# Virtual displacement

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{{Short description|Displacement in analytical mechanics}}
{{multiple image
|align = vertical
|total_width = 400
|image1 = Constraint force virtual displacement 1 dof.svg
|caption1 = One degree of freedom.
|image2 = Constraint force virtual displacement 2 dof.svg 
|caption2 = Two degrees of freedom.
|footer = Constraint force '''C''' and virtual displacement ''δ'''''r''' 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](/source/analytical_mechanics), a branch of [applied mathematics](/source/applied_mathematics) and [physics](/source/physics), a '''virtual displacement''' (or '''infinitesimal variation''') <math>\delta \gamma</math> shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very slightly from the actual trajectory <math>\gamma</math> of the system without violating the system's constraints.<ref name="TakhtajanClassicalFieldTheory2017">{{cite book|title=Classical Field Theory|last1=Takhtajan|first1=Leon A.|publisher=Department of Mathematics, Stony Brook University, Stony Brook, NY|year=2017|author-link=Leon Takhtajan |chapter=Part 1. Classical Mechanics |url= http://www.math.stonybrook.edu/~kirillov/mat560-fall19/MAT%20560.pdf }}</ref><ref name="Goldstein2001">{{cite book|title= Classical Mechanics|author1-link=Herbert Goldstein|author2-link=Charles P. Poole|last1= Goldstein|first1= H.|last2= Poole|first2= C. P.|last3= Safko|first3= J. L.|publisher= Addison-Wesley|year= 2001|isbn=  978-0-201-65702-9 | edition= 3rd|pages= 16}}</ref><ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref>{{rp|p=263}} For every time instant <math> t,</math> <math>\delta \gamma(t)</math> is a vector [tangential](/source/tangent_vector) to the [configuration space](/source/configuration_space_(physics)) at the point <math>\gamma(t).</math> The vectors <math>\delta \gamma(t)</math> show the directions in which <math>\gamma(t)</math> 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 <math>\gamma</math> pass through the given point <math>\mathbf{q}</math> at the given time <math>\tau,</math> i.e. <math>\gamma(\tau) = \mathbf{q},</math> then <math>\delta\gamma (\tau) = 0.</math>

==Notations==
Let <math>M</math> be the [configuration space](/source/Configuration_space_(physics)) of the mechanical system, <math>t_0,t_1 \in \mathbb{R}</math> be time instants, <math>q_0,q_1 \in M,</math> <math>C^\infty[t_0, t_1]</math> consists of [smooth function](/source/smooth_function)s on <math>[t_0, t_1]</math>, and

<math display="block">
P(M) = \{\gamma \in C^\infty([t_0,t_1], M) \mid  \gamma(t_0)=q_0,\ \gamma(t_1)=q_1\}.
</math>

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

==Definition==
For each path <math>\gamma \in P(M)</math> and <math>\epsilon_0 > 0,</math> a ''variation'' of <math>\gamma</math> is a smooth function <math>\Gamma : [t_0,t_1] \times [-\epsilon_0,\epsilon_0] \to M</math> such that, for every <math>\epsilon \in [-\epsilon_0,\epsilon_0],</math> <math>\Gamma(\cdot,\epsilon) \in P(M)</math> and <math>\Gamma(t,0) = \gamma(t).</math> The ''virtual displacement'' <math>\delta \gamma : [t_0,t_1] \to TM</math> <math>(TM</math> being the [tangent bundle](/source/tangent_bundle) of <math>M)</math> corresponding to the variation <math>\Gamma</math> assigns<ref name="TakhtajanClassicalFieldTheory2017"/> to every <math>t \in [t_0,t_1]</math> the [tangent vector](/source/tangent_vector)

<math display="block">\delta \gamma(t) = \left.\frac{d\Gamma(t,\epsilon)}{d\epsilon}\right|_{\epsilon=0} \in T_{\gamma(t)}M.</math>

In terms of the [tangent map](/source/tangent_map),

<math display="block">
\delta \gamma(t) = \Gamma^t_*\left(\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}\right).
</math>

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

==Properties==
* ''Coordinate representation.'' If <math>\{q_i\}^n_{i=1}</math> are the coordinates in an arbitrary chart on <math>M</math> and <math>n = \dim M,</math> then <math display="block">
\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)}.
</math>
* If, for some time instant <math>\tau</math> and every <math>\gamma \in P(M),</math> <math>\gamma(\tau)=\text{const},</math> then, for every <math>\gamma \in P(M),</math> <math>\delta \gamma (\tau) = 0.</math>
* If <math>\textstyle \gamma,\frac{d\gamma}{dt} \in P(M),</math> then <math>\delta \frac{d\gamma}{dt} = \frac{d}{dt}\delta \gamma.</math>

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

<math display="block">
\delta \gamma (t) = \left.\left(\frac{d}{d\epsilon} \Bigl(\gamma(t) + \sigma(t)\epsilon + o(\epsilon)\Bigr)\right)\right|_{\epsilon=0}
</math>

which leads to

<math display="block">
\delta \gamma (t) = \sigma(t) \in T_{\gamma(t)} \mathbb{R}^3.
</math>

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

<math display="block">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\},
</math>

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

<math display="block">
T_{(\mathbf{r}_1, \ldots, \mathbf{r}_N)} M = T_{\mathbf{r}_1}S \oplus \ldots \oplus T_{\mathbf{r}_N}S,
</math>

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

<math display="block">\gamma (t) = (\mathbf{r}_1(t),\ldots, \mathbf{r}_N(t)).</math>

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

<math display="block">\delta \gamma(t) = \delta \mathbf{r}_1(t) \oplus \ldots \oplus \delta \mathbf{r}_N(t),</math>

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

<math display="block">
\delta \gamma = (\delta \mathbf{r}_1, \ldots , \delta \mathbf{r}_N).</math>

===Rigid body rotating around fixed point===
A [rigid body](/source/rigid_body) rotating around a fixed point with no additional constraints has 3 degrees of freedom. The configuration space here is <math>M = SO(3),</math> the [special orthogonal group](/source/special_orthogonal_group) of dimension 3 (otherwise known as [3D rotation group](/source/3D_rotation_group)), and <math>P(M) = C^\infty([t_0,t_1], M).</math>  We use the standard notation <math> \mathfrak{so}(3) </math> to refer to the three-dimensional linear space of all [skew-symmetric](/source/Skew-symmetric_matrix) three-dimensional matrices. The [exponential map](/source/exponential_map_(Lie_theory)) <math>\exp : \mathfrak{so}(3) \to SO(3)</math> guarantees the existence of <math>\epsilon_0 > 0</math> such that, for every path <math>\gamma \in P(M),</math> its variation <math>\Gamma(t,\epsilon),</math> and <math>t \in [t_0,t_1],</math> there is a unique path <math> \Theta^t \in C^\infty([-\epsilon_0, \epsilon_0], \mathfrak{so}(3)) </math> such that <math>\Theta^t(0) = 0</math> and, for every <math>\epsilon \in [-\epsilon_0,\epsilon_0],</math> <math>\Gamma(t,\epsilon) = \gamma(t)\exp(\Theta^t(\epsilon)).</math> By the definition,

<math display="block">
\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}.
</math>

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

<math display="block">
\delta \gamma (t) = \gamma(t)\sigma(t) \in T_{\gamma(t)}\mathrm{SO}(3).
</math>

==See also==
*[D'Alembert principle](/source/D'Alembert_principle)
*[Virtual work](/source/Virtual_work)

==References==
<references/>

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

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Adapted from the Wikipedia article [Virtual displacement](https://en.wikipedia.org/wiki/Virtual_displacement) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Virtual_displacement?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.