Virtual displacement

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 $\mathbf$ at the given time $\tau,$ i.e. $\gamma(\tau) = \mathbf,$ then $\delta\gamma (\tau) = 0.$

Notations

Let $M$ be the configuration space of the mechanical system, $t_0,t_1 \in \mathbb$ be time instants, $q_0,q_1 \in M,$ C^\infty _0, t_1/math> consists of smooth functions on _0, t_1/math>, and

$P(M) = \.$

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 $\epsilon_0 > 0,$ a ''variation'' of $\gamma$ is a smooth function \Gamma : _0,t_1\times \epsilon_0,\epsilon_0\to M such that, for every \epsilon \in \epsilon_0,\epsilon_0 $\Gamma(\cdot,\epsilon) \in P(M)$ and $\Gamma(t,0) = \gamma(t).$ The ''virtual displacement'' \delta \gamma : _0,t_1\to TM $(TM$ being the tangent bundle of $M)$ corresponding to the variation $\Gamma$ assigns to every t \in _0,t_1/math> the tangent vector

$\delta \gamma(t) = \left.\frac\_ \in T_M.$

In terms of the tangent map,

$\delta \gamma(t) = \Gamma^t_*\left(\left.\frac\_\right).$

Here \Gamma^t_*: T_0 \epsilon,\epsilon\to T_M = T_M is the tangent map of \Gamma^t : \epsilon,\epsilon\to M, where $\Gamma^t(\epsilon) = \Gamma(t,\epsilon),$ and \textstyle \frac\Bigl, _ \in T_0 \epsilon,\epsilon

Properties

* ''Coordinate representation.'' If $\^n_$ are the coordinates in an arbitrary chart on $M$ and $n = \dim M,$ then $\delta \gamma(t) = \sum^n_ \frac\Biggl, _ \cdot \frac\Biggl, _.$
* If, for some time instant $\tau$ and every $\gamma \in P(M),$ $\gamma(\tau)=\text,$ then, for every $\gamma \in P(M),$ $\delta \gamma (\tau) = 0.$
* If $\textstyle \gamma,\frac \in P(M),$ then $\delta \frac = \frac\delta \gamma.$

Examples

Free particle in R³

A single particle freely moving in $\mathbb^3$ has 3 degrees of freedom. The configuration space is $M = \mathbb^3,$ and P(M) = C^\infty( _0,t_1 M). For every path $\gamma \in P(M)$ and a variation $\Gamma(t,\epsilon)$ of $\gamma,$ there exists a unique $\sigma \in T_0\mathbb^3$ such that $\Gamma(t,\epsilon) = \gamma(t) + \sigma(t) \epsilon + o(\epsilon),$ as $\epsilon \to 0.$
By the definition,

$\delta \gamma (t) = \left.\left(\frac \Bigl(\gamma(t) + \sigma(t)\epsilon + o(\epsilon)\Bigr)\right)\_$

which leads to

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

Free particles on a surface

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

$M = \,$

where $\mathbf_i \in \mathbb^3$ is the radius vector of the $i^\text$ particle. It follows that

$T_ M = T_S \oplus \ldots \oplus T_S,$

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

$\gamma (t) = (\mathbf_1(t),\ldots, \mathbf_N(t)).$

This implies that, for every $\delta \gamma(t) \in T_ M,$

$\delta \gamma(t) = \delta \mathbf_1(t) \oplus \ldots \oplus \delta \mathbf_N(t),$

where $\delta \mathbf_i(t) \in T_ S.$ Some authors express this as

$\delta \gamma = (\delta \mathbf_1, \ldots , \delta \mathbf_N).$

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^\infty( _0,t_1 M). We use the standard notation $\mathfrak(3)$ to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map $\exp : \mathfrak(3) \to SO(3)$ guarantees the existence of $\epsilon_0 > 0$ such that, for every path $\gamma \in P(M),$ its variation $\Gamma(t,\epsilon),$ and t \in _0,t_1 there is a unique path \Theta^t \in C^\infty( \epsilon_0, \epsilon_0 \mathfrak(3)) such that $\Theta^t(0) = 0$ and, for every \epsilon \in \epsilon_0,\epsilon_0 $\Gamma(t,\epsilon) = \gamma(t)\exp(\Theta^t(\epsilon)).$ By the definition,

$\delta \gamma (t) = \left.\left(\frac \Bigl(\gamma(t) \exp(\Theta^t(\epsilon))\Bigr)\right)\_ = \gamma(t) \left.\frac\_.$

Since, for some function \sigma : _0,t_1to \mathfrak(3), $\Theta^t(\epsilon) = \epsilon\sigma(t) + o(\epsilon)$, as $\epsilon \to 0$,

$\delta \gamma (t) = \gamma(t)\sigma(t) \in T_\mathrm(3).$

See also

* D'Alembert principle
* Virtual work

References

{{DEFAULTSORT:Virtual Displacement
Dynamical systems
Mechanics
Classical mechanics
Lagrangian mechanics