\; \backslash in\; T\_0[-\backslash epsilon,\backslash epsilon].$

• Coordinate representation. If $\backslash \{q\_i\backslash \}^n\_\{i=1\}$ are the coordinates in an arbitrary chart on $M$ and $n\; =\; \backslash dim\; M,$ then $$$$

\delta \gamma(t) = \sum^n_{i=1} \frac{d[q_i(\Gamma(t,\epsilon))]}{d\epsilon}\Biggl|_{\epsilon=0} \cdot \frac{d}{dq_i}\Biggl|_{\gamma(t)}.

• If, for some time instant $\backslash tau$ and every $\backslash gamma\; \backslash in\; P(M),$ $\backslash gamma(\backslash tau)=\backslash text\{const\},$ then, for every $\backslash gamma\; \backslash in\; P(M),$ $\backslash delta\; \backslash gamma\; (\backslash tau)\; =\; 0.$
• If $\backslash textstyle\; \backslash gamma,\backslash frac\{d\backslash gamma\}\{dt\}\; \backslash in\; P(M),$ then $\backslash delta\; \backslash frac\{d\backslash gamma\}\{dt\}\; =\; \backslash frac\{d\}\{dt\}\backslash delta\; \backslash gamma.$

A single particle freely moving in $\backslash mathbb\{R\}^3$ has 3 degrees of freedom. The configuration space is $M\; =\; \backslash mathbb\{R\}^3,$ and $P(M)\; =\; C^\backslash infty([t\_0,t\_1],\; M).$ For every path $\backslash gamma\; \backslash in\; P(M)$ and a variation $\backslash Gamma(t,\backslash epsilon)$ of $\backslash gamma,$ there exists a unique $\backslash sigma\; \backslash in\; T\_0\backslash mathbb\{R\}^3$ such that $\backslash Gamma(t,\backslash epsilon)\; =\; \backslash gamma(t)\; +\; \backslash sigma(t)\; \backslash epsilon\; +\; o(\backslash epsilon),$ as $\backslash epsilon\; \backslash to\; 0.$

By the definition,

\delta \gamma (t) = \left.\left(\frac{d}{d\epsilon} \Bigl(\gamma(t) + \sigma(t)\epsilon + o(\epsilon)\Bigr)\right)\right|_{\epsilon=0}

which leads to

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

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

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

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

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

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

$$\backslash gamma\; (t)\; =\; (\backslash mathbf\{r\}\_1(t),\backslash ldots,\; \backslash mathbf\{r\}\_N(t)).$$

This implies that, for every $\backslash delta\; \backslash gamma(t)\; \backslash in\; T\_\{(\backslash mathbf\{r\}\_1(t),\; \backslash ldots,\; \backslash mathbf\{r\}\_N(t))\}\; M,$

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

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

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

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 $P(M)\; =\; C^\backslash infty([t\_0,t\_1],\; M).$ We use the standard notation $\backslash mathfrak\{so\}(3)$ to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map $\backslash exp\; :\; \backslash mathfrak\{so\}(3)\; \backslash to\; SO(3)$ guarantees the existence of $\backslash epsilon\_0\; >\; 0$ such that, for every path $\backslash gamma\; \backslash in\; P(M),$ its variation $\backslash Gamma(t,\backslash epsilon),$ and $t\; \backslash in\; [t\_0,t\_1],$ there is a unique path $\backslash Theta^t\; \backslash in\; C^\backslash infty([-\backslash epsilon\_0,\; \backslash epsilon\_0],\; \backslash mathfrak\{so\}(3))$ such that $\backslash Theta^t(0)\; =\; 0$ and, for every $\backslash epsilon\; \backslash in\; [-\backslash epsilon\_0,\backslash epsilon\_0],$ $\backslash Gamma(t,\backslash epsilon)\; =\; \backslash gamma(t)\backslash exp(\backslash Theta^t(\backslash epsilon)).$ By the definition,

\delta \gamma (t) = \left.\left(\frac{d}{d\epsilon} \Bigl(\gamma(t) \exp(\Theta^t(\epsilon))\Bigr)\right)\right|_{\epsilon=0}

= \gamma(t) \left.\frac{d\Theta^t(\epsilon)}{d\epsilon}\right|_{\epsilon=0}.

Since, for some function $\backslash sigma\; :\; [t\_0,t\_1]\backslash to\; \backslash mathfrak\{so\}(3),$ $\backslash Theta^t(\backslash epsilon)\; =\; \backslash epsilon\backslash sigma(t)\; +\; o(\backslash epsilon)$, as $\backslash epsilon\; \backslash to\; 0$,

\delta \gamma (t) = \gamma(t)\sigma(t) \in T_{\gamma(t)}\mathrm{SO}(3).

• D'Alembert principle
• Virtual work

{{DEFAULTSORT:Virtual Displacement}}

Category:Dynamical systems

Category:Mechanics

Category:Classical mechanics

Category:Lagrangian mechanics