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 $$\mathbf{q}$$ 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, $$q_0,q_1 \in M,$$ $$C^\infty[t_0, t_1]$$ 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 $$\gamma(t_0)=q_0,$$ $$\gamma(t_1)=q_1$$ 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 function $$\Gamma : [t_0,t_1] \times [-\epsilon_0,\epsilon_0] \to M$$ 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 $$\delta \gamma : [t_0,t_1] \to TM$$ $$(TM$$ being the tangent bundle of $$M)$$ corresponding to the variation $$\Gamma$$ assigns to every $$t \in [t_0,t_1]$$ the tangent vector

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

In terms of the tangent map,

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

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

Properties

 * Coordinate representation. If $$\{q_i\}^n_{i=1}$$ are the coordinates in an arbitrary chart on $$M$$ and $$n = \mathop{\rm dim}M,$$ 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 $$\tau$$ and every $$\gamma \in P(M),$$ $$\gamma(\tau)=\text{const},$$ then, for every $$\gamma \in P(M),$$ $$\delta \gamma (\tau) = 0.$$


 * If $$\textstyle \gamma,\frac{d\gamma}{dt} \in P(M),$$ then $$\delta \frac{d\gamma}{dt} = \frac{d}{dt}\delta \gamma.$$

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 $$P(M)=C^\infty([t_0,t_1], M).$$ For every path $$ \gamma \in P(M)$$ and a variation $$\Gamma(t,\epsilon)$$ of $$ \gamma, $$ there exists a unique $$ \sigma \in T_0\mathbb{R}^3 $$ such that $$ \Gamma(t,\epsilon) = \gamma(t) + \sigma(t)\epsilon + o(\epsilon), $$ as $$\epsilon \to 0.$$ By the definition,

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

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

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

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

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

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

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

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

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

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

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

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 $$P(M)=C^\infty([t_0,t_1], M).$$ We use the standard notation $$ \mathfrak{so}(3) $$ to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map $$\exp : \mathfrak{so}(3) \to SO(3)$$ guarantees the existence of $$\epsilon_0 > 0$$ such that, for every path $$\gamma \in P(M),$$ its variation $$\Gamma(t,\epsilon),$$ and $$t \in [t_0,t_1],$$ there is a unique path $$ \Theta^t \in C^\infty([-\epsilon_0, \epsilon_0], \mathfrak{so}(3)) $$ 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,

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

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

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